LMFDB - Embedded newform 6039.2.a.k.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:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.436741$ of defining polynomial
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)

$f(q)$	$=$	$q+0.436741 q^{2} -1.80926 q^{4} -3.47884 q^{5} -1.93516 q^{7} -1.66366 q^{8} +O(q^{10})$	\(q+0.436741 q^{2} -1.80926 q^{4} -3.47884 q^{5} -1.93516 q^{7} -1.66366 q^{8} -1.51935 q^{10} -1.00000 q^{11} +4.56403 q^{13} -0.845165 q^{14} +2.89193 q^{16} -2.96582 q^{17} -4.35587 q^{19} +6.29411 q^{20} -0.436741 q^{22} -6.48962 q^{23} +7.10230 q^{25} +1.99330 q^{26} +3.50121 q^{28} -8.12635 q^{29} -7.92562 q^{31} +4.59034 q^{32} -1.29529 q^{34} +6.73212 q^{35} -1.05106 q^{37} -1.90239 q^{38} +5.78759 q^{40} -9.87724 q^{41} +0.495326 q^{43} +1.80926 q^{44} -2.83428 q^{46} -10.4314 q^{47} -3.25514 q^{49} +3.10186 q^{50} -8.25751 q^{52} -11.7850 q^{53} +3.47884 q^{55} +3.21945 q^{56} -3.54911 q^{58} +1.04645 q^{59} -1.00000 q^{61} -3.46144 q^{62} -3.77907 q^{64} -15.8775 q^{65} +6.10340 q^{67} +5.36593 q^{68} +2.94019 q^{70} -0.0603962 q^{71} +2.52552 q^{73} -0.459042 q^{74} +7.88089 q^{76} +1.93516 q^{77} +5.40838 q^{79} -10.0605 q^{80} -4.31379 q^{82} -10.9265 q^{83} +10.3176 q^{85} +0.216329 q^{86} +1.66366 q^{88} +18.5756 q^{89} -8.83215 q^{91} +11.7414 q^{92} -4.55580 q^{94} +15.1534 q^{95} -17.6748 q^{97} -1.42165 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * 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 $

$2$	0.436741	0.308822	0.154411	−	0.988007i	$-0.450652\pi$
$2$	0.436741	0.308822	0.154411	+	0.988007i	$0.450652\pi$
$3$	0	0
$3$	0	0
$4$	−1.80926	−0.904629
$5$	−3.47884	−1.55578	−0.777891	−	0.628399i	$-0.783711\pi$
$5$	−3.47884	−1.55578	−0.777891	+	0.628399i	$0.783711\pi$
$6$	0	0
$7$	−1.93516	−0.731424	−0.365712	−	0.930728i	$-0.619174\pi$
$7$	−1.93516	−0.731424	−0.365712	+	0.930728i	$0.619174\pi$
$8$	−1.66366	−0.588192
$9$	0	0
$10$	−1.51935	−0.480460
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.56403	1.26583	0.632917	−	0.774220i	$-0.281857\pi$
$13$	4.56403	1.26583	0.632917	+	0.774220i	$0.281857\pi$
$14$	−0.845165	−0.225880
$15$	0	0
$16$	2.89193	0.722982
$17$	−2.96582	−0.719316	−0.359658	−	0.933084i	$-0.617107\pi$
$17$	−2.96582	−0.719316	−0.359658	+	0.933084i	$0.617107\pi$
$18$	0	0
$19$	−4.35587	−0.999305	−0.499653	−	0.866226i	$-0.666539\pi$
$19$	−4.35587	−0.999305	−0.499653	+	0.866226i	$0.666539\pi$
$20$	6.29411	1.40741
$21$	0	0
$22$	−0.436741	−0.0931134
$23$	−6.48962	−1.35318	−0.676589	−	0.736360i	$-0.736543\pi$
$23$	−6.48962	−1.35318	−0.676589	+	0.736360i	$0.736543\pi$
$24$	0	0
$25$	7.10230	1.42046
$26$	1.99330	0.390918
$27$	0	0
$28$	3.50121	0.661667
$29$	−8.12635	−1.50903	−0.754513	−	0.656286i	$-0.772127\pi$
$29$	−8.12635	−1.50903	−0.754513	+	0.656286i	$0.772127\pi$
$30$	0	0
$31$	−7.92562	−1.42348	−0.711741	−	0.702442i	$-0.752093\pi$
$31$	−7.92562	−1.42348	−0.711741	+	0.702442i	$0.752093\pi$
$32$	4.59034	0.811465
$33$	0	0
$34$	−1.29529	−0.222141
$35$	6.73212	1.13794
$36$	0	0
$37$	−1.05106	−0.172794	−0.0863970	−	0.996261i	$-0.527535\pi$
$37$	−1.05106	−0.172794	−0.0863970	+	0.996261i	$0.527535\pi$
$38$	−1.90239	−0.308608
$39$	0	0
$40$	5.78759	0.915098
$41$	−9.87724	−1.54257	−0.771283	−	0.636493i	$-0.780385\pi$
$41$	−9.87724	−1.54257	−0.771283	+	0.636493i	$0.780385\pi$
$42$	0	0
$43$	0.495326	0.0755365	0.0377683	−	0.999287i	$-0.487975\pi$
$43$	0.495326	0.0755365	0.0377683	+	0.999287i	$0.487975\pi$
$44$	1.80926	0.272756
$45$	0	0
$46$	−2.83428	−0.417892
$47$	−10.4314	−1.52157	−0.760785	−	0.649004i	$-0.775186\pi$
$47$	−10.4314	−1.52157	−0.760785	+	0.649004i	$0.775186\pi$
$48$	0	0
$49$	−3.25514	−0.465020
$50$	3.10186	0.438669
$51$	0	0
$52$	−8.25751	−1.14511
$53$	−11.7850	−1.61879	−0.809395	−	0.587265i	$-0.800205\pi$
$53$	−11.7850	−1.61879	−0.809395	+	0.587265i	$0.800205\pi$
$54$	0	0
$55$	3.47884	0.469086
$56$	3.21945	0.430217
$57$	0	0
$58$	−3.54911	−0.466020
$59$	1.04645	0.136236	0.0681179	−	0.997677i	$-0.478301\pi$
$59$	1.04645	0.136236	0.0681179	+	0.997677i	$0.478301\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−3.46144	−0.439603
$63$	0	0
$64$	−3.77907	−0.472384
$65$	−15.8775	−1.96936
$66$	0	0
$67$	6.10340	0.745649	0.372825	−	0.927902i	$-0.378389\pi$
$67$	6.10340	0.745649	0.372825	+	0.927902i	$0.378389\pi$
$68$	5.36593	0.650714
$69$	0	0
$70$	2.94019	0.351420
$71$	−0.0603962	−0.00716771	−0.00358386	−	0.999994i	$-0.501141\pi$
$71$	−0.0603962	−0.00716771	−0.00358386	+	0.999994i	$0.501141\pi$
$72$	0	0
$73$	2.52552	0.295590	0.147795	−	0.989018i	$-0.452782\pi$
$73$	2.52552	0.295590	0.147795	+	0.989018i	$0.452782\pi$
$74$	−0.459042	−0.0533626
$75$	0	0
$76$	7.88089	0.904000
$77$	1.93516	0.220533
$78$	0	0
$79$	5.40838	0.608491	0.304245	−	0.952594i	$-0.401596\pi$
$79$	5.40838	0.608491	0.304245	+	0.952594i	$0.401596\pi$
