LMFDB - Embedded newform 6039.2.a.d.1.11

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 $ x^11 - 4x^10 - 6x^9 + 37x^8 - 2x^7 - 109x^6 + 55x^5 + 115x^4 - 76x^3 - 29x^2 + 14x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.70371$ 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)

$f(q)$	$=$	$q+2.70371 q^{2} +5.31003 q^{4} +2.83529 q^{5} +2.54457 q^{7} +8.94935 q^{8} +O(q^{10})$	\(q+2.70371 q^{2} +5.31003 q^{4} +2.83529 q^{5} +2.54457 q^{7} +8.94935 q^{8} +7.66580 q^{10} +1.00000 q^{11} -3.40952 q^{13} +6.87976 q^{14} +13.5763 q^{16} -3.34895 q^{17} +0.886587 q^{19} +15.0555 q^{20} +2.70371 q^{22} +2.73002 q^{23} +3.03890 q^{25} -9.21835 q^{26} +13.5117 q^{28} -2.08941 q^{29} -10.1317 q^{31} +18.8078 q^{32} -9.05459 q^{34} +7.21459 q^{35} +9.14208 q^{37} +2.39707 q^{38} +25.3740 q^{40} -8.50509 q^{41} -3.26815 q^{43} +5.31003 q^{44} +7.38117 q^{46} +6.27389 q^{47} -0.525187 q^{49} +8.21628 q^{50} -18.1047 q^{52} +2.19652 q^{53} +2.83529 q^{55} +22.7722 q^{56} -5.64914 q^{58} +10.7972 q^{59} +1.00000 q^{61} -27.3931 q^{62} +23.6980 q^{64} -9.66700 q^{65} -2.96631 q^{67} -17.7830 q^{68} +19.5061 q^{70} -12.3804 q^{71} -8.61886 q^{73} +24.7175 q^{74} +4.70780 q^{76} +2.54457 q^{77} -0.270134 q^{79} +38.4930 q^{80} -22.9953 q^{82} +8.98758 q^{83} -9.49527 q^{85} -8.83612 q^{86} +8.94935 q^{88} +12.9951 q^{89} -8.67576 q^{91} +14.4965 q^{92} +16.9628 q^{94} +2.51374 q^{95} -17.3230 q^{97} -1.41995 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) $ 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8	$ 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8} + 6 q^{10} + 11 q^{11} - 3 q^{13} + 9 q^{14} + 4 q^{16} + 7 q^{17} - 8 q^{19} + 25 q^{20} + 4 q^{22} + 15 q^{23} + 4 q^{25} + 2 q^{26} + 13 q^{28} + 8 q^{29} - 17 q^{31} + 27 q^{32} - 18 q^{34} + 2 q^{35} - 10 q^{37} + 30 q^{38} + 10 q^{40} + 25 q^{41} - 7 q^{43} + 6 q^{44} + 32 q^{46} + 30 q^{47} - 2 q^{49} - 11 q^{50} - 7 q^{52} + 18 q^{53} + 13 q^{55} + 20 q^{56} - 13 q^{58} + 43 q^{59} + 11 q^{61} - 7 q^{62} + 25 q^{64} + 27 q^{65} - 30 q^{67} - 10 q^{68} - 4 q^{70} + 7 q^{71} + 6 q^{73} + 44 q^{74} - 19 q^{76} - 5 q^{77} + 17 q^{79} + 22 q^{80} + 8 q^{82} + 34 q^{83} + 10 q^{85} - 2 q^{86} + 9 q^{88} + 41 q^{89} - 39 q^{91} + 32 q^{92} + 55 q^{94} + 9 q^{95} - 41 q^{97} + 29 q^{98}+O(q^{100}) $ 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8 + 6 * q^10 + 11 * q^11 - 3 * q^13 + 9 * q^14 + 4 * q^16 + 7 * q^17 - 8 * q^19 + 25 * q^20 + 4 * q^22 + 15 * q^23 + 4 * q^25 + 2 * q^26 + 13 * q^28 + 8 * q^29 - 17 * q^31 + 27 * q^32 - 18 * q^34 + 2 * q^35 - 10 * q^37 + 30 * q^38 + 10 * q^40 + 25 * q^41 - 7 * q^43 + 6 * q^44 + 32 * q^46 + 30 * q^47 - 2 * q^49 - 11 * q^50 - 7 * q^52 + 18 * q^53 + 13 * q^55 + 20 * q^56 - 13 * q^58 + 43 * q^59 + 11 * q^61 - 7 * q^62 + 25 * q^64 + 27 * q^65 - 30 * q^67 - 10 * q^68 - 4 * q^70 + 7 * q^71 + 6 * q^73 + 44 * q^74 - 19 * q^76 - 5 * q^77 + 17 * q^79 + 22 * q^80 + 8 * q^82 + 34 * q^83 + 10 * q^85 - 2 * q^86 + 9 * q^88 + 41 * q^89 - 39 * q^91 + 32 * q^92 + 55 * q^94 + 9 * q^95 - 41 * q^97 + 29 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.70371	1.91181	0.955905	−	0.293677i	$-0.0948791\pi$
$2$	2.70371	1.91181	0.955905	+	0.293677i	$0.0948791\pi$
$3$	0	0
$3$	0	0
$4$	5.31003	2.65501
$5$	2.83529	1.26798	0.633991	−	0.773340i	$-0.281416\pi$
$5$	2.83529	1.26798	0.633991	+	0.773340i	$0.281416\pi$
$6$	0	0
$7$	2.54457	0.961755	0.480878	−	0.876788i	$-0.340318\pi$
$7$	2.54457	0.961755	0.480878	+	0.876788i	$0.340318\pi$
$8$	8.94935	3.16407
$9$	0	0
$10$	7.66580	2.42414
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.40952	−0.945632	−0.472816	−	0.881161i	$-0.656762\pi$
$13$	−3.40952	−0.945632	−0.472816	+	0.881161i	$0.656762\pi$
$14$	6.87976	1.83869
$15$	0	0
$16$	13.5763	3.39409
$17$	−3.34895	−0.812240	−0.406120	−	0.913820i	$-0.633119\pi$
$17$	−3.34895	−0.812240	−0.406120	+	0.913820i	$0.633119\pi$
$18$	0	0
$19$	0.886587	0.203397	0.101699	−	0.994815i	$-0.467572\pi$
$19$	0.886587	0.203397	0.101699	+	0.994815i	$0.467572\pi$
$20$	15.0555	3.36651
$21$	0	0
$22$	2.70371	0.576432
$23$	2.73002	0.569248	0.284624	−	0.958639i	$-0.408131\pi$
$23$	2.73002	0.569248	0.284624	+	0.958639i	$0.408131\pi$
$24$	0	0
$25$	3.03890	0.607779
$26$	−9.21835	−1.80787
$27$	0	0
$28$	13.5117	2.55347
$29$	−2.08941	−0.387993	−0.193997	−	0.981002i	$-0.562145\pi$
$29$	−2.08941	−0.387993	−0.193997	+	0.981002i	$0.562145\pi$
$30$	0	0
$31$	−10.1317	−1.81970	−0.909851	−	0.414935i	$-0.863804\pi$
$31$	−10.1317	−1.81970	−0.909851	+	0.414935i	$0.863804\pi$
$32$	18.8078	3.32478
$33$	0	0
$34$	−9.05459	−1.55285
$35$	7.21459	1.21949
$36$	0	0
$37$	9.14208	1.50295	0.751475	−	0.659762i	$-0.229343\pi$
$37$	9.14208	1.50295	0.751475	+	0.659762i	$0.229343\pi$
$38$	2.39707	0.388856
$39$	0	0
$40$	25.3740	4.01199
$41$	−8.50509	−1.32827	−0.664136	−	0.747612i	$-0.731200\pi$
$41$	−8.50509	−1.32827	−0.664136	+	0.747612i	$0.731200\pi$
$42$	0	0
$43$	−3.26815	−0.498388	−0.249194	−	0.968454i	$-0.580166\pi$
$43$	−3.26815	−0.498388	−0.249194	+	0.968454i	$0.580166\pi$
$44$	5.31003	0.800517
$45$	0	0
$46$	7.38117	1.08829
$47$	6.27389	0.915141	0.457571	−	0.889173i	$-0.348720\pi$
