LMFDB - Embedded newform 8001.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.70725$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70725 q^{2} +0.914706 q^{4} -1.08728 q^{5} +1.00000 q^{7} +1.85287 q^{8} +O(q^{10})$	\(q-1.70725 q^{2} +0.914706 q^{4} -1.08728 q^{5} +1.00000 q^{7} +1.85287 q^{8} +1.85625 q^{10} -1.58139 q^{11} -5.55528 q^{13} -1.70725 q^{14} -4.99273 q^{16} +3.38289 q^{17} -4.84199 q^{19} -0.994538 q^{20} +2.69983 q^{22} -7.35331 q^{23} -3.81783 q^{25} +9.48427 q^{26} +0.914706 q^{28} -4.01548 q^{29} -8.96014 q^{31} +4.81810 q^{32} -5.77544 q^{34} -1.08728 q^{35} -3.94006 q^{37} +8.26650 q^{38} -2.01458 q^{40} -6.40603 q^{41} -8.32135 q^{43} -1.44651 q^{44} +12.5539 q^{46} +2.82123 q^{47} +1.00000 q^{49} +6.51800 q^{50} -5.08145 q^{52} +2.86232 q^{53} +1.71941 q^{55} +1.85287 q^{56} +6.85544 q^{58} +3.19939 q^{59} +15.1992 q^{61} +15.2972 q^{62} +1.75975 q^{64} +6.04013 q^{65} -14.4412 q^{67} +3.09435 q^{68} +1.85625 q^{70} -4.60022 q^{71} -11.4194 q^{73} +6.72667 q^{74} -4.42900 q^{76} -1.58139 q^{77} +5.60203 q^{79} +5.42847 q^{80} +10.9367 q^{82} +6.19625 q^{83} -3.67813 q^{85} +14.2066 q^{86} -2.93010 q^{88} +14.7613 q^{89} -5.55528 q^{91} -6.72612 q^{92} -4.81655 q^{94} +5.26458 q^{95} -4.99491 q^{97} -1.70725 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	−1.70725	−1.20721	−0.603604	−	0.797284i	$-0.706269\pi$
$2$	−1.70725	−1.20721	−0.603604	+	0.797284i	$0.706269\pi$
$3$	0	0
$3$	0	0
$4$	0.914706	0.457353
$5$	−1.08728	−0.486245	−0.243122	−	0.969996i	$-0.578172\pi$
$5$	−1.08728	−0.486245	−0.243122	+	0.969996i	$0.578172\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.85287	0.655088
$9$	0	0
$10$	1.85625	0.586999
$11$	−1.58139	−0.476806	−0.238403	−	0.971166i	$-0.576624\pi$
$11$	−1.58139	−0.476806	−0.238403	+	0.971166i	$0.576624\pi$
$12$	0	0
$13$	−5.55528	−1.54076	−0.770379	−	0.637586i	$-0.779933\pi$
$13$	−5.55528	−1.54076	−0.770379	+	0.637586i	$0.779933\pi$
$14$	−1.70725	−0.456282
$15$	0	0
$16$	−4.99273	−1.24818
$17$	3.38289	0.820471	0.410236	−	0.911980i	$-0.365446\pi$
$17$	3.38289	0.820471	0.410236	+	0.911980i	$0.365446\pi$
$18$	0	0
$19$	−4.84199	−1.11083	−0.555415	−	0.831573i	$-0.687441\pi$
$19$	−4.84199	−1.11083	−0.555415	+	0.831573i	$0.687441\pi$
$20$	−0.994538	−0.222386
$21$	0	0
$22$	2.69983	0.575605
$23$	−7.35331	−1.53327	−0.766636	−	0.642082i	$-0.778071\pi$
$23$	−7.35331	−1.53327	−0.766636	+	0.642082i	$0.778071\pi$
$24$	0	0
$25$	−3.81783	−0.763566
$26$	9.48427	1.86002
$27$	0	0
$28$	0.914706	0.172863
$29$	−4.01548	−0.745656	−0.372828	−	0.927900i	$-0.621612\pi$