$80$	−10.0605	−1.12480
$81$	0	0
$82$	−4.31379	−0.476378
$83$	−10.9265	−1.19934	−0.599672	−	0.800246i	$-0.704702\pi$
$83$	−10.9265	−1.19934	−0.599672	+	0.800246i	$0.704702\pi$
$84$	0	0
$85$	10.3176	1.11910
$86$	0.216329	0.0233274
$87$	0	0
$88$	1.66366	0.177346
$89$	18.5756	1.96901	0.984503	−	0.175368i	$-0.0561115\pi$
$89$	18.5756	1.96901	0.984503	+	0.175368i	$0.0561115\pi$
$90$	0	0
$91$	−8.83215	−0.925861
$92$	11.7414	1.22412
$93$	0	0
$94$	−4.55580	−0.469895
$95$	15.1534	1.55470
$96$	0	0
$97$	−17.6748	−1.79460	−0.897300	−	0.441421i	$-0.854474\pi$
$97$	−17.6748	−1.79460	−0.897300	+	0.441421i	$0.854474\pi$
$98$	−1.42165	−0.143608
$99$	0	0
$100$	−12.8499	−1.28499
$101$	5.63429	0.560633	0.280317	−	0.959908i	$-0.409561\pi$
$101$	5.63429	0.560633	0.280317	+	0.959908i	$0.409561\pi$
$102$	0	0
$103$	−16.7201	−1.64748	−0.823738	−	0.566971i	$-0.808115\pi$
$103$	−16.7201	−1.64748	−0.823738	+	0.566971i	$0.808115\pi$
$104$	−7.59298	−0.744553
$105$	0	0
$106$	−5.14697	−0.499918
$107$	10.2226	0.988261	0.494130	−	0.869388i	$-0.335487\pi$
$107$	10.2226	0.988261	0.494130	+	0.869388i	$0.335487\pi$
$108$	0	0
$109$	8.04155	0.770241	0.385120	−	0.922866i	$-0.374160\pi$
$109$	8.04155	0.770241	0.385120	+	0.922866i	$0.374160\pi$
$110$	1.51935	0.144864
$111$	0	0
$112$	−5.59636	−0.528806
$113$	5.03833	0.473967	0.236983	−	0.971514i	$-0.423841\pi$
$113$	5.03833	0.473967	0.236983	+	0.971514i	$0.423841\pi$
$114$	0	0
$115$	22.5763	2.10525
$116$	14.7027	1.36511
$117$	0	0
$118$	0.457026	0.0420726
$119$	5.73934	0.526125
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.436741	−0.0395406
$123$	0	0
$124$	14.3395	1.28772
$125$	−7.31354	−0.654143
$126$	0	0
$127$	12.0779	1.07174	0.535871	−	0.844300i	$-0.319983\pi$
$127$	12.0779	1.07174	0.535871	+	0.844300i	$0.319983\pi$
$128$	−10.8311	−0.957347
$129$	0	0
$130$	−6.93435	−0.608183
$131$	19.0023	1.66024	0.830119	−	0.557587i	$-0.188273\pi$
$131$	19.0023	1.66024	0.830119	+	0.557587i	$0.188273\pi$
$132$	0	0
$133$	8.42933	0.730915
$134$	2.66560	0.230273
$135$	0	0
$136$	4.93410	0.423096
$137$	22.4442	1.91754	0.958771	−	0.284180i	$-0.0917215\pi$
$137$	22.4442	1.91754	0.958771	+	0.284180i	$0.0917215\pi$
$138$	0	0
$139$	−12.2835	−1.04188	−0.520938	−	0.853594i	$-0.674418\pi$
$139$	−12.2835	−1.04188	−0.520938	+	0.853594i	$0.674418\pi$
$140$	−12.1801	−1.02941
$141$	0	0
$142$	−0.0263775	−0.00221355
$143$	−4.56403	−0.381663
$144$	0	0
$145$	28.2702	2.34771
$146$	1.10300	0.0912847
$147$	0	0
$148$	1.90165	0.156314
$149$	−19.8821	−1.62880	−0.814401	−	0.580302i	$-0.802935\pi$
$149$	−19.8821	−1.62880	−0.814401	+	0.580302i	$0.802935\pi$
$150$	0	0
$151$	−10.5885	−0.861678	−0.430839	−	0.902429i	$-0.641782\pi$
$151$	−10.5885	−0.861678	−0.430839	+	0.902429i	$0.641782\pi$
$152$	7.24668	0.587783
$153$	0	0
$154$	0.845165	0.0681053
$155$	27.5719	2.21463
$156$	0	0
$157$	−7.05272	−0.562868	−0.281434	−	0.959581i	$-0.590810\pi$
$157$	−7.05272	−0.562868	−0.281434	+	0.959581i	$0.590810\pi$
$158$	2.36206	0.187915
$159$	0	0
$160$	−15.9690	−1.26246
$161$	12.5585	0.989747
$162$	0	0
$163$	5.13074	0.401870	0.200935	−	0.979605i	$-0.435602\pi$
$163$	5.13074	0.401870	0.200935	+	0.979605i	$0.435602\pi$
$164$	17.8705	1.39545
$165$	0	0
$166$	−4.77207	−0.370384
$167$	−22.6642	−1.75381	−0.876903	−	0.480668i	$-0.840394\pi$
$167$	−22.6642	−1.75381	−0.876903	+	0.480668i	$0.840394\pi$
$168$	0	0
$169$	7.83036	0.602336
$170$	4.50611	0.345603
$171$	0	0
$172$	−0.896173	−0.0683325
$173$	−12.3401	−0.938201	−0.469100	−	0.883145i	$-0.655422\pi$
$173$	−12.3401	−0.938201	−0.469100	+	0.883145i	$0.655422\pi$
$174$	0	0
$175$	−13.7441	−1.03896
$176$	−2.89193	−0.217987
$177$	0	0
$178$	8.11270	0.608073
$179$	−4.14843	−0.310069	−0.155034	−	0.987909i	$-0.549549\pi$
$179$	−4.14843	−0.310069	−0.155034	+	0.987909i	$0.549549\pi$
$180$	0	0
$181$	15.7435	1.17021	0.585104	−	0.810958i	$-0.301054\pi$
$181$	15.7435	1.17021	0.585104	+	0.810958i	$0.301054\pi$
$182$	−3.85736	−0.285926
$183$	0	0
$184$	10.7965	0.795928
$185$	3.65648	0.268830
$186$	0	0
$187$	2.96582	0.216882
$188$	18.8730	1.37646
$189$	0	0
$190$	6.61808	0.480126
$191$	14.6638	1.06104	0.530518	−	0.847674i	$-0.321997\pi$
$191$	14.6638	1.06104	0.530518	+	0.847674i	$0.321997\pi$
$192$	0	0
$193$	−19.6530	−1.41466	−0.707328	−	0.706885i	$-0.750100\pi$
$193$	−19.6530	−1.41466	−0.707328	+	0.706885i	$0.750100\pi$
$194$	−7.71928	−0.554212
$195$	0	0
$196$	5.88938	0.420670
$197$	4.17761	0.297643	0.148821	−	0.988864i	$-0.452452\pi$
$197$	4.17761	0.297643	0.148821	+	0.988864i	$0.452452\pi$
$198$	0	0
$199$	−14.5828	−1.03375	−0.516875	−	0.856061i	$-0.672905\pi$
$199$	−14.5828	−1.03375	−0.516875	+	0.856061i	$0.672905\pi$
$200$	−11.8158	−0.835502
$201$	0	0
$202$	2.46072	0.173136
$203$	15.7258	1.10374
$204$	0	0
$205$	34.3613	2.39990
$206$	−7.30233	−0.508777
$207$	0	0
$208$	13.1988	0.915175
$209$	4.35587	0.301302