$47$	6.27389	0.915141	0.457571	+	0.889173i	$0.348720\pi$
$48$	0	0
$49$	−0.525187	−0.0750267
$50$	8.21628	1.16196
$51$	0	0
$52$	−18.1047	−2.51067
$53$	2.19652	0.301715	0.150857	−	0.988556i	$-0.451797\pi$
$53$	2.19652	0.301715	0.150857	+	0.988556i	$0.451797\pi$
$54$	0	0
$55$	2.83529	0.382311
$56$	22.7722	3.04306
$57$	0	0
$58$	−5.64914	−0.741769
$59$	10.7972	1.40568	0.702840	−	0.711348i	$-0.251915\pi$
$59$	10.7972	1.40568	0.702840	+	0.711348i	$0.251915\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−27.3931	−3.47892
$63$	0	0
$64$	23.6980	2.96225
$65$	−9.66700	−1.19904
$66$	0	0
$67$	−2.96631	−0.362392	−0.181196	−	0.983447i	$-0.557997\pi$
$67$	−2.96631	−0.362392	−0.181196	+	0.983447i	$0.557997\pi$
$68$	−17.7830	−2.15651
$69$	0	0
$70$	19.5061	2.33143
$71$	−12.3804	−1.46929	−0.734644	−	0.678453i	$-0.762651\pi$
$71$	−12.3804	−1.46929	−0.734644	+	0.678453i	$0.762651\pi$
$72$	0	0
$73$	−8.61886	−1.00876	−0.504381	−	0.863481i	$-0.668279\pi$
$73$	−8.61886	−1.00876	−0.504381	+	0.863481i	$0.668279\pi$
$74$	24.7175	2.87335
$75$	0	0
$76$	4.70780	0.540022
$77$	2.54457	0.289980
$78$	0	0
$79$	−0.270134	−0.0303925	−0.0151962	−	0.999885i	$-0.504837\pi$
$79$	−0.270134	−0.0303925	−0.0151962	+	0.999885i	$0.504837\pi$
$80$	38.4930	4.30364
$81$	0	0
$82$	−22.9953	−2.53940
$83$	8.98758	0.986515	0.493258	−	0.869883i	$-0.335806\pi$
$83$	8.98758	0.986515	0.493258	+	0.869883i	$0.335806\pi$
$84$	0	0
$85$	−9.49527	−1.02991
$86$	−8.83612	−0.952823
$87$	0	0
$88$	8.94935	0.954004
$89$	12.9951	1.37748	0.688739	−	0.725009i	$-0.258165\pi$
$89$	12.9951	1.37748	0.688739	+	0.725009i	$0.258165\pi$
$90$	0	0
$91$	−8.67576	−0.909466
$92$	14.4965	1.51136
$93$	0	0
$94$	16.9628	1.74958
$95$	2.51374	0.257904
$96$	0	0
$97$	−17.3230	−1.75888	−0.879441	−	0.476007i	$-0.842084\pi$
$97$	−17.3230	−1.75888	−0.879441	+	0.476007i	$0.842084\pi$
$98$	−1.41995	−0.143437
$99$	0	0
$100$	16.1366	1.61366
$101$	−7.54168	−0.750426	−0.375213	−	0.926939i	$-0.622430\pi$
$101$	−7.54168	−0.750426	−0.375213	+	0.926939i	$0.622430\pi$
$102$	0	0
$103$	−8.65239	−0.852546	−0.426273	−	0.904595i	$-0.640174\pi$
$103$	−8.65239	−0.852546	−0.426273	+	0.904595i	$0.640174\pi$
$104$	−30.5130	−2.99205
$105$	0	0
$106$	5.93873	0.576821
$107$	8.79929	0.850659	0.425330	−	0.905039i	$-0.360158\pi$
$107$	8.79929	0.850659	0.425330	+	0.905039i	$0.360158\pi$
$108$	0	0
$109$	16.1673	1.54855	0.774274	−	0.632851i	$-0.218115\pi$
$109$	16.1673	1.54855	0.774274	+	0.632851i	$0.218115\pi$
$110$	7.66580	0.730906
$111$	0	0
$112$	34.5459	3.26428
$113$	−18.2118	−1.71322	−0.856609	−	0.515967i	$-0.827433\pi$
$113$	−18.2118	−1.71322	−0.856609	+	0.515967i	$0.827433\pi$
$114$	0	0
$115$	7.74041	0.721797
$116$	−11.0948	−1.03013
$117$	0	0
$118$	29.1925	2.68739
$119$	−8.52163	−0.781177
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.70371	0.244782
$123$	0	0
$124$	−53.7995	−4.83134
$125$	−5.56031	−0.497329
$126$	0	0
$127$	16.6197	1.47476	0.737382	−	0.675476i	$-0.236062\pi$
$127$	16.6197	1.47476	0.737382	+	0.675476i	$0.236062\pi$
$128$	26.4569	2.33848
$129$	0	0
$130$	−26.1367	−2.29234
$131$	−6.26823	−0.547658	−0.273829	−	0.961778i	$-0.588290\pi$
$131$	−6.26823	−0.547658	−0.273829	+	0.961778i	$0.588290\pi$
$132$	0	0
$133$	2.25598	0.195618
$134$	−8.02002	−0.692824
$135$	0	0
$136$	−29.9709	−2.56999
$137$	−2.23987	−0.191365	−0.0956825	−	0.995412i	$-0.530503\pi$
$137$	−2.23987	−0.191365	−0.0956825	+	0.995412i	$0.530503\pi$
$138$	0	0
$139$	−12.4740	−1.05803	−0.529015	−	0.848613i	$-0.677438\pi$
$139$	−12.4740	−1.05803	−0.529015	+	0.848613i	$0.677438\pi$
$140$	38.3097	3.23776
$141$	0	0
$142$	−33.4731	−2.80900
$143$	−3.40952	−0.285119
$144$	0	0
$145$	−5.92409	−0.491969
$146$	−23.3029	−1.92856
$147$	0	0
$148$	48.5447	3.99035
$149$	6.56833	0.538099	0.269049	−	0.963126i	$-0.413291\pi$
$149$	6.56833	0.538099	0.269049	+	0.963126i	$0.413291\pi$
$150$	0	0
$151$	0.929439	0.0756367	0.0378184	−	0.999285i	$-0.487959\pi$
$151$	0.929439	0.0756367	0.0378184	+	0.999285i	$0.487959\pi$
$152$	7.93438	0.643563
$153$	0	0
$154$	6.87976	0.554387
$155$	−28.7263	−2.30735
$156$	0	0
$157$	−10.7070	−0.854513	−0.427257	−	0.904130i	$-0.640520\pi$
$157$	−10.7070	−0.854513	−0.427257	+	0.904130i	$0.640520\pi$
$158$	−0.730363	−0.0581046
$159$	0	0
$160$	53.3256	4.21576
$161$	6.94671	0.547477
$162$	0	0
$163$	−17.0270	−1.33366	−0.666829	−	0.745211i	$-0.732349\pi$
$163$	−17.0270	−1.33366	−0.666829	+	0.745211i	$0.732349\pi$
$164$	−45.1623	−3.52658
$165$	0	0
$166$	24.2998	1.88603
$167$	3.06679	0.237315	0.118657	−	0.992935i	$-0.462141\pi$
$167$	3.06679	0.237315	0.118657	+	0.992935i	$0.462141\pi$
$168$	0	0
$169$	−1.37515	−0.105781
$170$	−25.6724	−1.96898
$171$	0	0
$172$	−17.3540	−1.32323
$173$	9.81161	0.745963	0.372982	−	0.927839i	$-0.378335\pi$
$173$	9.81161	0.745963	0.372982	+	0.927839i	$0.378335\pi$
$174$	0	0
$175$	7.73267	0.584535
$176$	13.5763	1.02336
$177$	0	0
$178$	35.1349	2.63348
$179$	20.7737	1.55270	0.776349	−	0.630303i	$-0.217070\pi$
$179$	20.7737	1.55270	0.776349	+	0.630303i	$0.217070\pi$
$180$	0	0
$181$	−1.70688	−0.126871	−0.0634356	−	0.997986i	$-0.520206\pi$
$181$	−1.70688	−0.126871	−0.0634356	+	0.997986i	$0.520206\pi$
$182$	−23.4567	−1.73873
