LMFDB - Embedded newform 6039.2.a.k.1.14

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			1.14
Root			$-0.976920$ 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)

$f(q)$	$=$	$q+0.976920 q^{2} -1.04563 q^{4} +2.45987 q^{5} -4.72156 q^{7} -2.97533 q^{8} +O(q^{10})$	\(q+0.976920 q^{2} -1.04563 q^{4} +2.45987 q^{5} -4.72156 q^{7} -2.97533 q^{8} +2.40309 q^{10} -1.00000 q^{11} -3.64373 q^{13} -4.61259 q^{14} -0.815413 q^{16} -7.55245 q^{17} +3.11859 q^{19} -2.57210 q^{20} -0.976920 q^{22} -6.22515 q^{23} +1.05094 q^{25} -3.55963 q^{26} +4.93698 q^{28} +9.00853 q^{29} +6.83631 q^{31} +5.15408 q^{32} -7.37814 q^{34} -11.6144 q^{35} +1.37973 q^{37} +3.04662 q^{38} -7.31893 q^{40} -7.01523 q^{41} -2.05414 q^{43} +1.04563 q^{44} -6.08148 q^{46} -0.462126 q^{47} +15.2931 q^{49} +1.02669 q^{50} +3.80998 q^{52} +8.75381 q^{53} -2.45987 q^{55} +14.0482 q^{56} +8.80062 q^{58} +13.1468 q^{59} -1.00000 q^{61} +6.67853 q^{62} +6.66595 q^{64} -8.96309 q^{65} -2.33892 q^{67} +7.89704 q^{68} -11.3463 q^{70} +3.19609 q^{71} +6.46095 q^{73} +1.34788 q^{74} -3.26088 q^{76} +4.72156 q^{77} +7.48398 q^{79} -2.00581 q^{80} -6.85332 q^{82} -7.42750 q^{83} -18.5780 q^{85} -2.00673 q^{86} +2.97533 q^{88} -2.62699 q^{89} +17.2041 q^{91} +6.50918 q^{92} -0.451460 q^{94} +7.67132 q^{95} +14.0175 q^{97} +14.9401 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.976920	0.690787	0.345394	−	0.938458i	$-0.387745\pi$
$2$	0.976920	0.690787	0.345394	+	0.938458i	$0.387745\pi$
$3$	0	0
$3$	0	0
$4$	−1.04563	−0.522813
$5$	2.45987	1.10009	0.550043	−	0.835136i	$-0.314611\pi$
$5$	2.45987	1.10009	0.550043	+	0.835136i	$0.314611\pi$
$6$	0	0
$7$	−4.72156	−1.78458	−0.892290	−	0.451462i	$-0.850903\pi$
$7$	−4.72156	−1.78458	−0.892290	+	0.451462i	$0.850903\pi$
$8$	−2.97533	−1.05194
$9$	0	0
$10$	2.40309	0.759925
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.64373	−1.01059	−0.505294	−	0.862947i	$-0.668616\pi$
$13$	−3.64373	−1.01059	−0.505294	+	0.862947i	$0.668616\pi$
$14$	−4.61259	−1.23277
$15$	0	0
$16$	−0.815413	−0.203853
$17$	−7.55245	−1.83174	−0.915869	−	0.401477i	$-0.868497\pi$
$17$	−7.55245	−1.83174	−0.915869	+	0.401477i	$0.868497\pi$
$18$	0	0
$19$	3.11859	0.715454	0.357727	−	0.933826i	$-0.383552\pi$
$19$	3.11859	0.715454	0.357727	+	0.933826i	$0.383552\pi$
$20$	−2.57210	−0.575139
$21$	0	0
$22$	−0.976920	−0.208280
$23$	−6.22515	−1.29803	−0.649017	−	0.760774i	$-0.724819\pi$
$23$	−6.22515	−1.29803	−0.649017	+	0.760774i	$0.724819\pi$
$24$	0	0
$25$	1.05094	0.210188
$26$	−3.55963	−0.698102
$27$	0	0
$28$	4.93698	0.933002
$29$	9.00853	1.67284	0.836421	−	0.548087i	$-0.184644\pi$
$29$	9.00853	1.67284	0.836421	+	0.548087i	$0.184644\pi$
$30$	0	0
$31$	6.83631	1.22784	0.613918	−	0.789370i	$-0.289592\pi$
$31$	6.83631	1.22784	0.613918	+	0.789370i	$0.289592\pi$
$32$	5.15408	0.911121
$33$	0	0
$34$	−7.37814	−1.26534
$35$	−11.6144	−1.96319
$36$	0	0
$37$	1.37973	0.226825	0.113413	−	0.993548i	$-0.463822\pi$
$37$	1.37973	0.226825	0.113413	+	0.993548i	$0.463822\pi$
$38$	3.04662	0.494226
$39$	0	0
$40$	−7.31893	−1.15722
$41$	−7.01523	−1.09559	−0.547797	−	0.836611i	$-0.684534\pi$
$41$	−7.01523	−1.09559	−0.547797	+	0.836611i	$0.684534\pi$
$42$	0	0
$43$	−2.05414	−0.313253	−0.156626	−	0.987658i	$-0.550062\pi$
$43$	−2.05414	−0.313253	−0.156626	+	0.987658i	$0.550062\pi$
$44$	1.04563	0.157634
$45$	0	0
$46$	−6.08148	−0.896665
$47$	−0.462126	−0.0674080	−0.0337040	−	0.999432i	$-0.510730\pi$
$47$	−0.462126	−0.0674080	−0.0337040	+	0.999432i	$0.510730\pi$
$48$	0	0
$49$	15.2931	2.18473
$50$	1.02669	0.145195
$51$	0	0
$52$	3.80998	0.528349
$53$	8.75381	1.20243	0.601214	−	0.799088i	$-0.294684\pi$
$53$	8.75381	1.20243	0.601214	+	0.799088i	$0.294684\pi$
$54$	0	0
$55$	−2.45987	−0.331688
$56$	14.0482	1.87727
$57$	0	0
$58$	8.80062	1.15558
$59$	13.1468	1.71157	0.855786	−	0.517330i	$-0.173074\pi$
$59$	13.1468	1.71157	0.855786	+	0.517330i	$0.173074\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	6.67853	0.848174
$63$	0	0
$64$	6.66595	0.833244
$65$	−8.96309	−1.11173
$66$	0	0
$67$	−2.33892	−0.285744	−0.142872	−	0.989741i	$-0.545634\pi$
$67$	−2.33892	−0.285744	−0.142872	+	0.989741i	$0.545634\pi$
$68$	7.89704	0.957657
$69$	0	0
$70$	−11.3463	−1.35615
$71$	3.19609	0.379306	0.189653	−	0.981851i	$-0.439264\pi$
$71$	3.19609	0.379306	0.189653	+	0.981851i	$0.439264\pi$
$72$	0	0
$73$	6.46095	0.756198	0.378099	−	0.925765i	$-0.376578\pi$
$73$	6.46095	0.756198	0.378099	+	0.925765i	$0.376578\pi$
$74$	1.34788	0.156688
$75$	0	0
$76$	−3.26088	−0.374049
$77$	4.72156	0.538071
$78$	0	0
$79$	7.48398	0.842014	0.421007	−	0.907057i	$-0.361677\pi$
$79$	7.48398	0.842014	0.421007	+	0.907057i	$0.361677\pi$
$80$	−2.00581	−0.224256
$81$	0	0
$82$	−6.85332	−0.756823
$83$	−7.42750	−0.815274	−0.407637	−	0.913144i	$-0.633647\pi$