$210$	0	0
$211$	−11.4294	−0.786830	−0.393415	−	0.919361i	$-0.628706\pi$
$211$	−11.4294	−0.786830	−0.393415	+	0.919361i	$0.628706\pi$
$212$	21.3220	1.46440
$213$	0	0
$214$	4.46464	0.305197
$215$	−1.72316	−0.117518
$216$	0	0
$217$	15.3374	1.04117
$218$	3.51207	0.237868
$219$	0	0
$220$	−6.29411	−0.424349
$221$	−13.5361	−0.910535
$222$	0	0
$223$	3.82865	0.256385	0.128193	−	0.991749i	$-0.459082\pi$
$223$	3.82865	0.256385	0.128193	+	0.991749i	$0.459082\pi$
$224$	−8.88306	−0.593524
$225$	0	0
$226$	2.20044	0.146371
$227$	10.1929	0.676527	0.338264	−	0.941051i	$-0.390160\pi$
$227$	10.1929	0.676527	0.338264	+	0.941051i	$0.390160\pi$
$228$	0	0
$229$	4.69673	0.310369	0.155184	−	0.987886i	$-0.450403\pi$
$229$	4.69673	0.310369	0.155184	+	0.987886i	$0.450403\pi$
$230$	9.85999	0.650149
$231$	0	0
$232$	13.5195	0.887596
$233$	3.20209	0.209776	0.104888	−	0.994484i	$-0.466552\pi$
$233$	3.20209	0.209776	0.104888	+	0.994484i	$0.466552\pi$
$234$	0	0
$235$	36.2890	2.36723
$236$	−1.89329	−0.123243
$237$	0	0
$238$	2.50660	0.162479
$239$	15.0328	0.972388	0.486194	−	0.873851i	$-0.338385\pi$
$239$	15.0328	0.972388	0.486194	+	0.873851i	$0.338385\pi$
$240$	0	0
$241$	10.7597	0.693092	0.346546	−	0.938033i	$-0.387355\pi$
$241$	10.7597	0.693092	0.346546	+	0.938033i	$0.387355\pi$
$242$	0.436741	0.0280747
$243$	0	0
$244$	1.80926	0.115826
$245$	11.3241	0.723469
$246$	0	0
$247$	−19.8803	−1.26495
$248$	13.1855	0.837281
$249$	0	0
$250$	−3.19412	−0.202014
$251$	−25.8084	−1.62901	−0.814504	−	0.580157i	$-0.802991\pi$
$251$	−25.8084	−1.62901	−0.814504	+	0.580157i	$0.802991\pi$
$252$	0	0
$253$	6.48962	0.407999
$254$	5.27492	0.330978
$255$	0	0
$256$	2.82774	0.176734
$257$	23.8547	1.48801	0.744007	−	0.668172i	$-0.232923\pi$
$257$	23.8547	1.48801	0.744007	+	0.668172i	$0.232923\pi$
$258$	0	0
$259$	2.03398	0.126386
$260$	28.7265	1.78154
$261$	0	0
$262$	8.29907	0.512718
$263$	15.0382	0.927294	0.463647	−	0.886020i	$-0.346541\pi$
$263$	15.0382	0.927294	0.463647	+	0.886020i	$0.346541\pi$
$264$	0	0
$265$	40.9980	2.51848
$266$	3.68143	0.225723
$267$	0	0
$268$	−11.0426	−0.674536
$269$	−5.09501	−0.310648	−0.155324	−	0.987864i	$-0.549642\pi$
$269$	−5.09501	−0.310648	−0.155324	+	0.987864i	$0.549642\pi$
$270$	0	0
$271$	−13.6962	−0.831986	−0.415993	−	0.909368i	$-0.636566\pi$
$271$	−13.6962	−0.831986	−0.415993	+	0.909368i	$0.636566\pi$
$272$	−8.57693	−0.520053
$273$	0	0
$274$	9.80231	0.592179
$275$	−7.10230	−0.428285
$276$	0	0
$277$	12.0575	0.724465	0.362233	−	0.932088i	$-0.382015\pi$
$277$	12.0575	0.724465	0.362233	+	0.932088i	$0.382015\pi$
$278$	−5.36472	−0.321754
$279$	0	0
$280$	−11.1999	−0.669324
$281$	−28.6925	−1.71165	−0.855827	−	0.517263i	$-0.826951\pi$
$281$	−28.6925	−1.71165	−0.855827	+	0.517263i	$0.826951\pi$
$282$	0	0
$283$	−4.02652	−0.239352	−0.119676	−	0.992813i	$-0.538186\pi$
$283$	−4.02652	−0.239352	−0.119676	+	0.992813i	$0.538186\pi$
$284$	0.109272	0.00648412
$285$	0	0
$286$	−1.99330	−0.117866
$287$	19.1141	1.12827
$288$	0	0
$289$	−8.20393	−0.482584
$290$	12.3468	0.725026
$291$	0	0
$292$	−4.56932	−0.267399
$293$	−1.96841	−0.114996	−0.0574979	−	0.998346i	$-0.518312\pi$
$293$	−1.96841	−0.114996	−0.0574979	+	0.998346i	$0.518312\pi$
$294$	0	0
$295$	−3.64042	−0.211953
$296$	1.74861	0.101636
$297$	0	0
$298$	−8.68331	−0.503010
$299$	−29.6188	−1.71290
$300$	0	0
$301$	−0.958538	−0.0552492
$302$	−4.62442	−0.266105
$303$	0	0
$304$	−12.5969	−0.722480
$305$	3.47884	0.199198
$306$	0	0
$307$	23.5477	1.34394	0.671969	−	0.740580i	$-0.265449\pi$
$307$	23.5477	1.34394	0.671969	+	0.740580i	$0.265449\pi$
$308$	−3.50121	−0.199500
$309$	0	0
$310$	12.0418	0.683927
$311$	−21.8661	−1.23991	−0.619956	−	0.784636i	$-0.712850\pi$
$311$	−21.8661	−1.23991	−0.619956	+	0.784636i	$0.712850\pi$
$312$	0	0
$313$	−8.85138	−0.500310	−0.250155	−	0.968206i	$-0.580482\pi$
$313$	−8.85138	−0.500310	−0.250155	+	0.968206i	$0.580482\pi$
$314$	−3.08021	−0.173826
$315$	0	0
$316$	−9.78516	−0.550458
$317$	−0.553452	−0.0310849	−0.0155425	−	0.999879i	$-0.504948\pi$
$317$	−0.553452	−0.0310849	−0.0155425	+	0.999879i	$0.504948\pi$
$318$	0	0
$319$	8.12635	0.454988
$320$	13.1468	0.734927
$321$	0	0
$322$	5.48480	0.305656
$323$	12.9187	0.718816
$324$	0	0
$325$	32.4151	1.79807
$326$	2.24080	0.124106
$327$	0	0
$328$	16.4323	0.907324
$329$	20.1864	1.11291
$330$	0	0
$331$	−2.04463	−0.112383	−0.0561915	−	0.998420i	$-0.517896\pi$
$331$	−2.04463	−0.112383	−0.0561915	+	0.998420i	$0.517896\pi$
$332$	19.7689	1.08496
$333$	0	0
$334$	−9.89836	−0.541614
$335$	−21.2327	−1.16007
$336$	0	0
$337$	−4.08881	−0.222732	−0.111366	−	0.993779i	$-0.535523\pi$
$337$	−4.08881	−0.222732	−0.111366	+	0.993779i	$0.535523\pi$
$338$	3.41984	0.186015
$339$	0	0
$340$	−18.6672	−1.01237
$341$	7.92562	0.429196
$342$	0	0
$343$	19.8454	1.07155
$344$	−0.824053	−0.0444300
$345$	0	0
$346$	−5.38942	−0.289737