$183$	0	0
$184$	24.4319	1.80114
$185$	25.9205	1.90571
$186$	0	0
$187$	−3.34895	−0.244900
$188$	33.3145	2.42971
$189$	0	0
$190$	6.79641	0.493063
$191$	−9.68758	−0.700969	−0.350484	−	0.936569i	$-0.613983\pi$
$191$	−9.68758	−0.700969	−0.350484	+	0.936569i	$0.613983\pi$
$192$	0	0
$193$	−10.6191	−0.764383	−0.382192	−	0.924083i	$-0.624831\pi$
$193$	−10.6191	−0.764383	−0.382192	+	0.924083i	$0.624831\pi$
$194$	−46.8363	−3.36265
$195$	0	0
$196$	−2.78876	−0.199197
$197$	22.5058	1.60347	0.801736	−	0.597678i	$-0.203910\pi$
$197$	22.5058	1.60347	0.801736	+	0.597678i	$0.203910\pi$
$198$	0	0
$199$	3.15756	0.223833	0.111917	−	0.993718i	$-0.464301\pi$
$199$	3.15756	0.223833	0.111917	+	0.993718i	$0.464301\pi$
$200$	27.1961	1.92306
$201$	0	0
$202$	−20.3905	−1.43467
$203$	−5.31663	−0.373155
$204$	0	0
$205$	−24.1144	−1.68423
$206$	−23.3935	−1.62990
$207$	0	0
$208$	−46.2889	−3.20956
$209$	0.886587	0.0613265
$210$	0	0
$211$	19.6842	1.35512	0.677559	−	0.735468i	$-0.263038\pi$
$211$	19.6842	1.35512	0.677559	+	0.735468i	$0.263038\pi$
$212$	11.6636	0.801057
$213$	0	0
$214$	23.7907	1.62630
$215$	−9.26617	−0.631947
$216$	0	0
$217$	−25.7807	−1.75011
$218$	43.7117	2.96053
$219$	0	0
$220$	15.0555	1.01504
$221$	11.4183	0.768080
$222$	0	0
$223$	−16.4852	−1.10393	−0.551966	−	0.833866i	$-0.686122\pi$
$223$	−16.4852	−1.10393	−0.551966	+	0.833866i	$0.686122\pi$
$224$	47.8576	3.19762
$225$	0	0
$226$	−49.2392	−3.27534
$227$	13.2113	0.876865	0.438433	−	0.898764i	$-0.355534\pi$
$227$	13.2113	0.876865	0.438433	+	0.898764i	$0.355534\pi$
$228$	0	0
$229$	−22.3765	−1.47868	−0.739341	−	0.673331i	$-0.764863\pi$
$229$	−22.3765	−1.47868	−0.739341	+	0.673331i	$0.764863\pi$
$230$	20.9278	1.37994
$231$	0	0
$232$	−18.6988	−1.22764
$233$	−4.57514	−0.299727	−0.149864	−	0.988707i	$-0.547883\pi$
$233$	−4.57514	−0.299727	−0.149864	+	0.988707i	$0.547883\pi$
$234$	0	0
$235$	17.7883	1.16038
$236$	57.3336	3.73210
$237$	0	0
$238$	−23.0400	−1.49346
$239$	8.68459	0.561759	0.280880	−	0.959743i	$-0.409374\pi$
$239$	8.68459	0.561759	0.280880	+	0.959743i	$0.409374\pi$
$240$	0	0
$241$	−6.12102	−0.394289	−0.197145	−	0.980374i	$-0.563167\pi$
$241$	−6.12102	−0.394289	−0.197145	+	0.980374i	$0.563167\pi$
$242$	2.70371	0.173801
$243$	0	0
$244$	5.31003	0.339940
$245$	−1.48906	−0.0951325
$246$	0	0
$247$	−3.02284	−0.192339
$248$	−90.6719	−5.75767
$249$	0	0
$250$	−15.0334	−0.950798
$251$	22.4460	1.41678	0.708388	−	0.705823i	$-0.249423\pi$
$251$	22.4460	1.41678	0.708388	+	0.705823i	$0.249423\pi$
$252$	0	0
$253$	2.73002	0.171635
$254$	44.9349	2.81947
$255$	0	0
$256$	24.1357	1.50848
$257$	19.5805	1.22140	0.610700	−	0.791862i	$-0.290888\pi$
$257$	19.5805	1.22140	0.610700	+	0.791862i	$0.290888\pi$
$258$	0	0
$259$	23.2626	1.44547
$260$	−51.3321	−3.18348
$261$	0	0
$262$	−16.9474	−1.04702
$263$	6.01533	0.370921	0.185461	−	0.982652i	$-0.440622\pi$
$263$	6.01533	0.370921	0.185461	+	0.982652i	$0.440622\pi$
$264$	0	0
$265$	6.22777	0.382569
$266$	6.09951	0.373985
$267$	0	0
$268$	−15.7512	−0.962156
$269$	4.88819	0.298038	0.149019	−	0.988834i	$-0.452388\pi$
$269$	4.88819	0.298038	0.149019	+	0.988834i	$0.452388\pi$
$270$	0	0
$271$	17.3066	1.05130	0.525652	−	0.850700i	$-0.323821\pi$
$271$	17.3066	1.05130	0.525652	+	0.850700i	$0.323821\pi$
$272$	−45.4666	−2.75681
$273$	0	0
$274$	−6.05595	−0.365853
$275$	3.03890	0.183252
$276$	0	0
$277$	21.9422	1.31838	0.659189	−	0.751977i	$-0.270900\pi$
$277$	21.9422	1.31838	0.659189	+	0.751977i	$0.270900\pi$
$278$	−33.7260	−2.02275
$279$	0	0
$280$	64.5659	3.85855
$281$	9.66569	0.576607	0.288303	−	0.957539i	$-0.406909\pi$
$281$	9.66569	0.576607	0.288303	+	0.957539i	$0.406909\pi$
$282$	0	0
$283$	−11.4081	−0.678143	−0.339071	−	0.940761i	$-0.610113\pi$
$283$	−11.4081	−0.678143	−0.339071	+	0.940761i	$0.610113\pi$
$284$	−65.7405	−3.90098
$285$	0	0
$286$	−9.21835	−0.545092
$287$	−21.6418	−1.27747
$288$	0	0
$289$	−5.78451	−0.340266
$290$	−16.0170	−0.940550
$291$	0	0
$292$	−45.7664	−2.67828
$293$	22.3557	1.30603	0.653016	−	0.757344i	$-0.273503\pi$
$293$	22.3557	1.30603	0.653016	+	0.757344i	$0.273503\pi$
$294$	0	0
$295$	30.6133	1.78238
$296$	81.8157	4.75544
$297$	0	0
$298$	17.7588	1.02874
$299$	−9.30806	−0.538299
$300$	0	0
$301$	−8.31602	−0.479327
$302$	2.51293	0.144603
$303$	0	0
$304$	12.0366	0.690347
$305$	2.83529	0.162349
$306$	0	0
$307$	−15.3908	−0.878398	−0.439199	−	0.898390i	$-0.644738\pi$
$307$	−15.3908	−0.878398	−0.439199	+	0.898390i	$0.644738\pi$
$308$	13.5117	0.769901
$309$	0	0
$310$	−77.6674	−4.41121
$311$	−27.3603	−1.55146	−0.775730	−	0.631065i	$-0.782618\pi$
$311$	−27.3603	−1.55146	−0.775730	+	0.631065i	$0.782618\pi$
$312$	0	0
$313$	−18.0131	−1.01816	−0.509079	−	0.860720i	$-0.670014\pi$
$313$	−18.0131	−1.01816	−0.509079	+	0.860720i	$0.670014\pi$
$314$	−28.9486	−1.63367
$315$	0	0
$316$	−1.43442	−0.0806924
$317$	−1.82991	−0.102778	−0.0513889	−	0.998679i	$-0.516365\pi$
$317$	−1.82991	−0.102778	−0.0513889	+	0.998679i	$0.516365\pi$
$318$	0	0
$319$	−2.08941	−0.116984
$320$	67.1908	3.75608
$321$	0	0
$322$	18.7819	1.04667
$323$	−2.96914	−0.165207
$324$	0	0
$325$	−10.3612	−0.574735
$326$	−46.0360	−2.54970
$327$	0	0
$328$	−76.1150	−4.20275
$329$	15.9643	0.880142
$330$	0	0