$83$	−7.42750	−0.815274	−0.407637	+	0.913144i	$0.633647\pi$
$84$	0	0
$85$	−18.5780	−2.01507
$86$	−2.00673	−0.216391
$87$	0	0
$88$	2.97533	0.317172
$89$	−2.62699	−0.278461	−0.139230	−	0.990260i	$-0.544463\pi$
$89$	−2.62699	−0.278461	−0.139230	+	0.990260i	$0.544463\pi$
$90$	0	0
$91$	17.2041	1.80348
$92$	6.50918	0.678629
$93$	0	0
$94$	−0.451460	−0.0465645
$95$	7.67132	0.787061
$96$	0	0
$97$	14.0175	1.42326	0.711629	−	0.702556i	$-0.247958\pi$
$97$	14.0175	1.42326	0.711629	+	0.702556i	$0.247958\pi$
$98$	14.9401	1.50918
$99$	0	0
$100$	−1.09889	−0.109889
$101$	−14.0501	−1.39804	−0.699019	−	0.715103i	$-0.746380\pi$
$101$	−14.0501	−1.39804	−0.699019	+	0.715103i	$0.746380\pi$
$102$	0	0
$103$	5.18315	0.510711	0.255356	−	0.966847i	$-0.417807\pi$
$103$	5.18315	0.510711	0.255356	+	0.966847i	$0.417807\pi$
$104$	10.8413	1.06308
$105$	0	0
$106$	8.55178	0.830622
$107$	−7.13382	−0.689653	−0.344826	−	0.938666i	$-0.612062\pi$
$107$	−7.13382	−0.689653	−0.344826	+	0.938666i	$0.612062\pi$
$108$	0	0
$109$	13.7939	1.32122	0.660608	−	0.750731i	$-0.270298\pi$
$109$	13.7939	1.32122	0.660608	+	0.750731i	$0.270298\pi$
$110$	−2.40309	−0.229126
$111$	0	0
$112$	3.85002	0.363793
$113$	9.02749	0.849235	0.424617	−	0.905373i	$-0.360409\pi$
$113$	9.02749	0.849235	0.424617	+	0.905373i	$0.360409\pi$
$114$	0	0
$115$	−15.3130	−1.42795
$116$	−9.41956	−0.874584
$117$	0	0
$118$	12.8434	1.18233
$119$	35.6593	3.26888
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.976920	−0.0884462
$123$	0	0
$124$	−7.14822	−0.641929
$125$	−9.71415	−0.868860
$126$	0	0
$127$	−11.5367	−1.02372	−0.511858	−	0.859070i	$-0.671042\pi$
$127$	−11.5367	−1.02372	−0.511858	+	0.859070i	$0.671042\pi$
$128$	−3.79605	−0.335527
$129$	0	0
$130$	−8.75622	−0.767972
$131$	−12.3015	−1.07479	−0.537395	−	0.843331i	$-0.680592\pi$
$131$	−12.3015	−1.07479	−0.537395	+	0.843331i	$0.680592\pi$
$132$	0	0
$133$	−14.7246	−1.27679
$134$	−2.28494	−0.197389
$135$	0	0
$136$	22.4711	1.92688
$137$	18.4305	1.57463	0.787313	−	0.616554i	$-0.211472\pi$
$137$	18.4305	1.57463	0.787313	+	0.616554i	$0.211472\pi$
$138$	0	0
$139$	−4.71002	−0.399498	−0.199749	−	0.979847i	$-0.564013\pi$
$139$	−4.71002	−0.399498	−0.199749	+	0.979847i	$0.564013\pi$
$140$	12.1443	1.02638
$141$	0	0
$142$	3.12233	0.262020
$143$	3.64373	0.304704
$144$	0	0
$145$	22.1598	1.84027
$146$	6.31184	0.522372
$147$	0	0
$148$	−1.44268	−0.118587
$149$	3.70127	0.303220	0.151610	−	0.988440i	$-0.451554\pi$
$149$	3.70127	0.303220	0.151610	+	0.988440i	$0.451554\pi$
$150$	0	0
$151$	0.680484	0.0553770	0.0276885	−	0.999617i	$-0.491185\pi$
$151$	0.680484	0.0553770	0.0276885	+	0.999617i	$0.491185\pi$
$152$	−9.27886	−0.752615
$153$	0	0
$154$	4.61259	0.371693
$155$	16.8164	1.35073
$156$	0	0
$157$	−15.1246	−1.20707	−0.603536	−	0.797336i	$-0.706242\pi$
$157$	−15.1246	−1.20707	−0.603536	+	0.797336i	$0.706242\pi$
$158$	7.31126	0.581652
$159$	0	0
$160$	12.6783	1.00231
$161$	29.3924	2.31645
$162$	0	0
$163$	−1.68174	−0.131724	−0.0658621	−	0.997829i	$-0.520980\pi$
$163$	−1.68174	−0.131724	−0.0658621	+	0.997829i	$0.520980\pi$
$164$	7.33531	0.572791
$165$	0	0
$166$	−7.25608	−0.563181
$167$	8.21081	0.635372	0.317686	−	0.948196i	$-0.397094\pi$
$167$	8.21081	0.635372	0.317686	+	0.948196i	$0.397094\pi$
$168$	0	0
$169$	0.276763	0.0212895
$170$	−18.1492	−1.39198
$171$	0	0
$172$	2.14786	0.163773
$173$	10.1127	0.768853	0.384427	−	0.923156i	$-0.374399\pi$
$173$	10.1127	0.768853	0.384427	+	0.923156i	$0.374399\pi$
$174$	0	0
$175$	−4.96208	−0.375098
$176$	0.815413	0.0614641
$177$	0	0
$178$	−2.56636	−0.192357
$179$	−9.88089	−0.738532	−0.369266	−	0.929324i	$-0.620391\pi$
$179$	−9.88089	−0.738532	−0.369266	+	0.929324i	$0.620391\pi$
$180$	0	0
$181$	14.1305	1.05031	0.525157	−	0.851005i	$-0.324007\pi$
$181$	14.1305	1.05031	0.525157	+	0.851005i	$0.324007\pi$
$182$	16.8070	1.24582
$183$	0	0
$184$	18.5219	1.36545
$185$	3.39394	0.249527
$186$	0	0
$187$	7.55245	0.552290
$188$	0.483211	0.0352418
$189$	0	0
$190$	7.49427	0.543691
$191$	−14.6519	−1.06017	−0.530086	−	0.847944i	$-0.677840\pi$
$191$	−14.6519	−1.06017	−0.530086	+	0.847944i	$0.677840\pi$
$192$	0	0
$193$	22.1162	1.59196	0.795979	−	0.605324i	$-0.206957\pi$
$193$	22.1162	1.59196	0.795979	+	0.605324i	$0.206957\pi$
$194$	13.6939	0.983168
$195$	0	0
$196$	−15.9909	−1.14220
$197$	−12.5323	−0.892887	−0.446444	−	0.894812i	$-0.647310\pi$
$197$	−12.5323	−0.892887	−0.446444	+	0.894812i	$0.647310\pi$
$198$	0	0
$199$	−14.6887	−1.04126	−0.520628	−	0.853783i	$-0.674302\pi$
$199$	−14.6887	−1.04126	−0.520628	+	0.853783i	$0.674302\pi$
$200$	−3.12690	−0.221106
$201$	0	0
$202$	−13.7258	−0.965747
$203$	−42.5343	−2.98532
$204$	0	0
$205$	−17.2565	−1.20525
$206$	5.06353	0.352793
$207$	0	0
$208$	2.97114	0.206012
$209$	−3.11859	−0.215718
$210$	0	0
$211$	13.4080	0.923047	0.461524	−	0.887128i	$-0.347303\pi$