$347$	3.85730	0.207071	0.103535	−	0.994626i	$-0.466985\pi$
$347$	3.85730	0.207071	0.103535	+	0.994626i	$0.466985\pi$
$348$	0	0
$349$	22.3728	1.19759	0.598794	−	0.800903i	$-0.295647\pi$
$349$	22.3728	1.19759	0.598794	+	0.800903i	$0.295647\pi$
$350$	−6.00261	−0.320853
$351$	0	0
$352$	−4.59034	−0.244666
$353$	2.36325	0.125783	0.0628917	−	0.998020i	$-0.479968\pi$
$353$	2.36325	0.125783	0.0628917	+	0.998020i	$0.479968\pi$
$354$	0	0
$355$	0.210109	0.0111514
$356$	−33.6080	−1.78122
$357$	0	0
$358$	−1.81179	−0.0957560
$359$	−5.06403	−0.267269	−0.133635	−	0.991031i	$-0.542665\pi$
$359$	−5.06403	−0.267269	−0.133635	+	0.991031i	$0.542665\pi$
$360$	0	0
$361$	−0.0263928	−0.00138910
$362$	6.87584	0.361386
$363$	0	0
$364$	15.9796	0.837560
$365$	−8.78587	−0.459873
$366$	0	0
$367$	−3.23910	−0.169080	−0.0845399	−	0.996420i	$-0.526942\pi$
$367$	−3.23910	−0.169080	−0.0845399	+	0.996420i	$0.526942\pi$
$368$	−18.7675	−0.978324
$369$	0	0
$370$	1.59693	0.0830206
$371$	22.8059	1.18402
$372$	0	0
$373$	−3.70872	−0.192030	−0.0960150	−	0.995380i	$-0.530610\pi$
$373$	−3.70872	−0.192030	−0.0960150	+	0.995380i	$0.530610\pi$
$374$	1.29529	0.0669780
$375$	0	0
$376$	17.3542	0.894975
$377$	−37.0889	−1.91018
$378$	0	0
$379$	−21.3298	−1.09564	−0.547820	−	0.836596i	$-0.684542\pi$
$379$	−21.3298	−1.09564	−0.547820	+	0.836596i	$0.684542\pi$
$380$	−27.4163	−1.40643
$381$	0	0
$382$	6.40428	0.327671
$383$	−4.42601	−0.226158	−0.113079	−	0.993586i	$-0.536071\pi$
$383$	−4.42601	−0.226158	−0.113079	+	0.993586i	$0.536071\pi$
$384$	0	0
$385$	−6.73212	−0.343101
$386$	−8.58328	−0.436877
$387$	0	0
$388$	31.9782	1.62345
$389$	−18.0980	−0.917606	−0.458803	−	0.888538i	$-0.651722\pi$
$389$	−18.0980	−0.917606	−0.458803	+	0.888538i	$0.651722\pi$
$390$	0	0
$391$	19.2470	0.973363
$392$	5.41543	0.273521
$393$	0	0
$394$	1.82453	0.0919187
$395$	−18.8149	−0.946679
$396$	0	0
$397$	25.6066	1.28516	0.642579	−	0.766219i	$-0.277864\pi$
$397$	25.6066	1.28516	0.642579	+	0.766219i	$0.277864\pi$
$398$	−6.36892	−0.319245
$399$	0	0
$400$	20.5393	1.02697
$401$	24.9495	1.24592	0.622958	−	0.782255i	$-0.285931\pi$
$401$	24.9495	1.24592	0.622958	+	0.782255i	$0.285931\pi$
$402$	0	0
$403$	−36.1727	−1.80189
$404$	−10.1939	−0.507165
$405$	0	0
$406$	6.86811	0.340858
$407$	1.05106	0.0520993
$408$	0	0
$409$	26.3498	1.30292	0.651458	−	0.758685i	$-0.274158\pi$
$409$	26.3498	1.30292	0.651458	+	0.758685i	$0.274158\pi$
$410$	15.0070	0.741141
$411$	0	0
$412$	30.2509	1.49035
$413$	−2.02505	−0.0996460
$414$	0	0
$415$	38.0117	1.86592
$416$	20.9504	1.02718
$417$	0	0
$418$	1.90239	0.0930487
$419$	−21.0273	−1.02725	−0.513626	−	0.858014i	$-0.671698\pi$
$419$	−21.0273	−1.02725	−0.513626	+	0.858014i	$0.671698\pi$
$420$	0	0
$421$	−24.6599	−1.20185	−0.600924	−	0.799306i	$-0.705201\pi$
$421$	−24.6599	−1.20185	−0.600924	+	0.799306i	$0.705201\pi$
$422$	−4.99167	−0.242991
$423$	0	0
$424$	19.6061	0.952159
$425$	−21.0641	−1.02176
$426$	0	0
$427$	1.93516	0.0936492
$428$	−18.4954	−0.894009
$429$	0	0
$430$	−0.752573	−0.0362923
$431$	−34.0658	−1.64089	−0.820447	−	0.571723i	$-0.806275\pi$
$431$	−34.0658	−1.64089	−0.820447	+	0.571723i	$0.806275\pi$
$432$	0	0
$433$	−26.2258	−1.26033	−0.630165	−	0.776461i	$-0.717013\pi$
$433$	−26.2258	−1.26033	−0.630165	+	0.776461i	$0.717013\pi$
$434$	6.69845	0.321536
$435$	0	0
$436$	−14.5492	−0.696782
$437$	28.2679	1.35224
$438$	0	0
$439$	−31.2884	−1.49331	−0.746657	−	0.665209i	$-0.768343\pi$
$439$	−31.2884	−1.49331	−0.746657	+	0.665209i	$0.768343\pi$
$440$	−5.78759	−0.275913
$441$	0	0
$442$	−5.91175	−0.281193
$443$	4.24854	0.201854	0.100927	−	0.994894i	$-0.467819\pi$
$443$	4.24854	0.201854	0.100927	+	0.994894i	$0.467819\pi$
$444$	0	0
$445$	−64.6213	−3.06334
$446$	1.67213	0.0791775
$447$	0	0
$448$	7.31313	0.345513
$449$	−18.4688	−0.871594	−0.435797	−	0.900045i	$-0.643533\pi$
$449$	−18.4688	−0.871594	−0.435797	+	0.900045i	$0.643533\pi$
$450$	0	0
$451$	9.87724	0.465101
$452$	−9.11564	−0.428764
$453$	0	0
$454$	4.45166	0.208927
$455$	30.7256	1.44044
$456$	0	0
$457$	28.5285	1.33451	0.667254	−	0.744830i	$-0.267469\pi$
$457$	28.5285	1.33451	0.667254	+	0.744830i	$0.267469\pi$
$458$	2.05125	0.0958488
$459$	0	0
$460$	−40.8464	−1.90447
$461$	32.7053	1.52324	0.761620	−	0.648024i	$-0.224404\pi$
$461$	32.7053	1.52324	0.761620	+	0.648024i	$0.224404\pi$
$462$	0	0
$463$	−17.7971	−0.827102	−0.413551	−	0.910481i	$-0.635712\pi$
$463$	−17.7971	−0.827102	−0.413551	+	0.910481i	$0.635712\pi$
$464$	−23.5008	−1.09100
$465$	0	0
$466$	1.39848	0.0647834
$467$	−10.2527	−0.474439	−0.237220	−	0.971456i	$-0.576236\pi$
$467$	−10.2527	−0.474439	−0.237220	+	0.971456i	$0.576236\pi$
$468$	0	0
$469$	−11.8111	−0.545386
$470$	15.8489	0.731054
$471$	0	0
$472$	−1.74093	−0.0801327
$473$	−0.495326	−0.0227751
$474$	0	0
$475$	−30.9367	−1.41947
$476$	−10.3840	−0.475948
$477$	0	0
$478$	6.56541	0.300295
$479$	37.5407	1.71528	0.857639	−	0.514252i	$-0.171930\pi$