$331$	−26.7451	−1.47005	−0.735023	−	0.678042i	$-0.762829\pi$
$331$	−26.7451	−1.47005	−0.735023	+	0.678042i	$0.762829\pi$
$332$	47.7243	2.61921
$333$	0	0
$334$	8.29169	0.453701
$335$	−8.41035	−0.459507
$336$	0	0
$337$	−3.97635	−0.216605	−0.108303	−	0.994118i	$-0.534542\pi$
$337$	−3.97635	−0.216605	−0.108303	+	0.994118i	$0.534542\pi$
$338$	−3.71800	−0.202233
$339$	0	0
$340$	−50.4202	−2.73442
$341$	−10.1317	−0.548661
$342$	0	0
$343$	−19.1483	−1.03391
$344$	−29.2478	−1.57694
$345$	0	0
$346$	26.5277	1.42614
$347$	30.2180	1.62219	0.811094	−	0.584915i	$-0.198872\pi$
$347$	30.2180	1.62219	0.811094	+	0.584915i	$0.198872\pi$
$348$	0	0
$349$	34.8769	1.86692	0.933458	−	0.358687i	$-0.116775\pi$
$349$	34.8769	1.86692	0.933458	+	0.358687i	$0.116775\pi$
$350$	20.9069	1.11752
$351$	0	0
$352$	18.8078	1.00246
$353$	−5.82738	−0.310160	−0.155080	−	0.987902i	$-0.549564\pi$
$353$	−5.82738	−0.310160	−0.155080	+	0.987902i	$0.549564\pi$
$354$	0	0
$355$	−35.1022	−1.86303
$356$	69.0044	3.65722
$357$	0	0
$358$	56.1659	2.96846
$359$	−12.6523	−0.667765	−0.333882	−	0.942615i	$-0.608359\pi$
$359$	−12.6523	−0.667765	−0.333882	+	0.942615i	$0.608359\pi$
$360$	0	0
$361$	−18.2140	−0.958630
$362$	−4.61490	−0.242553
$363$	0	0
$364$	−46.0685	−2.41465
$365$	−24.4370	−1.27909
$366$	0	0
$367$	−4.96733	−0.259293	−0.129646	−	0.991560i	$-0.541384\pi$
$367$	−4.96733	−0.259293	−0.129646	+	0.991560i	$0.541384\pi$
$368$	37.0637	1.93208
$369$	0	0
$370$	70.0814	3.64336
$371$	5.58918	0.290176
$372$	0	0
$373$	17.1878	0.889951	0.444975	−	0.895543i	$-0.353212\pi$
$373$	17.1878	0.889951	0.444975	+	0.895543i	$0.353212\pi$
$374$	−9.05459	−0.468201
$375$	0	0
$376$	56.1472	2.89557
$377$	7.12388	0.366899
$378$	0	0
$379$	7.87717	0.404623	0.202312	−	0.979321i	$-0.435155\pi$
$379$	7.87717	0.404623	0.202312	+	0.979321i	$0.435155\pi$
$380$	13.3480	0.684739
$381$	0	0
$382$	−26.1924	−1.34012
$383$	19.1746	0.979777	0.489888	−	0.871785i	$-0.337037\pi$
$383$	19.1746	0.979777	0.489888	+	0.871785i	$0.337037\pi$
$384$	0	0
$385$	7.21459	0.367690
$386$	−28.7111	−1.46135
$387$	0	0
$388$	−91.9856	−4.66986
$389$	2.99467	0.151836	0.0759178	−	0.997114i	$-0.475811\pi$
$389$	2.99467	0.151836	0.0759178	+	0.997114i	$0.475811\pi$
$390$	0	0
$391$	−9.14270	−0.462366
$392$	−4.70008	−0.237390
$393$	0	0
$394$	60.8491	3.06553
$395$	−0.765910	−0.0385371
$396$	0	0
$397$	30.6079	1.53617	0.768084	−	0.640349i	$-0.221210\pi$
$397$	30.6079	1.53617	0.768084	+	0.640349i	$0.221210\pi$
$398$	8.53710	0.427926
$399$	0	0
$400$	41.2571	2.06286
$401$	−27.6816	−1.38235	−0.691176	−	0.722687i	$-0.742907\pi$
$401$	−27.6816	−1.38235	−0.691176	+	0.722687i	$0.742907\pi$
$402$	0	0
$403$	34.5442	1.72077
$404$	−40.0466	−1.99239
$405$	0	0
$406$	−14.3746	−0.713400
$407$	9.14208	0.453156
$408$	0	0
$409$	7.62016	0.376793	0.188396	−	0.982093i	$-0.439671\pi$
$409$	7.62016	0.376793	0.188396	+	0.982093i	$0.439671\pi$
$410$	−65.1983	−3.21992
$411$	0	0
$412$	−45.9445	−2.26352
$413$	27.4743	1.35192
$414$	0	0
$415$	25.4824	1.25088
$416$	−64.1255	−3.14401
$417$	0	0
$418$	2.39707	0.117245
$419$	−34.0267	−1.66232	−0.831158	−	0.556037i	$-0.812321\pi$
$419$	−34.0267	−1.66232	−0.831158	+	0.556037i	$0.812321\pi$
$420$	0	0
$421$	−2.78226	−0.135599	−0.0677996	−	0.997699i	$-0.521598\pi$
$421$	−2.78226	−0.135599	−0.0677996	+	0.997699i	$0.521598\pi$
$422$	53.2204	2.59073
$423$	0	0
$424$	19.6574	0.954647
$425$	−10.1771	−0.493663
$426$	0	0
$427$	2.54457	0.123140
$428$	46.7245	2.25851
$429$	0	0
$430$	−25.0530	−1.20816
$431$	−11.8226	−0.569472	−0.284736	−	0.958606i	$-0.591906\pi$
$431$	−11.8226	−0.569472	−0.284736	+	0.958606i	$0.591906\pi$
$432$	0	0
$433$	29.1839	1.40249	0.701244	−	0.712921i	$-0.252628\pi$
$433$	29.1839	1.40249	0.701244	+	0.712921i	$0.252628\pi$
$434$	−69.7035	−3.34587
$435$	0	0
$436$	85.8489	4.11142
$437$	2.42040	0.115783
$438$	0	0
$439$	4.71665	0.225114	0.112557	−	0.993645i	$-0.464096\pi$
$439$	4.71665	0.225114	0.112557	+	0.993645i	$0.464096\pi$
$440$	25.3740	1.20966
$441$	0	0
$442$	30.8718	1.46842
$443$	−1.15384	−0.0548206	−0.0274103	−	0.999624i	$-0.508726\pi$
$443$	−1.15384	−0.0548206	−0.0274103	+	0.999624i	$0.508726\pi$
$444$	0	0
$445$	36.8449	1.74662
$446$	−44.5712	−2.11051
$447$	0	0
$448$	60.3011	2.84896
$449$	20.6524	0.974649	0.487325	−	0.873221i	$-0.337973\pi$
$449$	20.6524	0.974649	0.487325	+	0.873221i	$0.337973\pi$
$450$	0	0
$451$	−8.50509	−0.400489
$452$	−96.7049	−4.54862
$453$	0	0
$454$	35.7195	1.67640
$455$	−24.5983	−1.15319
$456$	0	0
$457$	20.1264	0.941472	0.470736	−	0.882274i	$-0.343988\pi$
$457$	20.1264	0.941472	0.470736	+	0.882274i	$0.343988\pi$
$458$	−60.4995	−2.82696
$459$	0	0
$460$	41.1018	1.91638
$461$	−11.9937	−0.558602	−0.279301	−	0.960204i	$-0.590103\pi$
$461$	−11.9937	−0.558602	−0.279301	+	0.960204i	$0.590103\pi$
$462$	0	0
$463$	−3.31841	−0.154220	−0.0771099	−	0.997023i	$-0.524569\pi$
$463$	−3.31841	−0.154220	−0.0771099	+	0.997023i	$0.524569\pi$
$464$	−28.3665	−1.31688
$465$	0	0
$466$	−12.3698	−0.573021
$467$	22.9014	1.05975	0.529875	−	0.848076i	$-0.322239\pi$
$467$	22.9014	1.05975	0.529875	+	0.848076i	$0.322239\pi$
$468$	0	0
$469$	−7.54796	−0.348532
$470$	48.0944	2.21843
$471$	0	0
$472$	96.6282	4.44767