$211$	13.4080	0.923047	0.461524	+	0.887128i	$0.347303\pi$
$212$	−9.15322	−0.628646
$213$	0	0
$214$	−6.96918	−0.476403
$215$	−5.05290	−0.344605
$216$	0	0
$217$	−32.2780	−2.19117
$218$	13.4756	0.912679
$219$	0	0
$220$	2.57210	0.173411
$221$	27.5191	1.85113
$222$	0	0
$223$	−3.05622	−0.204660	−0.102330	−	0.994751i	$-0.532630\pi$
$223$	−3.05622	−0.204660	−0.102330	+	0.994751i	$0.532630\pi$
$224$	−24.3353	−1.62597
$225$	0	0
$226$	8.81914	0.586640
$227$	9.78960	0.649759	0.324879	−	0.945755i	$-0.394676\pi$
$227$	9.78960	0.649759	0.324879	+	0.945755i	$0.394676\pi$
$228$	0	0
$229$	−7.49245	−0.495115	−0.247557	−	0.968873i	$-0.579628\pi$
$229$	−7.49245	−0.495115	−0.247557	+	0.968873i	$0.579628\pi$
$230$	−14.9596	−0.986408
$231$	0	0
$232$	−26.8034	−1.75973
$233$	−15.2128	−0.996626	−0.498313	−	0.866997i	$-0.666047\pi$
$233$	−15.2128	−0.996626	−0.498313	+	0.866997i	$0.666047\pi$
$234$	0	0
$235$	−1.13677	−0.0741545
$236$	−13.7467	−0.894832
$237$	0	0
$238$	34.8363	2.25810
$239$	−28.5870	−1.84914	−0.924569	−	0.381015i	$-0.875575\pi$
$239$	−28.5870	−1.84914	−0.924569	+	0.381015i	$0.875575\pi$
$240$	0	0
$241$	−1.87641	−0.120870	−0.0604350	−	0.998172i	$-0.519249\pi$
$241$	−1.87641	−0.120870	−0.0604350	+	0.998172i	$0.519249\pi$
$242$	0.976920	0.0627988
$243$	0	0
$244$	1.04563	0.0669394
$245$	37.6190	2.40339
$246$	0	0
$247$	−11.3633	−0.723030
$248$	−20.3403	−1.29161
$249$	0	0
$250$	−9.48996	−0.600198
$251$	13.8777	0.875956	0.437978	−	0.898986i	$-0.355695\pi$
$251$	13.8777	0.875956	0.437978	+	0.898986i	$0.355695\pi$
$252$	0	0
$253$	6.22515	0.391372
$254$	−11.2704	−0.707169
$255$	0	0
$256$	−17.0403	−1.06502
$257$	−8.83137	−0.550886	−0.275443	−	0.961317i	$-0.588825\pi$
$257$	−8.83137	−0.550886	−0.275443	+	0.961317i	$0.588825\pi$
$258$	0	0
$259$	−6.51445	−0.404788
$260$	9.37204	0.581229
$261$	0	0
$262$	−12.0176	−0.742451
$263$	27.0329	1.66692	0.833461	−	0.552578i	$-0.186356\pi$
$263$	27.0329	1.66692	0.833461	+	0.552578i	$0.186356\pi$
$264$	0	0
$265$	21.5332	1.32277
$266$	−14.3848	−0.881987
$267$	0	0
$268$	2.44564	0.149391
$269$	−3.25490	−0.198455	−0.0992273	−	0.995065i	$-0.531637\pi$
$269$	−3.25490	−0.198455	−0.0992273	+	0.995065i	$0.531637\pi$
$270$	0	0
$271$	1.51697	0.0921492	0.0460746	−	0.998938i	$-0.485329\pi$
$271$	1.51697	0.0921492	0.0460746	+	0.998938i	$0.485329\pi$
$272$	6.15836	0.373406
$273$	0	0
$274$	18.0052	1.08773
$275$	−1.05094	−0.0633742
$276$	0	0
$277$	32.1804	1.93353	0.966766	−	0.255663i	$-0.0822937\pi$
$277$	32.1804	1.93353	0.966766	+	0.255663i	$0.0822937\pi$
$278$	−4.60131	−0.275968
$279$	0	0
$280$	34.5567	2.06516
$281$	−16.1157	−0.961380	−0.480690	−	0.876890i	$-0.659614\pi$
$281$	−16.1157	−0.961380	−0.480690	+	0.876890i	$0.659614\pi$
$282$	0	0
$283$	−4.12258	−0.245062	−0.122531	−	0.992465i	$-0.539101\pi$
$283$	−4.12258	−0.245062	−0.122531	+	0.992465i	$0.539101\pi$
$284$	−3.34192	−0.198306
$285$	0	0
$286$	3.55963	0.210486
$287$	33.1228	1.95518
$288$	0	0
$289$	40.0395	2.35526
$290$	21.6483	1.27123
$291$	0	0
$292$	−6.75574	−0.395350
$293$	−11.6904	−0.682961	−0.341480	−	0.939889i	$-0.610928\pi$
$293$	−11.6904	−0.682961	−0.341480	+	0.939889i	$0.610928\pi$
$294$	0	0
$295$	32.3394	1.88288
$296$	−4.10514	−0.238607
$297$	0	0
$298$	3.61585	0.209460
$299$	22.6828	1.31178
$300$	0	0
$301$	9.69872	0.559025
$302$	0.664779	0.0382537
$303$	0	0
$304$	−2.54294	−0.145848
$305$	−2.45987	−0.140852
$306$	0	0
$307$	−2.76858	−0.158011	−0.0790055	−	0.996874i	$-0.525174\pi$
$307$	−2.76858	−0.158011	−0.0790055	+	0.996874i	$0.525174\pi$
$308$	−4.93698	−0.281311
$309$	0	0
$310$	16.4283	0.933064
$311$	10.5903	0.600518	0.300259	−	0.953858i	$-0.402927\pi$
$311$	10.5903	0.600518	0.300259	+	0.953858i	$0.402927\pi$
$312$	0	0
$313$	−7.33658	−0.414688	−0.207344	−	0.978268i	$-0.566482\pi$
$313$	−7.33658	−0.414688	−0.207344	+	0.978268i	$0.566482\pi$
$314$	−14.7755	−0.833829
$315$	0	0
$316$	−7.82545	−0.440216
$317$	22.2305	1.24859	0.624296	−	0.781188i	$-0.285386\pi$
$317$	22.2305	1.24859	0.624296	+	0.781188i	$0.285386\pi$
$318$	0	0
$319$	−9.00853	−0.504381
$320$	16.3973	0.916639
$321$	0	0
$322$	28.7140	1.60017
$323$	−23.5530	−1.31052
$324$	0	0
$325$	−3.82935	−0.212414
$326$	−1.64293	−0.0909934
$327$	0	0
$328$	20.8727	1.15250
$329$	2.18195	0.120295
$330$	0	0
$331$	26.5447	1.45903	0.729515	−	0.683965i	$-0.239746\pi$
$331$	26.5447	1.45903	0.729515	+	0.683965i	$0.239746\pi$
$332$	7.76639	0.426236
$333$	0	0
$334$	8.02131	0.438907
$335$	−5.75343	−0.314343
$336$	0	0
$337$	20.7624	1.13100	0.565501	−	0.824748i	$-0.308683\pi$
$337$	20.7624	1.13100	0.565501	+	0.824748i	$0.308683\pi$
$338$	0.270376	0.0147065
$339$	0	0
$340$	19.4257	1.05350
$341$	−6.83631	−0.370207
$342$	0	0
$343$	−39.1563	−2.11424
$344$	6.11174	0.329523
$345$	0	0
$346$	9.87929	0.531114
$347$	13.6102	0.730632	0.365316	−	0.930884i	$-0.380961\pi$