$29$	−4.01548	−0.745656	−0.372828	+	0.927900i	$0.621612\pi$
$30$	0	0
$31$	−8.96014	−1.60929	−0.804644	−	0.593757i	$-0.797644\pi$
$31$	−8.96014	−1.60929	−0.804644	+	0.593757i	$0.797644\pi$
$32$	4.81810	0.851727
$33$	0	0
$34$	−5.77544	−0.990480
$35$	−1.08728	−0.183783
$36$	0	0
$37$	−3.94006	−0.647742	−0.323871	−	0.946101i	$-0.604984\pi$
$37$	−3.94006	−0.647742	−0.323871	+	0.946101i	$0.604984\pi$
$38$	8.26650	1.34100
$39$	0	0
$40$	−2.01458	−0.318533
$41$	−6.40603	−1.00045	−0.500227	−	0.865894i	$-0.666750\pi$
$41$	−6.40603	−1.00045	−0.500227	+	0.865894i	$0.666750\pi$
$42$	0	0
$43$	−8.32135	−1.26899	−0.634497	−	0.772925i	$-0.718793\pi$
$43$	−8.32135	−1.26899	−0.634497	+	0.772925i	$0.718793\pi$
$44$	−1.44651	−0.218069
$45$	0	0
$46$	12.5539	1.85098
$47$	2.82123	0.411519	0.205759	−	0.978603i	$-0.434034\pi$
$47$	2.82123	0.411519	0.205759	+	0.978603i	$0.434034\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.51800	0.921784
$51$	0	0
$52$	−5.08145	−0.704671
$53$	2.86232	0.393170	0.196585	−	0.980487i	$-0.437015\pi$
$53$	2.86232	0.393170	0.196585	+	0.980487i	$0.437015\pi$
$54$	0	0
$55$	1.71941	0.231845
$56$	1.85287	0.247600
$57$	0	0
$58$	6.85544	0.900163
$59$	3.19939	0.416525	0.208263	−	0.978073i	$-0.433219\pi$
$59$	3.19939	0.416525	0.208263	+	0.978073i	$0.433219\pi$
$60$	0	0
$61$	15.1992	1.94606	0.973028	−	0.230687i	$-0.0740975\pi$
$61$	15.1992	1.94606	0.973028	+	0.230687i	$0.0740975\pi$
$62$	15.2972	1.94275
$63$	0	0
$64$	1.75975	0.219968
$65$	6.04013	0.749186
$66$	0	0
$67$	−14.4412	−1.76427	−0.882137	−	0.470992i	$-0.843896\pi$
$67$	−14.4412	−1.76427	−0.882137	+	0.470992i	$0.843896\pi$
$68$	3.09435	0.375245
$69$	0	0
$70$	1.85625	0.221865
$71$	−4.60022	−0.545946	−0.272973	−	0.962022i	$-0.588007\pi$
$71$	−4.60022	−0.545946	−0.272973	+	0.962022i	$0.588007\pi$
$72$	0	0
$73$	−11.4194	−1.33654	−0.668272	−	0.743917i	$-0.732966\pi$
$73$	−11.4194	−1.33654	−0.668272	+	0.743917i	$0.732966\pi$
$74$	6.72667	0.781960
$75$	0	0
$76$	−4.42900	−0.508042
$77$	−1.58139	−0.180216
$78$	0	0
$79$	5.60203	0.630278	0.315139	−	0.949046i	$-0.397949\pi$
$79$	5.60203	0.630278	0.315139	+	0.949046i	$0.397949\pi$
$80$	5.42847	0.606921
$81$	0	0
$82$	10.9367	1.20776
$83$	6.19625	0.680127	0.340064	−	0.940402i	$-0.389551\pi$
$83$	6.19625	0.680127	0.340064	+	0.940402i	$0.389551\pi$
$84$	0	0
$85$	−3.67813	−0.398950
$86$	14.2066	1.53194
$87$	0	0
$88$	−2.93010	−0.312350
$89$	14.7613	1.56470	0.782349	−	0.622840i	$-0.214021\pi$
$89$	14.7613	1.56470	0.782349	+	0.622840i	$0.214021\pi$
$90$	0	0
$91$	−5.55528	−0.582352
$92$	−6.72612	−0.701247
$93$	0	0
$94$	−4.81655	−0.496789