$473$	−3.26815	−0.150270
$474$	0	0
$475$	2.69425	0.123621
$476$	−45.2501	−2.07404
$477$	0	0
$478$	23.4806	1.07398
$479$	13.7900	0.630082	0.315041	−	0.949078i	$-0.397982\pi$
$479$	13.7900	0.630082	0.315041	+	0.949078i	$0.397982\pi$
$480$	0	0
$481$	−31.1701	−1.42124
$482$	−16.5494	−0.753806
$483$	0	0
$484$	5.31003	0.241365
$485$	−49.1158	−2.23023
$486$	0	0
$487$	21.9929	0.996593	0.498296	−	0.867007i	$-0.333959\pi$
$487$	21.9929	0.996593	0.498296	+	0.867007i	$0.333959\pi$
$488$	8.94935	0.405118
$489$	0	0
$490$	−4.02598	−0.181875
$491$	17.4562	0.787789	0.393894	−	0.919156i	$-0.371128\pi$
$491$	17.4562	0.787789	0.393894	+	0.919156i	$0.371128\pi$
$492$	0	0
$493$	6.99733	0.315144
$494$	−8.17287	−0.367715
$495$	0	0
$496$	−137.551	−6.17623
$497$	−31.5028	−1.41309
$498$	0	0
$499$	−8.07512	−0.361492	−0.180746	−	0.983530i	$-0.557851\pi$
$499$	−8.07512	−0.361492	−0.180746	+	0.983530i	$0.557851\pi$
$500$	−29.5254	−1.32042
$501$	0	0
$502$	60.6873	2.70861
$503$	−17.1287	−0.763733	−0.381867	−	0.924217i	$-0.624719\pi$
$503$	−17.1287	−0.763733	−0.381867	+	0.924217i	$0.624719\pi$
$504$	0	0
$505$	−21.3829	−0.951526
$506$	7.38117	0.328133
$507$	0	0
$508$	88.2513	3.91552
$509$	17.5725	0.778889	0.389445	−	0.921050i	$-0.372667\pi$
$509$	17.5725	0.778889	0.389445	+	0.921050i	$0.372667\pi$
$510$	0	0
$511$	−21.9312	−0.970181
$512$	12.3420	0.545443
$513$	0	0
$514$	52.9400	2.33508
$515$	−24.5321	−1.08101
$516$	0	0
$517$	6.27389	0.275925
$518$	62.8953	2.76346
$519$	0	0
$520$	−86.5134	−3.79386
$521$	−11.3713	−0.498188	−0.249094	−	0.968479i	$-0.580133\pi$
$521$	−11.3713	−0.498188	−0.249094	+	0.968479i	$0.580133\pi$
$522$	0	0
$523$	−20.7630	−0.907904	−0.453952	−	0.891026i	$-0.649986\pi$
$523$	−20.7630	−0.907904	−0.453952	+	0.891026i	$0.649986\pi$
$524$	−33.2845	−1.45404
$525$	0	0
$526$	16.2637	0.709130
$527$	33.9305	1.47804
$528$	0	0
$529$	−15.5470	−0.675957
$530$	16.8381	0.731399
$531$	0	0
$532$	11.9793	0.519369
$533$	28.9983	1.25606
$534$	0	0
$535$	24.9486	1.07862
$536$	−26.5465	−1.14663
$537$	0	0
$538$	13.2162	0.569792
$539$	−0.525187	−0.0226214
$540$	0	0
$541$	−37.7938	−1.62488	−0.812441	−	0.583044i	$-0.801862\pi$
$541$	−37.7938	−1.62488	−0.812441	+	0.583044i	$0.801862\pi$
$542$	46.7921	2.00989
$543$	0	0
$544$	−62.9863	−2.70052
$545$	45.8391	1.96353
$546$	0	0
$547$	40.8728	1.74760	0.873798	−	0.486289i	$-0.161650\pi$
$547$	40.8728	1.74760	0.873798	+	0.486289i	$0.161650\pi$
$548$	−11.8938	−0.508077
$549$	0	0
$550$	8.21628	0.350343
$551$	−1.85244	−0.0789167
$552$	0	0
$553$	−0.687374	−0.0292301
$554$	59.3253	2.52049
$555$	0	0
$556$	−66.2372	−2.80908
$557$	16.4157	0.695557	0.347778	−	0.937577i	$-0.386936\pi$
$557$	16.4157	0.695557	0.347778	+	0.937577i	$0.386936\pi$
$558$	0	0
$559$	11.1428	0.471292
$560$	97.9478	4.13905
$561$	0	0
$562$	26.1332	1.10236
$563$	36.7114	1.54720	0.773601	−	0.633674i	$-0.218454\pi$
$563$	36.7114	1.54720	0.773601	+	0.633674i	$0.218454\pi$
$564$	0	0
$565$	−51.6357	−2.17233
$566$	−30.8442	−1.29648
$567$	0	0
$568$	−110.797	−4.64893
$569$	−26.4495	−1.10882	−0.554411	−	0.832243i	$-0.687056\pi$
$569$	−26.4495	−1.10882	−0.554411	+	0.832243i	$0.687056\pi$
$570$	0	0
$571$	23.1420	0.968462	0.484231	−	0.874940i	$-0.339099\pi$
$571$	23.1420	0.968462	0.484231	+	0.874940i	$0.339099\pi$
$572$	−18.1047	−0.756994
$573$	0	0
$574$	−58.5130	−2.44228
$575$	8.29624	0.345977
$576$	0	0
$577$	38.5517	1.60493	0.802464	−	0.596700i	$-0.203522\pi$
$577$	38.5517	1.60493	0.802464	+	0.596700i	$0.203522\pi$
$578$	−15.6396	−0.650523
$579$	0	0
$580$	−31.4571	−1.30618
$581$	22.8695	0.948786
$582$	0	0
$583$	2.19652	0.0909704
$584$	−77.1332	−3.19179
$585$	0	0
$586$	60.4432	2.49688
$587$	2.15784	0.0890637	0.0445318	−	0.999008i	$-0.485820\pi$
$587$	2.15784	0.0890637	0.0445318	+	0.999008i	$0.485820\pi$
$588$	0	0
$589$	−8.98261	−0.370122
$590$	82.7695	3.40756
$591$	0	0
$592$	124.116	5.10114
$593$	−37.9347	−1.55779	−0.778896	−	0.627153i	$-0.784220\pi$
$593$	−37.9347	−1.55779	−0.778896	+	0.627153i	$0.784220\pi$
$594$	0	0
$595$	−24.1613	−0.990518
$596$	34.8780	1.42866
$597$	0	0
$598$	−25.1663	−1.02912
$599$	−22.6408	−0.925076	−0.462538	−	0.886599i	$-0.653061\pi$
$599$	−22.6408	−0.925076	−0.462538	+	0.886599i	$0.653061\pi$
$600$	0	0
$601$	37.5261	1.53072	0.765361	−	0.643601i	$-0.222561\pi$
$601$	37.5261	1.53072	0.765361	+	0.643601i	$0.222561\pi$
$602$	−22.4841	−0.916383
$603$	0	0
$604$	4.93535	0.200817
$605$	2.83529	0.115271
$606$	0	0
$607$	−13.5437	−0.549721	−0.274860	−	0.961484i	$-0.588632\pi$
$607$	−13.5437	−0.549721	−0.274860	+	0.961484i	$0.588632\pi$
$608$	16.6747	0.676250
$609$	0	0
$610$	7.66580	0.310379
$611$	−21.3910	−0.865386
$612$	0	0
$613$	−22.4208	−0.905567	−0.452784	−	0.891620i	$-0.649569\pi$
$613$	−22.4208	−0.905567	−0.452784	+	0.891620i	$0.649569\pi$
$614$	−41.6122	−1.67933
$615$	0	0
$616$	22.7722	0.917518
$617$	23.8518	0.960239	0.480119	−	0.877203i	$-0.340593\pi$
$617$	23.8518	0.960239	0.480119	+	0.877203i	$0.340593\pi$
$618$	0	0
$619$	1.12889	0.0453740	0.0226870	−	0.999743i	$-0.492778\pi$
$619$	1.12889	0.0453740	0.0226870	+	0.999743i	$0.492778\pi$
$620$	−152.537	−6.12605
$621$	0	0
$622$	−73.9742	−2.96610
$623$	33.0669	1.32480
$624$	0	0
$625$	−30.9596	−1.23838