$347$	13.6102	0.730632	0.365316	+	0.930884i	$0.380961\pi$
$348$	0	0
$349$	13.0273	0.697334	0.348667	−	0.937247i	$-0.386634\pi$
$349$	13.0273	0.697334	0.348667	+	0.937247i	$0.386634\pi$
$350$	−4.84756	−0.259113
$351$	0	0
$352$	−5.15408	−0.274713
$353$	−26.5517	−1.41320	−0.706602	−	0.707611i	$-0.749773\pi$
$353$	−26.5517	−1.41320	−0.706602	+	0.707611i	$0.749773\pi$
$354$	0	0
$355$	7.86196	0.417269
$356$	2.74685	0.145583
$357$	0	0
$358$	−9.65284	−0.510169
$359$	−34.9557	−1.84489	−0.922445	−	0.386128i	$-0.873812\pi$
$359$	−34.9557	−1.84489	−0.922445	+	0.386128i	$0.873812\pi$
$360$	0	0
$361$	−9.27438	−0.488125
$362$	13.8044	0.725544
$363$	0	0
$364$	−17.9890	−0.942882
$365$	15.8931	0.831882
$366$	0	0
$367$	−8.20527	−0.428312	−0.214156	−	0.976799i	$-0.568700\pi$
$367$	−8.20527	−0.428312	−0.214156	+	0.976799i	$0.568700\pi$
$368$	5.07607	0.264608
$369$	0	0
$370$	3.31561	0.172370
$371$	−41.3316	−2.14583
$372$	0	0
$373$	33.1545	1.71667	0.858336	−	0.513087i	$-0.171498\pi$
$373$	33.1545	1.71667	0.858336	+	0.513087i	$0.171498\pi$
$374$	7.37814	0.381515
$375$	0	0
$376$	1.37498	0.0709091
$377$	−32.8246	−1.69056
$378$	0	0
$379$	−6.72850	−0.345620	−0.172810	−	0.984955i	$-0.555285\pi$
$379$	−6.72850	−0.345620	−0.172810	+	0.984955i	$0.555285\pi$
$380$	−8.02133	−0.411486
$381$	0	0
$382$	−14.3137	−0.732354
$383$	−7.57720	−0.387177	−0.193589	−	0.981083i	$-0.562013\pi$
$383$	−7.57720	−0.387177	−0.193589	+	0.981083i	$0.562013\pi$
$384$	0	0
$385$	11.6144	0.591925
$386$	21.6058	1.09970
$387$	0	0
$388$	−14.6570	−0.744098
$389$	28.4550	1.44273	0.721364	−	0.692556i	$-0.243516\pi$
$389$	28.4550	1.44273	0.721364	+	0.692556i	$0.243516\pi$
$390$	0	0
$391$	47.0151	2.37766
$392$	−45.5021	−2.29820
$393$	0	0
$394$	−12.2430	−0.616795
$395$	18.4096	0.926287
$396$	0	0
$397$	11.5866	0.581517	0.290758	−	0.956797i	$-0.406092\pi$
$397$	11.5866	0.581517	0.290758	+	0.956797i	$0.406092\pi$
$398$	−14.3497	−0.719287
$399$	0	0
$400$	−0.856952	−0.0428476
$401$	11.0846	0.553537	0.276768	−	0.960937i	$-0.410737\pi$
$401$	11.0846	0.553537	0.276768	+	0.960937i	$0.410737\pi$
$402$	0	0
$403$	−24.9096	−1.24084
$404$	14.6912	0.730913
$405$	0	0
$406$	−41.5526	−2.06222
$407$	−1.37973	−0.0683904
$408$	0	0
$409$	21.5998	1.06804	0.534020	−	0.845472i	$-0.320681\pi$
$409$	21.5998	1.06804	0.534020	+	0.845472i	$0.320681\pi$
$410$	−16.8583	−0.832570
$411$	0	0
$412$	−5.41964	−0.267006
$413$	−62.0735	−3.05444
$414$	0	0
$415$	−18.2707	−0.896871
$416$	−18.7801	−0.920768
$417$	0	0
$418$	−3.04662	−0.149015
$419$	17.4353	0.851772	0.425886	−	0.904777i	$-0.359962\pi$
$419$	17.4353	0.851772	0.425886	+	0.904777i	$0.359962\pi$
$420$	0	0
$421$	4.84909	0.236330	0.118165	−	0.992994i	$-0.462299\pi$
$421$	4.84909	0.236330	0.118165	+	0.992994i	$0.462299\pi$
$422$	13.0986	0.637629
$423$	0	0
$424$	−26.0455	−1.26488
$425$	−7.93719	−0.385010
$426$	0	0
$427$	4.72156	0.228492
$428$	7.45931	0.360560
$429$	0	0
$430$	−4.93628	−0.238049
$431$	25.7815	1.24185	0.620925	−	0.783870i	$-0.286757\pi$
$431$	25.7815	1.24185	0.620925	+	0.783870i	$0.286757\pi$
$432$	0	0
$433$	−0.873678	−0.0419863	−0.0209932	−	0.999780i	$-0.506683\pi$
$433$	−0.873678	−0.0419863	−0.0209932	+	0.999780i	$0.506683\pi$
$434$	−31.5330	−1.51363
$435$	0	0
$436$	−14.4233	−0.690749
$437$	−19.4137	−0.928683
$438$	0	0
$439$	9.44647	0.450855	0.225428	−	0.974260i	$-0.427622\pi$
$439$	9.44647	0.450855	0.225428	+	0.974260i	$0.427622\pi$
$440$	7.31893	0.348916
$441$	0	0
$442$	26.8840	1.27874
$443$	6.19118	0.294152	0.147076	−	0.989125i	$-0.453014\pi$
$443$	6.19118	0.294152	0.147076	+	0.989125i	$0.453014\pi$
$444$	0	0
$445$	−6.46205	−0.306331
$446$	−2.98569	−0.141376
$447$	0	0
$448$	−31.4737	−1.48699
$449$	2.22094	0.104813	0.0524063	−	0.998626i	$-0.483311\pi$
$449$	2.22094	0.104813	0.0524063	+	0.998626i	$0.483311\pi$
$450$	0	0
$451$	7.01523	0.330334
$452$	−9.43938	−0.443991
$453$	0	0
$454$	9.56366	0.448845
$455$	42.3197	1.98398
$456$	0	0
$457$	−2.54427	−0.119016	−0.0595081	−	0.998228i	$-0.518953\pi$
$457$	−2.54427	−0.119016	−0.0595081	+	0.998228i	$0.518953\pi$
$458$	−7.31952	−0.342019
$459$	0	0
$460$	16.0117	0.746550
$461$	23.0730	1.07462	0.537309	−	0.843385i	$-0.319441\pi$
$461$	23.0730	1.07462	0.537309	+	0.843385i	$0.319441\pi$
$462$	0	0
$463$	−24.1373	−1.12176	−0.560879	−	0.827898i	$-0.689537\pi$
$463$	−24.1373	−1.12176	−0.560879	+	0.827898i	$0.689537\pi$
$464$	−7.34567	−0.341014
$465$	0	0
$466$	−14.8617	−0.688456
$467$	−5.59507	−0.258909	−0.129454	−	0.991585i	$-0.541323\pi$
$467$	−5.59507	−0.258909	−0.129454	+	0.991585i	$0.541323\pi$
$468$	0	0
$469$	11.0433	0.509934
$470$	−1.11053	−0.0512250
$471$	0	0
$472$	−39.1162	−1.80047
$473$	2.05414	0.0944493
$474$	0	0
$475$	3.27746	0.150380
$476$	−37.2863	−1.70902
$477$	0	0
$478$	−27.9272	−1.27736
$479$	−27.8217	−1.27121	−0.635604	−	0.772015i	$-0.719249\pi$