$95$	5.26458	0.540135
$96$	0	0
$97$	−4.99491	−0.507157	−0.253578	−	0.967315i	$-0.581608\pi$
$97$	−4.99491	−0.507157	−0.253578	+	0.967315i	$0.581608\pi$
$98$	−1.70725	−0.172458
$99$	0	0
$100$	−3.49219	−0.349219
$101$	−15.3793	−1.53030	−0.765148	−	0.643854i	$-0.777334\pi$
$101$	−15.3793	−1.53030	−0.765148	+	0.643854i	$0.777334\pi$
$102$	0	0
$103$	18.4573	1.81866	0.909328	−	0.416080i	$-0.136596\pi$
$103$	18.4573	1.81866	0.909328	+	0.416080i	$0.136596\pi$
$104$	−10.2932	−1.00933
$105$	0	0
$106$	−4.88670	−0.474639
$107$	3.34462	0.323336	0.161668	−	0.986845i	$-0.448313\pi$
$107$	3.34462	0.323336	0.161668	+	0.986845i	$0.448313\pi$
$108$	0	0
$109$	−2.86219	−0.274148	−0.137074	−	0.990561i	$-0.543770\pi$
$109$	−2.86219	−0.274148	−0.137074	+	0.990561i	$0.543770\pi$
$110$	−2.93546	−0.279885
$111$	0	0
$112$	−4.99273	−0.471768
$113$	6.64072	0.624706	0.312353	−	0.949966i	$-0.398883\pi$
$113$	6.64072	0.624706	0.312353	+	0.949966i	$0.398883\pi$
$114$	0	0
$115$	7.99508	0.745545
$116$	−3.67299	−0.341028
$117$	0	0
$118$	−5.46217	−0.502833
$119$	3.38289	0.310109
$120$	0	0
$121$	−8.49921	−0.772656
$122$	−25.9488	−2.34930
$123$	0	0
$124$	−8.19590	−0.736013
$125$	9.58742	0.857525
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−12.6405	−1.11728
$129$	0	0
$130$	−10.3120	−0.904424
$131$	6.22354	0.543753	0.271876	−	0.962332i	$-0.412356\pi$
$131$	6.22354	0.543753	0.271876	+	0.962332i	$0.412356\pi$
$132$	0	0
$133$	−4.84199	−0.419854
$134$	24.6548	2.12985
$135$	0	0
$136$	6.26805	0.537481
$137$	−0.432704	−0.0369684	−0.0184842	−	0.999829i	$-0.505884\pi$
$137$	−0.432704	−0.0369684	−0.0184842	+	0.999829i	$0.505884\pi$
$138$	0	0
$139$	−6.17897	−0.524093	−0.262047	−	0.965055i	$-0.584397\pi$
$139$	−6.17897	−0.524093	−0.262047	+	0.965055i	$0.584397\pi$
$140$	−0.994538	−0.0840538
$141$	0	0
$142$	7.85373	0.659070
$143$	8.78506	0.734644
$144$	0	0
$145$	4.36594	0.362571
$146$	19.4958	1.61349
$147$	0	0
$148$	−3.60400	−0.296247
$149$	−13.4075	−1.09838	−0.549191	−	0.835697i	$-0.685064\pi$
$149$	−13.4075	−1.09838	−0.549191	+	0.835697i	$0.685064\pi$
$150$	0	0
$151$	11.2770	0.917708	0.458854	−	0.888512i	$-0.348260\pi$
$151$	11.2770	0.917708	0.458854	+	0.888512i	$0.348260\pi$
$152$	−8.97158	−0.727691
$153$	0	0
$154$	2.69983	0.217558
$155$	9.74214	0.782508
$156$	0	0
$157$	−20.5897	−1.64323	−0.821617	−	0.570040i	$-0.806928\pi$
$157$	−20.5897	−1.64323	−0.821617	+	0.570040i	$0.806928\pi$
$158$	−9.56408	−0.760877
$159$	0	0
$160$	−5.23860	−0.414148
$161$	−7.35331	−0.579522
$162$	0	0
$163$	13.6441	1.06869	0.534343	−	0.845268i	$-0.320559\pi$
$163$	13.6441	1.06869	0.534343	+	0.845268i	$0.320559\pi$