$626$	−48.7020	−1.94652
$627$	0	0
$628$	−56.8546	−2.26875
$629$	−30.6164	−1.22076
$630$	0	0
$631$	45.7452	1.82109	0.910543	−	0.413414i	$-0.135664\pi$
$631$	45.7452	1.82109	0.910543	+	0.413414i	$0.135664\pi$
$632$	−2.41752	−0.0961639
$633$	0	0
$634$	−4.94753	−0.196491
$635$	47.1219	1.86997
$636$	0	0
$637$	1.79064	0.0709476
$638$	−5.64914	−0.223652
$639$	0	0
$640$	75.0131	2.96515
$641$	−30.7435	−1.21430	−0.607148	−	0.794589i	$-0.707686\pi$
$641$	−30.7435	−1.21430	−0.607148	+	0.794589i	$0.707686\pi$
$642$	0	0
$643$	13.7128	0.540781	0.270390	−	0.962751i	$-0.412847\pi$
$643$	13.7128	0.540781	0.270390	+	0.962751i	$0.412847\pi$
$644$	36.8872	1.45356
$645$	0	0
$646$	−8.02768	−0.315845
$647$	−34.2123	−1.34503	−0.672513	−	0.740085i	$-0.734785\pi$
$647$	−34.2123	−1.34503	−0.672513	+	0.740085i	$0.734785\pi$
$648$	0	0
$649$	10.7972	0.423828
$650$	−28.0136	−1.09878
$651$	0	0
$652$	−90.4139	−3.54088
$653$	13.4851	0.527713	0.263857	−	0.964562i	$-0.415005\pi$
$653$	13.4851	0.527713	0.263857	+	0.964562i	$0.415005\pi$
$654$	0	0
$655$	−17.7723	−0.694420
$656$	−115.468	−4.50827
$657$	0	0
$658$	43.1629	1.68266
$659$	−10.7366	−0.418240	−0.209120	−	0.977890i	$-0.567060\pi$
$659$	−10.7366	−0.418240	−0.209120	+	0.977890i	$0.567060\pi$
$660$	0	0
$661$	39.1172	1.52148	0.760742	−	0.649054i	$-0.224835\pi$
$661$	39.1172	1.52148	0.760742	+	0.649054i	$0.224835\pi$
$662$	−72.3110	−2.81045
$663$	0	0
$664$	80.4330	3.12141
$665$	6.39637	0.248040
$666$	0	0
$667$	−5.70412	−0.220864
$668$	16.2847	0.630075
$669$	0	0
$670$	−22.7391	−0.878489
$671$	1.00000	0.0386046
$672$	0	0
$673$	−23.7957	−0.917256	−0.458628	−	0.888628i	$-0.651659\pi$
$673$	−23.7957	−0.917256	−0.458628	+	0.888628i	$0.651659\pi$
$674$	−10.7509	−0.414108
$675$	0	0
$676$	−7.30209	−0.280850
$677$	−37.6164	−1.44572	−0.722858	−	0.690996i	$-0.757172\pi$
$677$	−37.6164	−1.44572	−0.722858	+	0.690996i	$0.757172\pi$
$678$	0	0
$679$	−44.0795	−1.69161
$680$	−84.9765	−3.25870
$681$	0	0
$682$	−27.3931	−1.04893
$683$	5.00868	0.191652	0.0958260	−	0.995398i	$-0.469451\pi$
$683$	5.00868	0.191652	0.0958260	+	0.995398i	$0.469451\pi$
$684$	0	0
$685$	−6.35069	−0.242647
$686$	−51.7715	−1.97664
$687$	0	0
$688$	−44.3695	−1.69157
$689$	−7.48907	−0.285311
$690$	0	0
$691$	−10.5248	−0.400383	−0.200192	−	0.979757i	$-0.564156\pi$
$691$	−10.5248	−0.400383	−0.200192	+	0.979757i	$0.564156\pi$
$692$	52.0999	1.98054
$693$	0	0
$694$	81.7007	3.10131
$695$	−35.3674	−1.34156
$696$	0	0
$697$	28.4831	1.07888
$698$	94.2968	3.56919
$699$	0	0
$700$	41.0607	1.55195
$701$	27.2911	1.03077	0.515385	−	0.856959i	$-0.327649\pi$
$701$	27.2911	1.03077	0.515385	+	0.856959i	$0.327649\pi$
$702$	0	0
$703$	8.10525	0.305695
$704$	23.6980	0.893152
$705$	0	0
$706$	−15.7555	−0.592967
$707$	−19.1903	−0.721726
$708$	0	0
$709$	−25.4021	−0.953997	−0.476999	−	0.878904i	$-0.658275\pi$
$709$	−25.4021	−0.953997	−0.476999	+	0.878904i	$0.658275\pi$
$710$	−94.9060	−3.56176
$711$	0	0
$712$	116.298	4.35844
$713$	−27.6597	−1.03586
$714$	0	0
$715$	−9.66700	−0.361525
$716$	110.309	4.12244
$717$	0	0
$718$	−34.2082	−1.27664
$719$	6.51099	0.242819	0.121410	−	0.992602i	$-0.461259\pi$
$719$	6.51099	0.242819	0.121410	+	0.992602i	$0.461259\pi$
$720$	0	0
$721$	−22.0166	−0.819940
$722$	−49.2452	−1.83272
$723$	0	0
$724$	−9.06357	−0.336845
$725$	−6.34949	−0.235814
$726$	0	0
$727$	−3.54365	−0.131427	−0.0657134	−	0.997839i	$-0.520932\pi$
$727$	−3.54365	−0.131427	−0.0657134	+	0.997839i	$0.520932\pi$
$728$	−77.6423	−2.87762
$729$	0	0
$730$	−66.0705	−2.44538
$731$	10.9449	0.404811
$732$	0	0
$733$	−29.0335	−1.07238	−0.536189	−	0.844098i	$-0.680137\pi$
$733$	−29.0335	−1.07238	−0.536189	+	0.844098i	$0.680137\pi$
$734$	−13.4302	−0.495718
$735$	0	0
$736$	51.3456	1.89262
$737$	−2.96631	−0.109265
$738$	0	0
$739$	−40.0793	−1.47434	−0.737170	−	0.675707i	$-0.763838\pi$
$739$	−40.0793	−1.47434	−0.737170	+	0.675707i	$0.763838\pi$
$740$	137.639	5.05970
$741$	0	0
$742$	15.1115	0.554761
$743$	−5.55093	−0.203644	−0.101822	−	0.994803i	$-0.532467\pi$
$743$	−5.55093	−0.203644	−0.101822	+	0.994803i	$0.532467\pi$
$744$	0	0
$745$	18.6232	0.682300
$746$	46.4708	1.70142
$747$	0	0
$748$	−17.7830	−0.650212
$749$	22.3904	0.818126
$750$	0	0
$751$	2.71659	0.0991298	0.0495649	−	0.998771i	$-0.484217\pi$
$751$	2.71659	0.0991298	0.0495649	+	0.998771i	$0.484217\pi$
$752$	85.1765	3.10607
$753$	0	0
$754$	19.2609	0.701440
$755$	2.63523	0.0959060
$756$	0	0
$757$	27.0786	0.984190	0.492095	−	0.870542i	$-0.336231\pi$
$757$	27.0786	0.984190	0.492095	+	0.870542i	$0.336231\pi$
$758$	21.2976	0.773562
$759$	0	0
$760$	22.4963	0.816026
$761$	36.1572	1.31070	0.655349	−	0.755326i	$-0.272522\pi$
$761$	36.1572	1.31070	0.655349	+	0.755326i	$0.272522\pi$
$762$	0	0
$763$	41.1388	1.48932
$764$	−51.4413	−1.86108
$765$	0	0
$766$	51.8425	1.87315
$767$	−36.8134	−1.32926
$768$	0	0
$769$	8.01720	0.289108	0.144554	−	0.989497i	$-0.453825\pi$
$769$	8.01720	0.289108	0.144554	+	0.989497i	$0.453825\pi$
$770$	19.5061	0.702953
$771$	0	0
$772$	−56.3880	−2.02945
$773$	4.66875	0.167923	0.0839617	−	0.996469i	$-0.473243\pi$
$773$	4.66875	0.167923	0.0839617	+	0.996469i	$0.473243\pi$
$774$	0	0
$775$	−30.7891	−1.10598
$776$	−155.029	−5.56523