$479$	−27.8217	−1.27121	−0.635604	+	0.772015i	$0.719249\pi$
$480$	0	0
$481$	−5.02734	−0.229227
$482$	−1.83310	−0.0834954
$483$	0	0
$484$	−1.04563	−0.0475285
$485$	34.4811	1.56571
$486$	0	0
$487$	28.2998	1.28238	0.641192	−	0.767380i	$-0.278440\pi$
$487$	28.2998	1.28238	0.641192	+	0.767380i	$0.278440\pi$
$488$	2.97533	0.134687
$489$	0	0
$490$	36.7508	1.66023
$491$	3.70645	0.167270	0.0836348	−	0.996496i	$-0.473347\pi$
$491$	3.70645	0.167270	0.0836348	+	0.996496i	$0.473347\pi$
$492$	0	0
$493$	−68.0365	−3.06421
$494$	−11.1010	−0.499460
$495$	0	0
$496$	−5.57441	−0.250298
$497$	−15.0905	−0.676903
$498$	0	0
$499$	12.4246	0.556203	0.278102	−	0.960552i	$-0.410295\pi$
$499$	12.4246	0.556203	0.278102	+	0.960552i	$0.410295\pi$
$500$	10.1574	0.454252
$501$	0	0
$502$	13.5575	0.605099
$503$	39.4627	1.75955	0.879777	−	0.475387i	$-0.157692\pi$
$503$	39.4627	1.75955	0.879777	+	0.475387i	$0.157692\pi$
$504$	0	0
$505$	−34.5614	−1.53796
$506$	6.08148	0.270355
$507$	0	0
$508$	12.0631	0.535212
$509$	−38.9460	−1.72625	−0.863125	−	0.504991i	$-0.831496\pi$
$509$	−38.9460	−1.72625	−0.863125	+	0.504991i	$0.831496\pi$
$510$	0	0
$511$	−30.5058	−1.34950
$512$	−9.05495	−0.400176
$513$	0	0
$514$	−8.62755	−0.380545
$515$	12.7499	0.561826
$516$	0	0
$517$	0.462126	0.0203243
$518$	−6.36410	−0.279623
$519$	0	0
$520$	26.6682	1.16948
$521$	6.44332	0.282287	0.141143	−	0.989989i	$-0.454922\pi$
$521$	6.44332	0.282287	0.141143	+	0.989989i	$0.454922\pi$
$522$	0	0
$523$	−31.6099	−1.38220	−0.691101	−	0.722758i	$-0.742874\pi$
$523$	−31.6099	−1.38220	−0.691101	+	0.722758i	$0.742874\pi$
$524$	12.8628	0.561914
$525$	0	0
$526$	26.4090	1.15149
$527$	−51.6308	−2.24908
$528$	0	0
$529$	15.7525	0.684890
$530$	21.0362	0.913756
$531$	0	0
$532$	15.3964	0.667520
$533$	25.5616	1.10720
$534$	0	0
$535$	−17.5483	−0.758677
$536$	6.95907	0.300586
$537$	0	0
$538$	−3.17977	−0.137090
$539$	−15.2931	−0.658720
$540$	0	0
$541$	−0.190222	−0.00817828	−0.00408914	−	0.999992i	$-0.501302\pi$
$541$	−0.190222	−0.00817828	−0.00408914	+	0.999992i	$0.501302\pi$
$542$	1.48196	0.0636555
$543$	0	0
$544$	−38.9259	−1.66893
$545$	33.9312	1.45345
$546$	0	0
$547$	−11.5701	−0.494702	−0.247351	−	0.968926i	$-0.579560\pi$
$547$	−11.5701	−0.494702	−0.247351	+	0.968926i	$0.579560\pi$
$548$	−19.2714	−0.823235
$549$	0	0
$550$	−1.02669	−0.0437781
$551$	28.0939	1.19684
$552$	0	0
$553$	−35.3361	−1.50264
$554$	31.4377	1.33566
$555$	0	0
$556$	4.92492	0.208863
$557$	−0.363924	−0.0154199	−0.00770997	−	0.999970i	$-0.502454\pi$
$557$	−0.363924	−0.0154199	−0.00770997	+	0.999970i	$0.502454\pi$
$558$	0	0
$559$	7.48472	0.316570
$560$	9.47053	0.400203
$561$	0	0
$562$	−15.7437	−0.664109
$563$	27.7077	1.16774	0.583871	−	0.811847i	$-0.301537\pi$
$563$	27.7077	1.16774	0.583871	+	0.811847i	$0.301537\pi$
$564$	0	0
$565$	22.2064	0.934231
$566$	−4.02743	−0.169286
$567$	0	0
$568$	−9.50944	−0.399007
$569$	−39.0534	−1.63720	−0.818602	−	0.574361i	$-0.805251\pi$
$569$	−39.0534	−1.63720	−0.818602	+	0.574361i	$0.805251\pi$
$570$	0	0
$571$	27.8561	1.16574	0.582871	−	0.812564i	$-0.301929\pi$
$571$	27.8561	1.16574	0.582871	+	0.812564i	$0.301929\pi$
$572$	−3.80998	−0.159303
$573$	0	0
$574$	32.3583	1.35061
$575$	−6.54227	−0.272831
$576$	0	0
$577$	−15.7908	−0.657381	−0.328691	−	0.944438i	$-0.606607\pi$
$577$	−15.7908	−0.657381	−0.328691	+	0.944438i	$0.606607\pi$
$578$	39.1154	1.62699
$579$	0	0
$580$	−23.1708	−0.962117
$581$	35.0694	1.45492
$582$	0	0
$583$	−8.75381	−0.362546
$584$	−19.2235	−0.795474
$585$	0	0
$586$	−11.4206	−0.471780
$587$	32.5907	1.34516	0.672581	−	0.740024i	$-0.265186\pi$
$587$	32.5907	1.34516	0.672581	+	0.740024i	$0.265186\pi$
$588$	0	0
$589$	21.3196	0.878461
$590$	31.5931	1.30067
$591$	0	0
$592$	−1.12505	−0.0462391
$593$	−4.80499	−0.197317	−0.0986586	−	0.995121i	$-0.531455\pi$
$593$	−4.80499	−0.197317	−0.0986586	+	0.995121i	$0.531455\pi$
$594$	0	0
$595$	87.7172	3.59605
$596$	−3.87015	−0.158527
$597$	0	0
$598$	22.1592	0.906159
$599$	3.29562	0.134655	0.0673277	−	0.997731i	$-0.478553\pi$
$599$	3.29562	0.134655	0.0673277	+	0.997731i	$0.478553\pi$
$600$	0	0
$601$	22.3621	0.912170	0.456085	−	0.889936i	$-0.349251\pi$
$601$	22.3621	0.912170	0.456085	+	0.889936i	$0.349251\pi$
$602$	9.47488	0.386167
$603$	0	0
$604$	−0.711532	−0.0289518
$605$	2.45987	0.100008
$606$	0	0
$607$	−20.9391	−0.849893	−0.424947	−	0.905218i	$-0.639707\pi$
$607$	−20.9391	−0.849893	−0.424947	+	0.905218i	$0.639707\pi$
$608$	16.0735	0.651865
$609$	0	0
$610$	−2.40309	−0.0972984
$611$	1.68386	0.0681217
$612$	0	0
$613$	31.0793	1.25528	0.627640	−	0.778504i	$-0.284021\pi$
$613$	31.0793	1.25528	0.627640	+	0.778504i	$0.284021\pi$
$614$	−2.70468	−0.109152
$615$	0	0
$616$	−14.0482	−0.566019
$617$	−27.3936	−1.10282	−0.551412	−	0.834233i	$-0.685911\pi$
$617$	−27.3936	−1.10282	−0.551412	+	0.834233i	$0.685911\pi$