$479$	37.5407	1.71528	0.857639	+	0.514252i	$0.171930\pi$
$480$	0	0
$481$	−4.79709	−0.218728
$482$	4.69919	0.214042
$483$	0	0
$484$	−1.80926	−0.0822390
$485$	61.4876	2.79201
$486$	0	0
$487$	−23.1356	−1.04837	−0.524186	−	0.851604i	$-0.675630\pi$
$487$	−23.1356	−1.04837	−0.524186	+	0.851604i	$0.675630\pi$
$488$	1.66366	0.0753102
$489$	0	0
$490$	4.94569	0.223423
$491$	3.37400	0.152267	0.0761333	−	0.997098i	$-0.475743\pi$
$491$	3.37400	0.152267	0.0761333	+	0.997098i	$0.475743\pi$
$492$	0	0
$493$	24.1013	1.08547
$494$	−8.68254	−0.390646
$495$	0	0
$496$	−22.9203	−1.02915
$497$	0.116877	0.00524263
$498$	0	0
$499$	−3.50167	−0.156756	−0.0783781	−	0.996924i	$-0.524974\pi$
$499$	−3.50167	−0.156756	−0.0783781	+	0.996924i	$0.524974\pi$
$500$	13.2321	0.591757
$501$	0	0
$502$	−11.2716	−0.503074
$503$	−33.4010	−1.48928	−0.744639	−	0.667468i	$-0.767378\pi$
$503$	−33.4010	−1.48928	−0.744639	+	0.667468i	$0.767378\pi$
$504$	0	0
$505$	−19.6008	−0.872223
$506$	2.83428	0.125999
$507$	0	0
$508$	−21.8521	−0.969530
$509$	−4.81578	−0.213456	−0.106728	−	0.994288i	$-0.534037\pi$
$509$	−4.81578	−0.213456	−0.106728	+	0.994288i	$0.534037\pi$
$510$	0	0
$511$	−4.88730	−0.216201
$512$	22.8973	1.01193
$513$	0	0
$514$	10.4183	0.459532
$515$	58.1663	2.56311
$516$	0	0
$517$	10.4314	0.458771
$518$	0.888323	0.0390307
$519$	0	0
$520$	26.4147	1.15836
$521$	14.9388	0.654481	0.327241	−	0.944941i	$-0.393881\pi$
$521$	14.9388	0.654481	0.327241	+	0.944941i	$0.393881\pi$
$522$	0	0
$523$	−11.8191	−0.516812	−0.258406	−	0.966036i	$-0.583197\pi$
$523$	−11.8191	−0.516812	−0.258406	+	0.966036i	$0.583197\pi$
$524$	−34.3800	−1.50190
$525$	0	0
$526$	6.56778	0.286369
$527$	23.5059	1.02393
$528$	0	0
$529$	19.1151	0.831093
$530$	17.9055	0.777764
$531$	0	0
$532$	−15.2508	−0.661207
$533$	−45.0800	−1.95263
$534$	0	0
$535$	−35.5629	−1.53752
$536$	−10.1540	−0.438585
$537$	0	0
$538$	−2.22520	−0.0959351
$539$	3.25514	0.140209
$540$	0	0
$541$	23.8977	1.02744	0.513721	−	0.857957i	$-0.328267\pi$
$541$	23.8977	1.02744	0.513721	+	0.857957i	$0.328267\pi$
$542$	−5.98169	−0.256936
$543$	0	0
$544$	−13.6141	−0.583700
$545$	−27.9752	−1.19833
$546$	0	0
$547$	3.31882	0.141902	0.0709512	−	0.997480i	$-0.477397\pi$
$547$	3.31882	0.141902	0.0709512	+	0.997480i	$0.477397\pi$
$548$	−40.6074	−1.73466
$549$	0	0
$550$	−3.10186	−0.132264
$551$	35.3973	1.50798
$552$	0	0
$553$	−10.4661	−0.445065
$554$	5.26600	0.223731
$555$	0	0
$556$	22.2241	0.942511
$557$	−5.44841	−0.230856	−0.115428	−	0.993316i	$-0.536824\pi$
$557$	−5.44841	−0.230856	−0.115428	+	0.993316i	$0.536824\pi$
$558$	0	0
$559$	2.26068	0.0956167
$560$	19.4688	0.822708
$561$	0	0
$562$	−12.5312	−0.528597
$563$	2.97850	0.125529	0.0627643	−	0.998028i	$-0.480008\pi$
$563$	2.97850	0.125529	0.0627643	+	0.998028i	$0.480008\pi$
$564$	0	0
$565$	−17.5275	−0.737389
$566$	−1.75854	−0.0739171
$567$	0	0
$568$	0.100479	0.00421599
$569$	15.1867	0.636658	0.318329	−	0.947980i	$-0.396878\pi$
$569$	15.1867	0.636658	0.318329	+	0.947980i	$0.396878\pi$
$570$	0	0
$571$	−0.197248	−0.00825458	−0.00412729	−	0.999991i	$-0.501314\pi$
$571$	−0.197248	−0.00825458	−0.00412729	+	0.999991i	$0.501314\pi$
$572$	8.25751	0.345264
$573$	0	0
$574$	8.34790	0.348434
$575$	−46.0912	−1.92214
$576$	0	0
$577$	−23.8728	−0.993839	−0.496919	−	0.867797i	$-0.665536\pi$
$577$	−23.8728	−0.993839	−0.496919	+	0.867797i	$0.665536\pi$
$578$	−3.58299	−0.149033
$579$	0	0
$580$	−51.1481	−2.12381
$581$	21.1447	0.877229
$582$	0	0
$583$	11.7850	0.488083
$584$	−4.20160	−0.173863
$585$	0	0
$586$	−0.859685	−0.0355132
$587$	25.9108	1.06945	0.534727	−	0.845025i	$-0.320415\pi$
$587$	25.9108	1.06945	0.534727	+	0.845025i	$0.320415\pi$
$588$	0	0
$589$	34.5230	1.42249
$590$	−1.58992	−0.0654558
$591$	0	0
$592$	−3.03960	−0.124927
$593$	−0.815514	−0.0334891	−0.0167446	−	0.999860i	$-0.505330\pi$
$593$	−0.815514	−0.0334891	−0.0167446	+	0.999860i	$0.505330\pi$
$594$	0	0
$595$	−19.9662	−0.818536
$596$	35.9718	1.47346
$597$	0	0
$598$	−12.9357	−0.528981
$599$	−6.38692	−0.260963	−0.130481	−	0.991451i	$-0.541652\pi$
$599$	−6.38692	−0.260963	−0.130481	+	0.991451i	$0.541652\pi$
$600$	0	0
$601$	−29.4015	−1.19931	−0.599656	−	0.800258i	$-0.704696\pi$
$601$	−29.4015	−1.19931	−0.599656	+	0.800258i	$0.704696\pi$
$602$	−0.418632	−0.0170622
$603$	0	0
$604$	19.1573	0.779499
$605$	−3.47884	−0.141435
$606$	0	0
$607$	9.41610	0.382188	0.191094	−	0.981572i	$-0.438797\pi$
$607$	9.41610	0.382188	0.191094	+	0.981572i	$0.438797\pi$
$608$	−19.9949	−0.810901
$609$	0	0
$610$	1.51935	0.0615166
$611$	−47.6090	−1.92606
$612$	0	0
$613$	−3.24986	−0.131261	−0.0656304	−	0.997844i	$-0.520906\pi$
$613$	−3.24986	−0.131261	−0.0656304	+	0.997844i	$0.520906\pi$
$614$	10.2842	0.415038
$615$	0	0
$616$	−3.21945	−0.129715
$617$	8.37130	0.337016	0.168508	−	0.985700i	$-0.446105\pi$
$617$	8.37130	0.337016	0.168508	+	0.985700i	$0.446105\pi$