$777$	0	0
$778$	8.09670	0.290281
$779$	−7.54050	−0.270167
$780$	0	0
$781$	−12.3804	−0.443007
$782$	−24.7192	−0.883956
$783$	0	0
$784$	−7.13012	−0.254647
$785$	−30.3576	−1.08351
$786$	0	0
$787$	−14.5293	−0.517913	−0.258956	−	0.965889i	$-0.583379\pi$
$787$	−14.5293	−0.517913	−0.258956	+	0.965889i	$0.583379\pi$
$788$	119.507	4.25724
$789$	0	0
$790$	−2.07080	−0.0736756
$791$	−46.3410	−1.64770
$792$	0	0
$793$	−3.40952	−0.121076
$794$	82.7549	2.93686
$795$	0	0
$796$	16.7667	0.594280
$797$	−34.5351	−1.22329	−0.611647	−	0.791131i	$-0.709493\pi$
$797$	−34.5351	−1.22329	−0.611647	+	0.791131i	$0.709493\pi$
$798$	0	0
$799$	−21.0110	−0.743315
$800$	57.1549	2.02073
$801$	0	0
$802$	−74.8429	−2.64279
$803$	−8.61886	−0.304153
$804$	0	0
$805$	19.6960	0.694192
$806$	93.3973	3.28978
$807$	0	0
$808$	−67.4931	−2.37440
$809$	−14.9159	−0.524415	−0.262208	−	0.965012i	$-0.584450\pi$
$809$	−14.9159	−0.524415	−0.262208	+	0.965012i	$0.584450\pi$
$810$	0	0
$811$	24.0571	0.844758	0.422379	−	0.906419i	$-0.361195\pi$
$811$	24.0571	0.844758	0.422379	+	0.906419i	$0.361195\pi$
$812$	−28.2315	−0.990731
$813$	0	0
$814$	24.7175	0.866348
$815$	−48.2766	−1.69106
$816$	0	0
$817$	−2.89750	−0.101371
$818$	20.6027	0.720356
$819$	0	0
$820$	−128.048	−4.47164
$821$	14.7592	0.515099	0.257550	−	0.966265i	$-0.417085\pi$
$821$	14.7592	0.515099	0.257550	+	0.966265i	$0.417085\pi$
$822$	0	0
$823$	34.0054	1.18536	0.592678	−	0.805440i	$-0.298071\pi$
$823$	34.0054	1.18536	0.592678	+	0.805440i	$0.298071\pi$
$824$	−77.4333	−2.69752
$825$	0	0
$826$	74.2823	2.58461
$827$	−37.0827	−1.28949	−0.644746	−	0.764397i	$-0.723037\pi$
$827$	−37.0827	−1.28949	−0.644746	+	0.764397i	$0.723037\pi$
$828$	0	0
$829$	1.60842	0.0558627	0.0279314	−	0.999610i	$-0.491108\pi$
$829$	1.60842	0.0558627	0.0279314	+	0.999610i	$0.491108\pi$
$830$	68.8971	2.39145
$831$	0	0
$832$	−80.7989	−2.80120
$833$	1.75883	0.0609397
$834$	0	0
$835$	8.69524	0.300911
$836$	4.70780	0.162823
$837$	0	0
$838$	−91.9983	−3.17803
$839$	−40.1514	−1.38618	−0.693090	−	0.720851i	$-0.743751\pi$
$839$	−40.1514	−1.38618	−0.693090	+	0.720851i	$0.743751\pi$
$840$	0	0
$841$	−24.6344	−0.849461
$842$	−7.52242	−0.259240
$843$	0	0
$844$	104.524	3.59786
$845$	−3.89896	−0.134128
$846$	0	0
$847$	2.54457	0.0874323
$848$	29.8207	1.02405
$849$	0	0
$850$	−27.5159	−0.943789
$851$	24.9580	0.855551
$852$	0	0
$853$	−56.2311	−1.92532	−0.962659	−	0.270718i	$-0.912739\pi$
$853$	−56.2311	−1.92532	−0.962659	+	0.270718i	$0.912739\pi$
$854$	6.87976	0.235420
$855$	0	0
$856$	78.7479	2.69155
$857$	−19.4700	−0.665081	−0.332541	−	0.943089i	$-0.607906\pi$
$857$	−19.4700	−0.665081	−0.332541	+	0.943089i	$0.607906\pi$
$858$	0	0
$859$	−23.4586	−0.800397	−0.400199	−	0.916428i	$-0.631059\pi$
$859$	−23.4586	−0.800397	−0.400199	+	0.916428i	$0.631059\pi$
$860$	−49.2036	−1.67783
$861$	0	0
$862$	−31.9647	−1.08872
$863$	21.4170	0.729042	0.364521	−	0.931195i	$-0.381233\pi$
$863$	21.4170	0.729042	0.364521	+	0.931195i	$0.381233\pi$
$864$	0	0
$865$	27.8188	0.945868
$866$	78.9047	2.68129
$867$	0	0
$868$	−136.896	−4.64656
$869$	−0.270134	−0.00916367
$870$	0	0
$871$	10.1137	0.342689
$872$	144.687	4.89972
$873$	0	0
$874$	6.54405	0.221356
$875$	−14.1486	−0.478309
$876$	0	0
$877$	53.5938	1.80973	0.904867	−	0.425695i	$-0.139970\pi$
$877$	53.5938	1.80973	0.904867	+	0.425695i	$0.139970\pi$
$878$	12.7524	0.430374
$879$	0	0
$880$	38.4930	1.29760
$881$	−52.4715	−1.76781	−0.883906	−	0.467665i	$-0.845095\pi$
$881$	−52.4715	−1.76781	−0.883906	+	0.467665i	$0.845095\pi$
$882$	0	0
$883$	7.35426	0.247491	0.123745	−	0.992314i	$-0.460509\pi$
$883$	7.35426	0.247491	0.123745	+	0.992314i	$0.460509\pi$
$884$	60.6317	2.03926
$885$	0	0
$886$	−3.11964	−0.104807
$887$	39.2495	1.31787	0.658934	−	0.752201i	$-0.271008\pi$
$887$	39.2495	1.31787	0.658934	+	0.752201i	$0.271008\pi$
$888$	0	0
$889$	42.2900	1.41836
$890$	99.6179	3.33920
$891$	0	0
$892$	−87.5371	−2.93096
$893$	5.56235	0.186137
$894$	0	0
$895$	58.8995	1.96879
$896$	67.3213	2.24905
$897$	0	0
$898$	55.8381	1.86334
$899$	21.1692	0.706032
$900$	0	0
$901$	−7.35603	−0.245065
$902$	−22.9953	−0.765659
$903$	0	0
$904$	−162.983	−5.42074
$905$	−4.83950	−0.160870
$906$	0	0
$907$	−54.4496	−1.80797	−0.903985	−	0.427564i	$-0.859372\pi$
$907$	−54.4496	−1.80797	−0.903985	+	0.427564i	$0.859372\pi$
$908$	70.1524	2.32809
$909$	0	0
$910$	−66.5066	−2.20467
$911$	−16.9754	−0.562421	−0.281211	−	0.959646i	$-0.590736\pi$
$911$	−16.9754	−0.562421	−0.281211	+	0.959646i	$0.590736\pi$
$912$	0	0
$913$	8.98758	0.297446
$914$	54.4158	1.79991
$915$	0	0
$916$	−118.820	−3.92592
$917$	−15.9499	−0.526713
$918$	0	0
$919$	−25.9451	−0.855850	−0.427925	−	0.903814i	$-0.640755\pi$
$919$	−25.9451	−0.855850	−0.427925	+	0.903814i	$0.640755\pi$
$920$	69.2716	2.28382
$921$	0	0
$922$	−32.4274	−1.06794
$923$	42.2114	1.38940
$924$	0	0
$925$	27.7818	0.913461
$926$	−8.97201	−0.294839
$927$	0	0
$928$	−39.2971	−1.28999
$929$	−2.59288	−0.0850695	−0.0425347	−	0.999095i	$-0.513543\pi$
$929$	−2.59288	−0.0850695	−0.0425347	+	0.999095i	$0.513543\pi$
$930$	0	0
$931$	−0.465624	−0.0152602
$932$	−24.2941	−0.795780
$933$	0	0
$934$	61.9186	2.02604
$935$	−9.49527	−0.310528
$936$	0	0