$164$	−5.85964	−0.457561
$165$	0	0
$166$	−10.5786	−0.821055
$167$	24.9692	1.93218	0.966089	−	0.258210i	$-0.0831325\pi$
$167$	24.9692	1.93218	0.966089	+	0.258210i	$0.0831325\pi$
$168$	0	0
$169$	17.8612	1.37394
$170$	6.27950	0.481616
$171$	0	0
$172$	−7.61159	−0.580378
$173$	7.80876	0.593689	0.296845	−	0.954926i	$-0.404066\pi$
$173$	7.80876	0.593689	0.296845	+	0.954926i	$0.404066\pi$
$174$	0	0
$175$	−3.81783	−0.288601
$176$	7.89544	0.595141
$177$	0	0
$178$	−25.2013	−1.88892
$179$	20.4507	1.52856	0.764279	−	0.644886i	$-0.223095\pi$
$179$	20.4507	1.52856	0.764279	+	0.644886i	$0.223095\pi$
$180$	0	0
$181$	−4.32280	−0.321311	−0.160656	−	0.987011i	$-0.551361\pi$
$181$	−4.32280	−0.321311	−0.160656	+	0.987011i	$0.551361\pi$
$182$	9.48427	0.703021
$183$	0	0
$184$	−13.6247	−1.00443
$185$	4.28393	0.314961
$186$	0	0
$187$	−5.34966	−0.391206
$188$	2.58060	0.188210
$189$	0	0
$190$	−8.98797	−0.652056
$191$	−18.8230	−1.36198	−0.680992	−	0.732291i	$-0.738451\pi$
$191$	−18.8230	−1.36198	−0.680992	+	0.732291i	$0.738451\pi$
$192$	0	0
$193$	4.03163	0.290203	0.145102	−	0.989417i	$-0.453649\pi$
$193$	4.03163	0.290203	0.145102	+	0.989417i	$0.453649\pi$
$194$	8.52757	0.612244
$195$	0	0
$196$	0.914706	0.0653362
$197$	−8.77313	−0.625060	−0.312530	−	0.949908i	$-0.601176\pi$
$197$	−8.77313	−0.625060	−0.312530	+	0.949908i	$0.601176\pi$
$198$	0	0
$199$	−16.1994	−1.14835	−0.574174	−	0.818734i	$-0.694677\pi$
$199$	−16.1994	−1.14835	−0.574174	+	0.818734i	$0.694677\pi$
$200$	−7.07394	−0.500203
$201$	0	0
$202$	26.2563	1.84739
$203$	−4.01548	−0.281832
$204$	0	0
$205$	6.96513	0.486465
$206$	−31.5113	−2.19550
$207$	0	0
$208$	27.7360	1.92315
$209$	7.65707	0.529651
$210$	0	0
$211$	−6.64122	−0.457200	−0.228600	−	0.973520i	$-0.573415\pi$
$211$	−6.64122	−0.457200	−0.228600	+	0.973520i	$0.573415\pi$
$212$	2.61818	0.179818
$213$	0	0
$214$	−5.71010	−0.390334
$215$	9.04760	0.617041
$216$	0	0
$217$	−8.96014	−0.608254
$218$	4.88648	0.330954
$219$	0	0
$220$	1.57275	0.106035
$221$	−18.7929	−1.26415
$222$	0	0
$223$	18.9937	1.27192	0.635958	−	0.771724i	$-0.280605\pi$
$223$	18.9937	1.27192	0.635958	+	0.771724i	$0.280605\pi$
$224$	4.81810	0.321923
$225$	0	0
$226$	−11.3374	−0.754151
$227$	−9.18958	−0.609934	−0.304967	−	0.952363i	$-0.598645\pi$
$227$	−9.18958	−0.609934	−0.304967	+	0.952363i	$0.598645\pi$
$228$	0	0
$229$	0.707717	0.0467672	0.0233836	−	0.999727i	$-0.492556\pi$
$229$	0.707717	0.0467672	0.0233836	+	0.999727i	$0.492556\pi$
$230$	−13.6496	−0.900029
$231$	0	0
$232$	−7.44016	−0.488471
$233$	13.4302	0.879842	0.439921	−	0.898036i	$-0.355006\pi$