$618$	0	0
$619$	11.3488	0.456145	0.228073	−	0.973644i	$-0.426758\pi$
$619$	11.3488	0.456145	0.228073	+	0.973644i	$0.426758\pi$
$620$	−17.5837	−0.706177
$621$	0	0
$622$	10.3458	0.414830
$623$	12.4035	0.496936
$624$	0	0
$625$	−29.1502	−1.16601
$626$	−7.16726	−0.286461
$627$	0	0
$628$	15.8146	0.631073
$629$	−10.4203	−0.415485
$630$	0	0
$631$	35.5391	1.41479	0.707394	−	0.706819i	$-0.249871\pi$
$631$	35.5391	1.41479	0.707394	+	0.706819i	$0.249871\pi$
$632$	−22.2674	−0.885748
$633$	0	0
$634$	21.7175	0.862511
$635$	−28.3787	−1.12617
$636$	0	0
$637$	−55.7239	−2.20786
$638$	−8.80062	−0.348420
$639$	0	0
$640$	−9.33778	−0.369108
$641$	8.50764	0.336032	0.168016	−	0.985784i	$-0.446264\pi$
$641$	8.50764	0.336032	0.168016	+	0.985784i	$0.446264\pi$
$642$	0	0
$643$	−37.3113	−1.47141	−0.735707	−	0.677300i	$-0.763150\pi$
$643$	−37.3113	−1.47141	−0.735707	+	0.677300i	$0.763150\pi$
$644$	−30.7335	−1.21107
$645$	0	0
$646$	−23.0094	−0.905293
$647$	−7.30155	−0.287053	−0.143527	−	0.989646i	$-0.545844\pi$
$647$	−7.30155	−0.287053	−0.143527	+	0.989646i	$0.545844\pi$
$648$	0	0
$649$	−13.1468	−0.516058
$650$	−3.74097	−0.146733
$651$	0	0
$652$	1.75847	0.0688672
$653$	25.8048	1.00982	0.504910	−	0.863172i	$-0.331526\pi$
$653$	25.8048	1.00982	0.504910	+	0.863172i	$0.331526\pi$
$654$	0	0
$655$	−30.2601	−1.18236
$656$	5.72031	0.223340
$657$	0	0
$658$	2.13159	0.0830982
$659$	7.01770	0.273371	0.136685	−	0.990615i	$-0.456355\pi$
$659$	7.01770	0.273371	0.136685	+	0.990615i	$0.456355\pi$
$660$	0	0
$661$	25.9209	1.00821	0.504103	−	0.863644i	$-0.331823\pi$
$661$	25.9209	1.00821	0.504103	+	0.863644i	$0.331823\pi$
$662$	25.9321	1.00788
$663$	0	0
$664$	22.0993	0.857619
$665$	−36.2206	−1.40457
$666$	0	0
$667$	−56.0794	−2.17140
$668$	−8.58544	−0.332181
$669$	0	0
$670$	−5.62064	−0.217144
$671$	1.00000	0.0386046
$672$	0	0
$673$	−17.3693	−0.669538	−0.334769	−	0.942300i	$-0.608658\pi$
$673$	−17.3693	−0.669538	−0.334769	+	0.942300i	$0.608658\pi$
$674$	20.2833	0.781282
$675$	0	0
$676$	−0.289391	−0.0111304
$677$	27.9109	1.07270	0.536351	−	0.843995i	$-0.319802\pi$
$677$	27.9109	1.07270	0.536351	+	0.843995i	$0.319802\pi$
$678$	0	0
$679$	−66.1843	−2.53992
$680$	55.2758	2.11973
$681$	0	0
$682$	−6.67853	−0.255734
$683$	−29.0389	−1.11114	−0.555572	−	0.831468i	$-0.687501\pi$
$683$	−29.0389	−1.11114	−0.555572	+	0.831468i	$0.687501\pi$
$684$	0	0
$685$	45.3366	1.73222
$686$	−38.2526	−1.46049
$687$	0	0
$688$	1.67497	0.0638576
$689$	−31.8965	−1.21516
$690$	0	0
$691$	35.9848	1.36893	0.684463	−	0.729048i	$-0.260037\pi$
$691$	35.9848	1.36893	0.684463	+	0.729048i	$0.260037\pi$
$692$	−10.5741	−0.401967
$693$	0	0
$694$	13.2960	0.504711
$695$	−11.5860	−0.439482
$696$	0	0
$697$	52.9822	2.00684
$698$	12.7266	0.481709
$699$	0	0
$700$	5.18848	0.196106
$701$	7.79083	0.294255	0.147128	−	0.989117i	$-0.452997\pi$
$701$	7.79083	0.294255	0.147128	+	0.989117i	$0.452997\pi$
$702$	0	0
$703$	4.30280	0.162283
$704$	−6.66595	−0.251232
$705$	0	0
$706$	−25.9389	−0.976223
$707$	66.3384	2.49491
$708$	0	0
$709$	29.8202	1.11992	0.559960	−	0.828520i	$-0.310816\pi$
$709$	29.8202	1.11992	0.559960	+	0.828520i	$0.310816\pi$
$710$	7.68051	0.288244
$711$	0	0
$712$	7.81618	0.292924
$713$	−42.5570	−1.59377
$714$	0	0
$715$	8.96309	0.335200
$716$	10.3317	0.386115
$717$	0	0
$718$	−34.1489	−1.27443
$719$	26.0584	0.971816	0.485908	−	0.874010i	$-0.338489\pi$
$719$	26.0584	0.971816	0.485908	+	0.874010i	$0.338489\pi$
$720$	0	0
$721$	−24.4725	−0.911405
$722$	−9.06034	−0.337191
$723$	0	0
$724$	−14.7753	−0.549118
$725$	9.46744	0.351612
$726$	0	0
$727$	−1.87217	−0.0694349	−0.0347175	−	0.999397i	$-0.511053\pi$
$727$	−1.87217	−0.0694349	−0.0347175	+	0.999397i	$0.511053\pi$
$728$	−51.1879	−1.89715
$729$	0	0
$730$	15.5263	0.574653
$731$	15.5138	0.573797
$732$	0	0
$733$	16.7809	0.619815	0.309908	−	0.950767i	$-0.399702\pi$
$733$	16.7809	0.619815	0.309908	+	0.950767i	$0.399702\pi$
$734$	−8.01590	−0.295872
$735$	0	0
$736$	−32.0849	−1.18266
$737$	2.33892	0.0861552
$738$	0	0
$739$	2.00232	0.0736567	0.0368283	−	0.999322i	$-0.488275\pi$
$739$	2.00232	0.0736567	0.0368283	+	0.999322i	$0.488275\pi$
$740$	−3.54879	−0.130456
$741$	0	0
$742$	−40.3777	−1.48231
$743$	31.3472	1.15002	0.575008	−	0.818148i	$-0.304999\pi$
$743$	31.3472	1.15002	0.575008	+	0.818148i	$0.304999\pi$
$744$	0	0
$745$	9.10463	0.333568
$746$	32.3893	1.18586
$747$	0	0
$748$	−7.89704	−0.288744
$749$	33.6828	1.23074
$750$	0	0
$751$	−0.559148	−0.0204036	−0.0102018	−	0.999948i	$-0.503247\pi$
$751$	−0.559148	−0.0204036	−0.0102018	+	0.999948i	$0.503247\pi$
$752$	0.376823	0.0137413
$753$	0	0
$754$	−32.0671	−1.16781
$755$	1.67390	0.0609195
$756$	0	0
$757$	−2.20187	−0.0800285	−0.0400142	−	0.999199i	$-0.512740\pi$
$757$	−2.20187	−0.0800285	−0.0400142	+	0.999199i	$0.512740\pi$
$758$	−6.57321	−0.238750
$759$	0	0