$937$	45.6680	1.49191	0.745954	−	0.665997i	$-0.231994\pi$
$937$	45.6680	1.49191	0.745954	+	0.665997i	$0.231994\pi$
$938$	−20.4075	−0.666328
$939$	0	0
$940$	94.4566	3.08083
$941$	−41.0172	−1.33712	−0.668562	−	0.743657i	$-0.733090\pi$
$941$	−41.0172	−1.33712	−0.668562	+	0.743657i	$0.733090\pi$
$942$	0	0
$943$	−23.2190	−0.756116
$944$	146.587	4.77100
$945$	0	0
$946$	−8.83612	−0.287287
$947$	−32.9919	−1.07209	−0.536047	−	0.844188i	$-0.680083\pi$
$947$	−32.9919	−1.07209	−0.536047	+	0.844188i	$0.680083\pi$
$948$	0	0
$949$	29.3862	0.953916
$950$	7.28445	0.236339
$951$	0	0
$952$	−76.2630	−2.47170
$953$	25.2512	0.817967	0.408984	−	0.912542i	$-0.365883\pi$
$953$	25.2512	0.817967	0.408984	+	0.912542i	$0.365883\pi$
$954$	0	0
$955$	−27.4672	−0.888816
$956$	46.1154	1.49148
$957$	0	0
$958$	37.2841	1.20460
$959$	−5.69950	−0.184046
$960$	0	0
$961$	71.6508	2.31132
$962$	−84.2749	−2.71713
$963$	0	0
$964$	−32.5028	−1.04684
$965$	−30.1084	−0.969224
$966$	0	0
$967$	−35.4783	−1.14090	−0.570452	−	0.821331i	$-0.693232\pi$
$967$	−35.4783	−1.14090	−0.570452	+	0.821331i	$0.693232\pi$
$968$	8.94935	0.287643
$969$	0	0
$970$	−132.795	−4.26378
$971$	−58.7515	−1.88543	−0.942713	−	0.333605i	$-0.891735\pi$
$971$	−58.7515	−1.88543	−0.942713	+	0.333605i	$0.891735\pi$
$972$	0	0
$973$	−31.7409	−1.01757
$974$	59.4623	1.90530
$975$	0	0
$976$	13.5763	0.434568
$977$	20.8909	0.668360	0.334180	−	0.942509i	$-0.391541\pi$
$977$	20.8909	0.668360	0.334180	+	0.942509i	$0.391541\pi$
$978$	0	0
$979$	12.9951	0.415325
$980$	−7.90695	−0.252578
$981$	0	0
$982$	47.1965	1.50610
$983$	−21.5657	−0.687838	−0.343919	−	0.938999i	$-0.611755\pi$
$983$	−21.5657	−0.687838	−0.343919	+	0.938999i	$0.611755\pi$
$984$	0	0
$985$	63.8106	2.03318
$986$	18.9187	0.602495
$987$	0	0
$988$	−16.0514	−0.510662
$989$	−8.92211	−0.283706
$990$	0	0
$991$	−14.4640	−0.459463	−0.229731	−	0.973254i	$-0.573785\pi$
$991$	−14.4640	−0.459463	−0.229731	+	0.973254i	$0.573785\pi$
$992$	−190.554	−6.05010
$993$	0	0
$994$	−85.1744	−2.70157
$995$	8.95260	0.283816
$996$	0	0
$997$	−9.45956	−0.299587	−0.149794	−	0.988717i	$-0.547861\pi$
$997$	−9.45956	−0.299587	−0.149794	+	0.988717i	$0.547861\pi$
$998$	−21.8327	−0.691103
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.d.1.11		11
3.2	odd	2		2013.2.a.a.1.1	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.a.1.1	✓	11	3.2	odd	2
6039.2.a.d.1.11		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.70371 q^{2} +5.31003 q^{4} +2.83529 q^{5} +2.54457 q^{7} +8.94935 q^{8} +O(q^{10})\)	\(q+2.70371 q^{2} +5.31003 q^{4} +2.83529 q^{5} +2.54457 q^{7} +8.94935 q^{8} +7.66580 q^{10} +1.00000 q^{11} -3.40952 q^{13} +6.87976 q^{14} +13.5763 q^{16} -3.34895 q^{17} +0.886587 q^{19} +15.0555 q^{20} +2.70371 q^{22} +2.73002 q^{23} +3.03890 q^{25} -9.21835 q^{26} +13.5117 q^{28} -2.08941 q^{29} -10.1317 q^{31} +18.8078 q^{32} -9.05459 q^{34} +7.21459 q^{35} +9.14208 q^{37} +2.39707 q^{38} +25.3740 q^{40} -8.50509 q^{41} -3.26815 q^{43} +5.31003 q^{44} +7.38117 q^{46} +6.27389 q^{47} -0.525187 q^{49} +8.21628 q^{50} -18.1047 q^{52} +2.19652 q^{53} +2.83529 q^{55} +22.7722 q^{56} -5.64914 q^{58} +10.7972 q^{59} +1.00000 q^{61} -27.3931 q^{62} +23.6980 q^{64} -9.66700 q^{65} -2.96631 q^{67} -17.7830 q^{68} +19.5061 q^{70} -12.3804 q^{71} -8.61886 q^{73} +24.7175 q^{74} +4.70780 q^{76} +2.54457 q^{77} -0.270134 q^{79} +38.4930 q^{80} -22.9953 q^{82} +8.98758 q^{83} -9.49527 q^{85} -8.83612 q^{86} +8.94935 q^{88} +12.9951 q^{89} -8.67576 q^{91} +14.4965 q^{92} +16.9628 q^{94} +2.51374 q^{95} -17.3230 q^{97} -1.41995 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8	\( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8} + 6 q^{10} + 11 q^{11} - 3 q^{13} + 9 q^{14} + 4 q^{16} + 7 q^{17} - 8 q^{19} + 25 q^{20} + 4 q^{22} + 15 q^{23} + 4 q^{25} + 2 q^{26} + 13 q^{28} + 8 q^{29} - 17 q^{31} + 27 q^{32} - 18 q^{34} + 2 q^{35} - 10 q^{37} + 30 q^{38} + 10 q^{40} + 25 q^{41} - 7 q^{43} + 6 q^{44} + 32 q^{46} + 30 q^{47} - 2 q^{49} - 11 q^{50} - 7 q^{52} + 18 q^{53} + 13 q^{55} + 20 q^{56} - 13 q^{58} + 43 q^{59} + 11 q^{61} - 7 q^{62} + 25 q^{64} + 27 q^{65} - 30 q^{67} - 10 q^{68} - 4 q^{70} + 7 q^{71} + 6 q^{73} + 44 q^{74} - 19 q^{76} - 5 q^{77} + 17 q^{79} + 22 q^{80} + 8 q^{82} + 34 q^{83} + 10 q^{85} - 2 q^{86} + 9 q^{88} + 41 q^{89} - 39 q^{91} + 32 q^{92} + 55 q^{94} + 9 q^{95} - 41 q^{97} + 29 q^{98}+O(q^{100}) \) 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8 + 6 * q^10 + 11 * q^11 - 3 * q^13 + 9 * q^14 + 4 * q^16 + 7 * q^17 - 8 * q^19 + 25 * q^20 + 4 * q^22 + 15 * q^23 + 4 * q^25 + 2 * q^26 + 13 * q^28 + 8 * q^29 - 17 * q^31 + 27 * q^32 - 18 * q^34 + 2 * q^35 - 10 * q^37 + 30 * q^38 + 10 * q^40 + 25 * q^41 - 7 * q^43 + 6 * q^44 + 32 * q^46 + 30 * q^47 - 2 * q^49 - 11 * q^50 - 7 * q^52 + 18 * q^53 + 13 * q^55 + 20 * q^56 - 13 * q^58 + 43 * q^59 + 11 * q^61 - 7 * q^62 + 25 * q^64 + 27 * q^65 - 30 * q^67 - 10 * q^68 - 4 * q^70 + 7 * q^71 + 6 * q^73 + 44 * q^74 - 19 * q^76 - 5 * q^77 + 17 * q^79 + 22 * q^80 + 8 * q^82 + 34 * q^83 + 10 * q^85 - 2 * q^86 + 9 * q^88 + 41 * q^89 - 39 * q^91 + 32 * q^92 + 55 * q^94 + 9 * q^95 - 41 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more