$618$	0	0
$619$	−8.62770	−0.346777	−0.173388	−	0.984854i	$-0.555472\pi$
$619$	−8.62770	−0.346777	−0.173388	+	0.984854i	$0.555472\pi$
$620$	−49.8847	−2.00342
$621$	0	0
$622$	−9.54981	−0.382913
$623$	−35.9468	−1.44018
$624$	0	0
$625$	−10.0689	−0.402755
$626$	−3.86576	−0.154507
$627$	0	0
$628$	12.7602	0.509187
$629$	3.11726	0.124293
$630$	0	0
$631$	28.1700	1.12143	0.560716	−	0.828008i	$-0.310526\pi$
$631$	28.1700	1.12143	0.560716	+	0.828008i	$0.310526\pi$
$632$	−8.99770	−0.357909
$633$	0	0
$634$	−0.241715	−0.00959972
$635$	−42.0171	−1.66740
$636$	0	0
$637$	−14.8565	−0.588637
$638$	3.54911	0.140510
$639$	0	0
$640$	37.6798	1.48942
$641$	4.34993	0.171812	0.0859059	−	0.996303i	$-0.472622\pi$
$641$	4.34993	0.171812	0.0859059	+	0.996303i	$0.472622\pi$
$642$	0	0
$643$	33.0246	1.30236	0.651181	−	0.758923i	$-0.274274\pi$
$643$	33.0246	1.30236	0.651181	+	0.758923i	$0.274274\pi$
$644$	−22.7215	−0.895354
$645$	0	0
$646$	5.64213	0.221986
$647$	21.7241	0.854061	0.427031	−	0.904237i	$-0.359560\pi$
$647$	21.7241	0.854061	0.427031	+	0.904237i	$0.359560\pi$
$648$	0	0
$649$	−1.04645	−0.0410766
$650$	14.1570	0.555283
$651$	0	0
$652$	−9.28283	−0.363543
$653$	17.0226	0.666146	0.333073	−	0.942901i	$-0.391915\pi$
$653$	17.0226	0.666146	0.333073	+	0.942901i	$0.391915\pi$
$654$	0	0
$655$	−66.1058	−2.58297
$656$	−28.5643	−1.11525
$657$	0	0
$658$	8.81622	0.343692
$659$	−45.0710	−1.75572	−0.877858	−	0.478921i	$-0.841028\pi$
$659$	−45.0710	−1.75572	−0.877858	+	0.478921i	$0.841028\pi$
$660$	0	0
$661$	17.6243	0.685504	0.342752	−	0.939426i	$-0.388641\pi$
$661$	17.6243	0.685504	0.342752	+	0.939426i	$0.388641\pi$
$662$	−0.892972	−0.0347063
$663$	0	0
$664$	18.1780	0.705444
$665$	−29.3242	−1.13715
$666$	0	0
$667$	52.7369	2.04198
$668$	41.0053	1.58654
$669$	0	0
$670$	−9.27320	−0.358255
$671$	1.00000	0.0386046
$672$	0	0
$673$	−9.44548	−0.364096	−0.182048	−	0.983290i	$-0.558273\pi$
$673$	−9.44548	−0.364096	−0.182048	+	0.983290i	$0.558273\pi$
$674$	−1.78575	−0.0687845
$675$	0	0
$676$	−14.1671	−0.544890
$677$	42.3833	1.62892	0.814461	−	0.580218i	$-0.197033\pi$
$677$	42.3833	1.62892	0.814461	+	0.580218i	$0.197033\pi$
$678$	0	0
$679$	34.2036	1.31261
$680$	−17.1649	−0.658245
$681$	0	0
$682$	3.46144	0.132545
$683$	7.58574	0.290260	0.145130	−	0.989413i	$-0.453640\pi$
$683$	7.58574	0.290260	0.145130	+	0.989413i	$0.453640\pi$
$684$	0	0
$685$	−78.0798	−2.98328
$686$	8.66728	0.330918
$687$	0	0
$688$	1.43245	0.0546116
$689$	−53.7869	−2.04912
$690$	0	0
$691$	6.46933	0.246105	0.123052	−	0.992400i	$-0.460732\pi$
$691$	6.46933	0.246105	0.123052	+	0.992400i	$0.460732\pi$
$692$	22.3264	0.848723
$693$	0	0
$694$	1.68464	0.0639480
$695$	42.7324	1.62093
$696$	0	0
$697$	29.2941	1.10959
$698$	9.77111	0.369842
$699$	0	0
$700$	24.8666	0.939871
$701$	−13.5910	−0.513325	−0.256663	−	0.966501i	$-0.582623\pi$
$701$	−13.5910	−0.513325	−0.256663	+	0.966501i	$0.582623\pi$
$702$	0	0
$703$	4.57830	0.172674
$704$	3.77907	0.142429
$705$	0	0
$706$	1.03213	0.0388447
$707$	−10.9033	−0.410060
$708$	0	0
$709$	9.54923	0.358629	0.179314	−	0.983792i	$-0.442612\pi$
$709$	9.54923	0.358629	0.179314	+	0.983792i	$0.442612\pi$
$710$	0.0917629	0.00344380
$711$	0	0
$712$	−30.9034	−1.15815
$713$	51.4342	1.92623
$714$	0	0
$715$	15.8775	0.593785
$716$	7.50559	0.280497
$717$	0	0
$718$	−2.21167	−0.0825387
$719$	6.51267	0.242881	0.121441	−	0.992599i	$-0.461249\pi$
$719$	6.51267	0.242881	0.121441	+	0.992599i	$0.461249\pi$
$720$	0	0
$721$	32.3561	1.20500
$722$	−0.0115268	−0.000428984 0
$723$	0	0
$724$	−28.4841	−1.05860
$725$	−57.7157	−2.14351
$726$	0	0
$727$	3.58509	0.132964	0.0664818	−	0.997788i	$-0.478823\pi$
$727$	3.58509	0.132964	0.0664818	+	0.997788i	$0.478823\pi$
$728$	14.6937	0.544584
$729$	0	0
$730$	−3.83714	−0.142019
$731$	−1.46905	−0.0543347
$732$	0	0
$733$	−17.0361	−0.629243	−0.314622	−	0.949217i	$-0.601878\pi$
$733$	−17.0361	−0.629243	−0.314622	+	0.949217i	$0.601878\pi$
$734$	−1.41465	−0.0522156
$735$	0	0
$736$	−29.7895	−1.09806
$737$	−6.10340	−0.224822
$738$	0	0
$739$	41.5324	1.52779	0.763897	−	0.645339i	$-0.223284\pi$
$739$	41.5324	1.52779	0.763897	+	0.645339i	$0.223284\pi$
$740$	−6.61552	−0.243191
$741$	0	0
$742$	9.96024	0.365652
$743$	−18.1772	−0.666855	−0.333428	−	0.942776i	$-0.608205\pi$
$743$	−18.1772	−0.666855	−0.333428	+	0.942776i	$0.608205\pi$
$744$	0	0
$745$	69.1665	2.53406
$746$	−1.61975	−0.0593031
$747$	0	0
$748$	−5.36593	−0.196198
$749$	−19.7825	−0.722837
$750$	0	0
$751$	45.6729	1.66663	0.833314	−	0.552799i	$-0.186440\pi$
$751$	45.6729	1.66663	0.833314	+	0.552799i	$0.186440\pi$
$752$	−30.1668	−1.10007
$753$	0	0
$754$	−16.1982	−0.589905
$755$	36.8356	1.34058
$756$	0	0
$757$	−22.2550	−0.808873	−0.404436	−	0.914566i	$-0.632532\pi$
$757$	−22.2550	−0.808873	−0.404436	+	0.914566i	$0.632532\pi$
$758$	−9.31560	−0.338358
$759$	0	0
$760$	−25.2100	−0.914462