$760$	−22.8247	−0.827940
$761$	4.55252	0.165029	0.0825143	−	0.996590i	$-0.473705\pi$
$761$	4.55252	0.165029	0.0825143	+	0.996590i	$0.473705\pi$
$762$	0	0
$763$	−65.1287	−2.35782
$764$	15.3204	0.554272
$765$	0	0
$766$	−7.40233	−0.267457
$767$	−47.9035	−1.72969
$768$	0	0
$769$	15.0961	0.544380	0.272190	−	0.962243i	$-0.412252\pi$
$769$	15.0961	0.544380	0.272190	+	0.962243i	$0.412252\pi$
$770$	11.3463	0.408894
$771$	0	0
$772$	−23.1253	−0.832297
$773$	35.0842	1.26189	0.630945	−	0.775828i	$-0.282667\pi$
$773$	35.0842	1.26189	0.630945	+	0.775828i	$0.282667\pi$
$774$	0	0
$775$	7.18456	0.258077
$776$	−41.7066	−1.49718
$777$	0	0
$778$	27.7983	0.996618
$779$	−21.8776	−0.783848
$780$	0	0
$781$	−3.19609	−0.114365
$782$	45.9300	1.64245
$783$	0	0
$784$	−12.4702	−0.445364
$785$	−37.2044	−1.32788
$786$	0	0
$787$	−43.7479	−1.55944	−0.779722	−	0.626125i	$-0.784640\pi$
$787$	−43.7479	−1.55944	−0.779722	+	0.626125i	$0.784640\pi$
$788$	13.1041	0.466813
$789$	0	0
$790$	17.9847	0.639867
$791$	−42.6238	−1.51553
$792$	0	0
$793$	3.64373	0.129393
$794$	11.3192	0.401704
$795$	0	0
$796$	15.3589	0.544383
$797$	4.04978	0.143450	0.0717252	−	0.997424i	$-0.477150\pi$
$797$	4.04978	0.143450	0.0717252	+	0.997424i	$0.477150\pi$
$798$	0	0
$799$	3.49018	0.123474
$800$	5.41663	0.191507
$801$	0	0
$802$	10.8287	0.382376
$803$	−6.46095	−0.228002
$804$	0	0
$805$	72.3014	2.54829
$806$	−24.3347	−0.857155
$807$	0	0
$808$	41.8038	1.47065
$809$	−49.8548	−1.75280	−0.876401	−	0.481582i	$-0.840062\pi$
$809$	−49.8548	−1.75280	−0.876401	+	0.481582i	$0.840062\pi$
$810$	0	0
$811$	−18.2786	−0.641850	−0.320925	−	0.947105i	$-0.603994\pi$
$811$	−18.2786	−0.641850	−0.320925	+	0.947105i	$0.603994\pi$
$812$	44.4750	1.56077
$813$	0	0
$814$	−1.34788	−0.0472432
$815$	−4.13686	−0.144908
$816$	0	0
$817$	−6.40601	−0.224118
$818$	21.1013	0.737788
$819$	0	0
$820$	18.0439	0.630120
$821$	45.7437	1.59646	0.798232	−	0.602350i	$-0.205769\pi$
$821$	45.7437	1.59646	0.798232	+	0.602350i	$0.205769\pi$
$822$	0	0
$823$	−10.5923	−0.369225	−0.184613	−	0.982811i	$-0.559103\pi$
$823$	−10.5923	−0.369225	−0.184613	+	0.982811i	$0.559103\pi$
$824$	−15.4216	−0.537237
$825$	0	0
$826$	−60.6409	−2.10997
$827$	8.50237	0.295656	0.147828	−	0.989013i	$-0.452772\pi$
$827$	8.50237	0.295656	0.147828	+	0.989013i	$0.452772\pi$
$828$	0	0
$829$	41.9221	1.45602	0.728008	−	0.685569i	$-0.240447\pi$
$829$	41.9221	1.45602	0.728008	+	0.685569i	$0.240447\pi$
$830$	−17.8490	−0.619547
$831$	0	0
$832$	−24.2889	−0.842066
$833$	−115.500	−4.00185
$834$	0	0
$835$	20.1975	0.698963
$836$	3.26088	0.112780
$837$	0	0
$838$	17.0329	0.588393
$839$	−49.2667	−1.70087	−0.850437	−	0.526077i	$-0.823662\pi$
$839$	−49.2667	−1.70087	−0.850437	+	0.526077i	$0.823662\pi$
$840$	0	0
$841$	52.1536	1.79840
$842$	4.73718	0.163254
$843$	0	0
$844$	−14.0198	−0.482581
$845$	0.680801	0.0234203
$846$	0	0
$847$	−4.72156	−0.162235
$848$	−7.13797	−0.245119
$849$	0	0
$850$	−7.75400	−0.265960
$851$	−8.58899	−0.294427
$852$	0	0
$853$	−16.1055	−0.551442	−0.275721	−	0.961238i	$-0.588917\pi$
$853$	−16.1055	−0.551442	−0.275721	+	0.961238i	$0.588917\pi$
$854$	4.61259	0.157839
$855$	0	0
$856$	21.2255	0.725473
$857$	54.7150	1.86903	0.934515	−	0.355925i	$-0.115834\pi$
$857$	54.7150	1.86903	0.934515	+	0.355925i	$0.115834\pi$
$858$	0	0
$859$	46.3457	1.58130	0.790648	−	0.612271i	$-0.209744\pi$
$859$	46.3457	1.58130	0.790648	+	0.612271i	$0.209744\pi$
$860$	5.28345	0.180164
$861$	0	0
$862$	25.1865	0.857854
$863$	−23.7776	−0.809398	−0.404699	−	0.914450i	$-0.632624\pi$
$863$	−23.7776	−0.809398	−0.404699	+	0.914450i	$0.632624\pi$
$864$	0	0
$865$	24.8759	0.845805
$866$	−0.853514	−0.0290036
$867$	0	0
$868$	33.7507	1.14557
$869$	−7.48398	−0.253877
$870$	0	0
$871$	8.52239	0.288770
$872$	−41.0415	−1.38984
$873$	0	0
$874$	−18.9656	−0.641522
$875$	45.8659	1.55055
$876$	0	0
$877$	12.3722	0.417781	0.208890	−	0.977939i	$-0.433015\pi$
$877$	12.3722	0.417781	0.208890	+	0.977939i	$0.433015\pi$
$878$	9.22845	0.311445
$879$	0	0
$880$	2.00581	0.0676157
$881$	−48.3684	−1.62957	−0.814786	−	0.579761i	$-0.803146\pi$
$881$	−48.3684	−1.62957	−0.814786	+	0.579761i	$0.803146\pi$
$882$	0	0
$883$	−18.0643	−0.607913	−0.303957	−	0.952686i	$-0.598308\pi$
$883$	−18.0643	−0.607913	−0.303957	+	0.952686i	$0.598308\pi$
$884$	−28.7747	−0.967797
$885$	0	0
$886$	6.04829	0.203196
$887$	−6.81182	−0.228718	−0.114359	−	0.993439i	$-0.536481\pi$
$887$	−6.81182	−0.228718	−0.114359	+	0.993439i	$0.536481\pi$
$888$	0	0
$889$	54.4711	1.82690
$890$	−6.31291	−0.211609
$891$	0	0
$892$	3.19567	0.106999
$893$	−1.44118	−0.0482273
$894$	0	0
$895$	−24.3057	−0.812449
$896$	17.9233	0.598774
$897$	0	0
$898$	2.16968	0.0724031
$899$	61.5851	2.05398
$900$	0	0
$901$	−66.1127	−2.20253
$902$	6.85332	0.228191
$903$	0	0
$904$	−26.8598	−0.893344
$905$	34.7592	1.15544
$906$	0	0