$761$	14.1456	0.512779	0.256390	−	0.966573i	$-0.417467\pi$
$761$	14.1456	0.512779	0.256390	+	0.966573i	$0.417467\pi$
$762$	0	0
$763$	−15.5617	−0.563372
$764$	−26.5306	−0.959844
$765$	0	0
$766$	−1.93302	−0.0698427
$767$	4.77601	0.172452
$768$	0	0
$769$	−13.7367	−0.495357	−0.247678	−	0.968842i	$-0.579668\pi$
$769$	−13.7367	−0.495357	−0.247678	+	0.968842i	$0.579668\pi$
$770$	−2.94019	−0.105957
$771$	0	0
$772$	35.5574	1.27974
$773$	36.9060	1.32742	0.663709	−	0.747991i	$-0.268981\pi$
$773$	36.9060	1.32742	0.663709	+	0.747991i	$0.268981\pi$
$774$	0	0
$775$	−56.2901	−2.02200
$776$	29.4047	1.05557
$777$	0	0
$778$	−7.90414	−0.283377
$779$	43.0240	1.54149
$780$	0	0
$781$	0.0603962	0.00216115
$782$	8.40595	0.300596
$783$	0	0
$784$	−9.41362	−0.336201
$785$	24.5353	0.875701
$786$	0	0
$787$	−50.6675	−1.80610	−0.903051	−	0.429534i	$-0.858678\pi$
$787$	−50.6675	−1.80610	−0.903051	+	0.429534i	$0.858678\pi$
$788$	−7.55838	−0.269256
$789$	0	0
$790$	−8.21722	−0.292356
$791$	−9.75001	−0.346670
$792$	0	0
$793$	−4.56403	−0.162073
$794$	11.1834	0.396885
$795$	0	0
$796$	26.3841	0.935160
$797$	−47.1764	−1.67107	−0.835536	−	0.549435i	$-0.814843\pi$
$797$	−47.1764	−1.67107	−0.835536	+	0.549435i	$0.814843\pi$
$798$	0	0
$799$	30.9375	1.09449
$800$	32.6019	1.15265
$801$	0	0
$802$	10.8964	0.384767
$803$	−2.52552	−0.0891236
$804$	0	0
$805$	−43.6889	−1.53983
$806$	−15.7981	−0.556464
$807$	0	0
$808$	−9.37353	−0.329760
$809$	9.36222	0.329158	0.164579	−	0.986364i	$-0.447373\pi$
$809$	9.36222	0.329158	0.164579	+	0.986364i	$0.447373\pi$
$810$	0	0
$811$	−48.1058	−1.68922	−0.844611	−	0.535380i	$-0.820168\pi$
$811$	−48.1058	−1.68922	−0.844611	+	0.535380i	$0.820168\pi$
$812$	−28.4521	−0.998472
$813$	0	0
$814$	0.459042	0.0160894
$815$	−17.8490	−0.625223
$816$	0	0
$817$	−2.15758	−0.0754841
$818$	11.5080	0.402369
$819$	0	0
$820$	−62.1684	−2.17102
$821$	−12.1493	−0.424014	−0.212007	−	0.977268i	$-0.568000\pi$
$821$	−12.1493	−0.424014	−0.212007	+	0.977268i	$0.568000\pi$
$822$	0	0
$823$	−52.0806	−1.81541	−0.907707	−	0.419604i	$-0.862169\pi$
$823$	−52.0806	−1.81541	−0.907707	+	0.419604i	$0.862169\pi$
$824$	27.8164	0.969032
$825$	0	0
$826$	−0.884420	−0.0307729
$827$	17.3050	0.601753	0.300876	−	0.953663i	$-0.402721\pi$
$827$	17.3050	0.601753	0.300876	+	0.953663i	$0.402721\pi$
$828$	0	0
$829$	−32.5453	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$829$	−32.5453	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$830$	16.6012	0.576237
$831$	0	0
$832$	−17.2478	−0.597960
$833$	9.65414	0.334496
$834$	0	0
$835$	78.8449	2.72854
$836$	−7.88089	−0.272566
$837$	0	0
$838$	−9.18348	−0.317238
$839$	10.1679	0.351035	0.175517	−	0.984476i	$-0.443840\pi$
$839$	10.1679	0.351035	0.175517	+	0.984476i	$0.443840\pi$
$840$	0	0
$841$	37.0375	1.27716
$842$	−10.7700	−0.371157
$843$	0	0
$844$	20.6787	0.711789
$845$	−27.2405	−0.937103
$846$	0	0
$847$	−1.93516	−0.0664931
$848$	−34.0813	−1.17036
$849$	0	0
$850$	−9.19955	−0.315542
$851$	6.82101	0.233821
$852$	0	0
$853$	−10.7159	−0.366907	−0.183453	−	0.983028i	$-0.558728\pi$
$853$	−10.7159	−0.366907	−0.183453	+	0.983028i	$0.558728\pi$
$854$	0.845165	0.0289210
$855$	0	0
$856$	−17.0070	−0.581287
$857$	10.1298	0.346027	0.173014	−	0.984919i	$-0.444650\pi$
$857$	10.1298	0.346027	0.173014	+	0.984919i	$0.444650\pi$
$858$	0	0
$859$	−8.76656	−0.299111	−0.149556	−	0.988753i	$-0.547784\pi$
$859$	−8.76656	−0.299111	−0.149556	+	0.988753i	$0.547784\pi$
$860$	3.11764	0.106311
$861$	0	0
$862$	−14.8779	−0.506744
$863$	−0.329472	−0.0112153	−0.00560767	−	0.999984i	$-0.501785\pi$
$863$	−0.329472	−0.0112153	−0.00560767	+	0.999984i	$0.501785\pi$
$864$	0	0
$865$	42.9292	1.45964
$866$	−11.4539	−0.389218
$867$	0	0
$868$	−27.7493	−0.941871
$869$	−5.40838	−0.183467
$870$	0	0
$871$	27.8561	0.943868
$872$	−13.3784	−0.453049
$873$	0	0
$874$	12.3458	0.417601
$875$	14.1529	0.478456
$876$	0	0
$877$	19.8266	0.669497	0.334748	−	0.942308i	$-0.391349\pi$
$877$	19.8266	0.669497	0.334748	+	0.942308i	$0.391349\pi$
$878$	−13.6649	−0.461169
$879$	0	0
$880$	10.0605	0.339141
$881$	−43.9146	−1.47952	−0.739761	−	0.672870i	$-0.765061\pi$
$881$	−43.9146	−1.47952	−0.739761	+	0.672870i	$0.765061\pi$
$882$	0	0
$883$	−1.12961	−0.0380143	−0.0190071	−	0.999819i	$-0.506051\pi$
$883$	−1.12961	−0.0380143	−0.0190071	+	0.999819i	$0.506051\pi$
$884$	24.4902	0.823696
$885$	0	0
$886$	1.85551	0.0623370
$887$	30.5280	1.02503	0.512515	−	0.858678i	$-0.328714\pi$
$887$	30.5280	1.02503	0.512515	+	0.858678i	$0.328714\pi$
$888$	0	0
$889$	−23.3728	−0.783898
$890$	−28.2228	−0.946029
$891$	0	0
$892$	−6.92702	−0.231934
$893$	45.4377	1.52051
$894$	0	0
$895$	14.4317	0.482399
$896$	20.9601	0.700226
$897$	0	0
$898$	−8.06605	−0.269168
$899$	64.4063	2.14807
$900$	0	0
$901$	34.9520	1.16442
$902$	4.31379	0.143634
$903$	0	0
$904$	−8.38206	−0.278783
$905$	−54.7692	−1.82059
$906$	0	0
$907$	−6.59359	−0.218937	−0.109468	−	0.993990i	$-0.534915\pi$
$907$	−6.59359	−0.218937	−0.109468	+	0.993990i	$0.534915\pi$
$908$	−18.4416	−0.612006
$909$	0	0
$910$	13.4191	0.444839
$911$	25.2887	0.837850	0.418925	−	0.908021i	$-0.362407\pi$
$911$	25.2887	0.837850	0.418925	+	0.908021i	$0.362407\pi$
$912$	0	0
$913$	10.9265	0.361616
$914$	12.4596	0.412126
$915$	0	0
$916$	−8.49760	−0.280769
$917$	−36.7725	−1.21434
$918$	0	0
$919$	−39.4731	−1.30210	−0.651049	−	0.759036i	$-0.725671\pi$
$919$	−39.4731	−1.30210	−0.651049	+	0.759036i	$0.725671\pi$
$920$	−37.5592	−1.23829
$921$	0	0
$922$	14.2837	0.470410
$923$	−0.275650	−0.00907313
$924$	0	0
$925$	−7.46497	−0.245447
$926$	−7.77272	−0.255428
$927$	0	0
$928$	−37.3027	−1.22452
$929$	−51.4929	−1.68943	−0.844713	−	0.535219i	$-0.820229\pi$
$929$	−51.4929	−1.68943	−0.844713	+	0.535219i	$0.820229\pi$
$930$	0	0
$931$	14.1790	0.464696
$932$	−5.79340	−0.189769
$933$	0	0
$934$	−4.47778	−0.146517
$935$	−10.3176	−0.337421
$936$	0	0
$937$	5.68848	0.185835	0.0929173	−	0.995674i	$-0.470381\pi$
$937$	5.68848	0.185835	0.0929173	+	0.995674i	$0.470381\pi$
$938$	−5.15838	−0.168427
$939$	0	0
$940$	−65.6561	−2.14147
$941$	9.84589	0.320967	0.160483	−	0.987039i	$-0.448695\pi$
$941$	9.84589	0.320967	0.160483	+	0.987039i	$0.448695\pi$
$942$	0	0
$943$	64.0995	2.08737
$944$	3.02625	0.0984960
$945$	0	0
$946$	−0.216329	−0.00703346
$947$	9.33207	0.303251	0.151626	−	0.988438i	$-0.451549\pi$
$947$	9.33207	0.303251	0.151626	+	0.988438i	$0.451549\pi$
$948$	0	0
$949$	11.5265	0.374167
$950$	−13.5113	−0.438365
$951$	0	0
$952$	−9.54830	−0.309462
$953$	2.64276	0.0856074	0.0428037	−	0.999084i	$-0.486371\pi$
$953$	2.64276	0.0856074	0.0428037	+	0.999084i	$0.486371\pi$
$954$	0	0
$955$	−51.0130	−1.65074
$956$	−27.1981	−0.879650
$957$	0	0
$958$	16.3955	0.529716
$959$	−43.4333	−1.40254
$960$	0	0
$961$	31.8154	1.02630
$962$	−2.09508	−0.0675482
$963$	0	0
$964$	−19.4670	−0.626991
$965$	68.3697	2.20090
$966$	0	0
$967$	−38.0052	−1.22217	−0.611083	−	0.791567i	$-0.709266\pi$
$967$	−38.0052	−1.22217	−0.611083	+	0.791567i	$0.709266\pi$
$968$	−1.66366	−0.0534720
$969$	0	0
$970$	26.8541	0.862234
$971$	0.160161	0.00513981	0.00256991	−	0.999997i	$-0.499182\pi$
$971$	0.160161	0.00513981	0.00256991	+	0.999997i	$0.499182\pi$
$972$	0	0
$973$	23.7707	0.762053
$974$	−10.1042	−0.323761
$975$	0	0
$976$	−2.89193	−0.0925684
$977$	−20.4963	−0.655735	−0.327867	−	0.944724i	$-0.606330\pi$
$977$	−20.4963	−0.655735	−0.327867	+	0.944724i	$0.606330\pi$
$978$	0	0
$979$	−18.5756	−0.593678
$980$	−20.4882	−0.654471
$981$	0	0
$982$	1.47356	0.0470233
$983$	−10.2845	−0.328024	−0.164012	−	0.986458i	$-0.552444\pi$
$983$	−10.2845	−0.328024	−0.164012	+	0.986458i	$0.552444\pi$
$984$	0	0
$985$	−14.5332	−0.463067
$986$	10.5260	0.335216
$987$	0	0
$988$	35.9686	1.14431
$989$	−3.21448	−0.102214
$990$	0	0
$991$	9.85979	0.313207	0.156603	−	0.987662i	$-0.449946\pi$
$991$	9.85979	0.313207	0.156603	+	0.987662i	$0.449946\pi$
$992$	−36.3812	−1.15511
$993$	0	0
$994$	0.0510448	0.00161904
$995$	50.7313	1.60829
$996$	0	0
$997$	−26.1240	−0.827356	−0.413678	−	0.910423i	$-0.635756\pi$
$997$	−26.1240	−0.827356	−0.413678	+	0.910423i	$0.635756\pi$
$998$	−1.52932	−0.0484098
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.12		19
3.2	odd	2		671.2.a.c.1.8	✓	19
33.32	even	2		7381.2.a.i.1.12		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.8	✓	19	3.2	odd	2
6039.2.a.k.1.12		19	1.1	even	1	trivial
7381.2.a.i.1.12		19	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+0.436741 q^{2} -1.80926 q^{4} -3.47884 q^{5} -1.93516 q^{7} -1.66366 q^{8} +O(q^{10})\)	\(q+0.436741 q^{2} -1.80926 q^{4} -3.47884 q^{5} -1.93516 q^{7} -1.66366 q^{8} -1.51935 q^{10} -1.00000 q^{11} +4.56403 q^{13} -0.845165 q^{14} +2.89193 q^{16} -2.96582 q^{17} -4.35587 q^{19} +6.29411 q^{20} -0.436741 q^{22} -6.48962 q^{23} +7.10230 q^{25} +1.99330 q^{26} +3.50121 q^{28} -8.12635 q^{29} -7.92562 q^{31} +4.59034 q^{32} -1.29529 q^{34} +6.73212 q^{35} -1.05106 q^{37} -1.90239 q^{38} +5.78759 q^{40} -9.87724 q^{41} +0.495326 q^{43} +1.80926 q^{44} -2.83428 q^{46} -10.4314 q^{47} -3.25514 q^{49} +3.10186 q^{50} -8.25751 q^{52} -11.7850 q^{53} +3.47884 q^{55} +3.21945 q^{56} -3.54911 q^{58} +1.04645 q^{59} -1.00000 q^{61} -3.46144 q^{62} -3.77907 q^{64} -15.8775 q^{65} +6.10340 q^{67} +5.36593 q^{68} +2.94019 q^{70} -0.0603962 q^{71} +2.52552 q^{73} -0.459042 q^{74} +7.88089 q^{76} +1.93516 q^{77} +5.40838 q^{79} -10.0605 q^{80} -4.31379 q^{82} -10.9265 q^{83} +10.3176 q^{85} +0.216329 q^{86} +1.66366 q^{88} +18.5756 q^{89} -8.83215 q^{91} +11.7414 q^{92} -4.55580 q^{94} +15.1534 q^{95} -17.6748 q^{97} -1.42165 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

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Database

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