$907$	−18.3398	−0.608963	−0.304481	−	0.952518i	$-0.598483\pi$
$907$	−18.3398	−0.608963	−0.304481	+	0.952518i	$0.598483\pi$
$908$	−10.2363	−0.339702
$909$	0	0
$910$	41.3430	1.37051
$911$	−21.7646	−0.721092	−0.360546	−	0.932741i	$-0.617410\pi$
$911$	−21.7646	−0.721092	−0.360546	+	0.932741i	$0.617410\pi$
$912$	0	0
$913$	7.42750	0.245814
$914$	−2.48555	−0.0822148
$915$	0	0
$916$	7.83430	0.258853
$917$	58.0824	1.91805
$918$	0	0
$919$	−25.4171	−0.838433	−0.419217	−	0.907886i	$-0.637695\pi$
$919$	−25.4171	−0.838433	−0.419217	+	0.907886i	$0.637695\pi$
$920$	45.5614	1.50211
$921$	0	0
$922$	22.5405	0.742333
$923$	−11.6457	−0.383323
$924$	0	0
$925$	1.45001	0.0476761
$926$	−23.5803	−0.774896
$927$	0	0
$928$	46.4307	1.52416
$929$	29.1102	0.955076	0.477538	−	0.878611i	$-0.341529\pi$
$929$	29.1102	0.955076	0.477538	+	0.878611i	$0.341529\pi$
$930$	0	0
$931$	47.6929	1.56307
$932$	15.9069	0.521049
$933$	0	0
$934$	−5.46594	−0.178851
$935$	18.5780	0.607566
$936$	0	0
$937$	42.2703	1.38091	0.690455	−	0.723375i	$-0.257410\pi$
$937$	42.2703	1.38091	0.690455	+	0.723375i	$0.257410\pi$
$938$	10.7885	0.352256
$939$	0	0
$940$	1.18863	0.0387690
$941$	−44.2341	−1.44199	−0.720996	−	0.692940i	$-0.756315\pi$
$941$	−44.2341	−1.44199	−0.720996	+	0.692940i	$0.756315\pi$
$942$	0	0
$943$	43.6708	1.42212
$944$	−10.7201	−0.348909
$945$	0	0
$946$	2.00673	0.0652443
$947$	16.7129	0.543096	0.271548	−	0.962425i	$-0.412464\pi$
$947$	16.7129	0.543096	0.271548	+	0.962425i	$0.412464\pi$
$948$	0	0
$949$	−23.5420	−0.764205
$950$	3.20182	0.103881
$951$	0	0
$952$	−106.098	−3.43867
$953$	−3.86490	−0.125197	−0.0625983	−	0.998039i	$-0.519939\pi$
$953$	−3.86490	−0.125197	−0.0625983	+	0.998039i	$0.519939\pi$
$954$	0	0
$955$	−36.0417	−1.16628
$956$	29.8913	0.966753
$957$	0	0
$958$	−27.1796	−0.878134
$959$	−87.0208	−2.81005
$960$	0	0
$961$	15.7351	0.507583
$962$	−4.91132	−0.158347
$963$	0	0
$964$	1.96202	0.0631924
$965$	54.4029	1.75129
$966$	0	0
$967$	−22.8594	−0.735107	−0.367554	−	0.930002i	$-0.619805\pi$
$967$	−22.8594	−0.735107	−0.367554	+	0.930002i	$0.619805\pi$
$968$	−2.97533	−0.0956309
$969$	0	0
$970$	33.6853	1.08157
$971$	15.7356	0.504980	0.252490	−	0.967600i	$-0.418751\pi$
$971$	15.7356	0.504980	0.252490	+	0.967600i	$0.418751\pi$
$972$	0	0
$973$	22.2386	0.712937
$974$	27.6466	0.885855
$975$	0	0
$976$	0.815413	0.0261007
$977$	−1.70450	−0.0545319	−0.0272660	−	0.999628i	$-0.508680\pi$
$977$	−1.70450	−0.0545319	−0.0272660	+	0.999628i	$0.508680\pi$
$978$	0	0
$979$	2.62699	0.0839591
$980$	−39.3354	−1.25652
$981$	0	0
$982$	3.62090	0.115548
$983$	−52.3102	−1.66844	−0.834218	−	0.551435i	$-0.814081\pi$
$983$	−52.3102	−1.66844	−0.834218	+	0.551435i	$0.814081\pi$
$984$	0	0
$985$	−30.8277	−0.982252
$986$	−66.4662	−2.11672
$987$	0	0
$988$	11.8818	0.378009
$989$	12.7873	0.406613
$990$	0	0
$991$	−11.8436	−0.376224	−0.188112	−	0.982148i	$-0.560237\pi$
$991$	−11.8436	−0.376224	−0.188112	+	0.982148i	$0.560237\pi$
$992$	35.2348	1.11871
$993$	0	0
$994$	−14.7422	−0.467596
$995$	−36.1323	−1.14547
$996$	0	0
$997$	19.9226	0.630954	0.315477	−	0.948933i	$-0.397836\pi$
$997$	19.9226	0.630954	0.315477	+	0.948933i	$0.397836\pi$
$998$	12.1379	0.384218
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.14		19
3.2	odd	2		671.2.a.c.1.6	✓	19
33.32	even	2		7381.2.a.i.1.14		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.6	✓	19	3.2	odd	2
6039.2.a.k.1.14		19	1.1	even	1	trivial
7381.2.a.i.1.14		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)

\(f(q)\)	\(=\)	\(q+0.976920 q^{2} -1.04563 q^{4} +2.45987 q^{5} -4.72156 q^{7} -2.97533 q^{8} +O(q^{10})\)	\(q+0.976920 q^{2} -1.04563 q^{4} +2.45987 q^{5} -4.72156 q^{7} -2.97533 q^{8} +2.40309 q^{10} -1.00000 q^{11} -3.64373 q^{13} -4.61259 q^{14} -0.815413 q^{16} -7.55245 q^{17} +3.11859 q^{19} -2.57210 q^{20} -0.976920 q^{22} -6.22515 q^{23} +1.05094 q^{25} -3.55963 q^{26} +4.93698 q^{28} +9.00853 q^{29} +6.83631 q^{31} +5.15408 q^{32} -7.37814 q^{34} -11.6144 q^{35} +1.37973 q^{37} +3.04662 q^{38} -7.31893 q^{40} -7.01523 q^{41} -2.05414 q^{43} +1.04563 q^{44} -6.08148 q^{46} -0.462126 q^{47} +15.2931 q^{49} +1.02669 q^{50} +3.80998 q^{52} +8.75381 q^{53} -2.45987 q^{55} +14.0482 q^{56} +8.80062 q^{58} +13.1468 q^{59} -1.00000 q^{61} +6.67853 q^{62} +6.66595 q^{64} -8.96309 q^{65} -2.33892 q^{67} +7.89704 q^{68} -11.3463 q^{70} +3.19609 q^{71} +6.46095 q^{73} +1.34788 q^{74} -3.26088 q^{76} +4.72156 q^{77} +7.48398 q^{79} -2.00581 q^{80} -6.85332 q^{82} -7.42750 q^{83} -18.5780 q^{85} -2.00673 q^{86} +2.97533 q^{88} -2.62699 q^{89} +17.2041 q^{91} +6.50918 q^{92} -0.451460 q^{94} +7.67132 q^{95} +14.0175 q^{97} +14.9401 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 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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