$233$	13.4302	0.879842	0.439921	+	0.898036i	$0.355006\pi$
$234$	0	0
$235$	−3.06746	−0.200099
$236$	2.92650	0.190499
$237$	0	0
$238$	−5.77544	−0.374366
$239$	0.623102	0.0403051	0.0201526	−	0.999797i	$-0.493585\pi$
$239$	0.623102	0.0403051	0.0201526	+	0.999797i	$0.493585\pi$
$240$	0	0
$241$	0.924991	0.0595839	0.0297920	−	0.999556i	$-0.490516\pi$
$241$	0.924991	0.0595839	0.0297920	+	0.999556i	$0.490516\pi$
$242$	14.5103	0.932757
$243$	0	0
$244$	13.9028	0.890035
$245$	−1.08728	−0.0694635
$246$	0	0
$247$	26.8987	1.71152
$248$	−16.6020	−1.05423
$249$	0	0
$250$	−16.3681	−1.03521
$251$	−11.3142	−0.714146	−0.357073	−	0.934077i	$-0.616225\pi$
$251$	−11.3142	−0.714146	−0.357073	+	0.934077i	$0.616225\pi$
$252$	0	0
$253$	11.6284	0.731074
$254$	−1.70725	−0.107122
$255$	0	0
$256$	18.0611	1.12882
$257$	−22.4836	−1.40249	−0.701244	−	0.712921i	$-0.747372\pi$
$257$	−22.4836	−1.40249	−0.701244	+	0.712921i	$0.747372\pi$
$258$	0	0
$259$	−3.94006	−0.244823
$260$	5.52494	0.342642
$261$	0	0
$262$	−10.6251	−0.656423
$263$	−7.25932	−0.447629	−0.223814	−	0.974632i	$-0.571851\pi$
$263$	−7.25932	−0.447629	−0.223814	+	0.974632i	$0.571851\pi$
$264$	0	0
$265$	−3.11213	−0.191177
$266$	8.26650	0.506852
$267$	0	0
$268$	−13.2095	−0.806897
$269$	5.61611	0.342420	0.171210	−	0.985235i	$-0.445232\pi$
$269$	5.61611	0.342420	0.171210	+	0.985235i	$0.445232\pi$
$270$	0	0
$271$	15.4371	0.937737	0.468869	−	0.883268i	$-0.344662\pi$
$271$	15.4371	0.937737	0.468869	+	0.883268i	$0.344662\pi$
$272$	−16.8898	−1.02410
$273$	0	0
$274$	0.738734	0.0446286
$275$	6.03747	0.364073
$276$	0	0
$277$	21.4310	1.28767	0.643833	−	0.765166i	$-0.277343\pi$
$277$	21.4310	1.28767	0.643833	+	0.765166i	$0.277343\pi$
$278$	10.5491	0.632690
$279$	0	0
$280$	−2.01458	−0.120394
$281$	−18.1891	−1.08507	−0.542535	−	0.840033i	$-0.682535\pi$
$281$	−18.1891	−1.08507	−0.542535	+	0.840033i	$0.682535\pi$
$282$	0	0
$283$	−6.98641	−0.415299	−0.207649	−	0.978203i	$-0.566581\pi$
$283$	−6.98641	−0.415299	−0.207649	+	0.978203i	$0.566581\pi$
$284$	−4.20785	−0.249690
$285$	0	0
$286$	−14.9983	−0.886868
$287$	−6.40603	−0.378136
$288$	0	0
$289$	−5.55606	−0.326827
$290$	−7.45375	−0.437699
$291$	0	0
$292$	−10.4454	−0.611272
$293$	−3.67122	−0.214475	−0.107237	−	0.994233i	$-0.534201\pi$
$293$	−3.67122	−0.214475	−0.107237	+	0.994233i	$0.534201\pi$
$294$	0	0
$295$	−3.47862	−0.202533
$296$	−7.30042	−0.424328
$297$	0	0
$298$	22.8899	1.32598
$299$	40.8497	2.36240
$300$	0	0
$301$	−8.32135	−0.479634
$302$	−19.2526	−1.10786
$303$	0	0
$304$	24.1747	1.38652
$305$	−16.5257	−0.946259
$306$	0	0
$307$	0.782621	0.0446666	0.0223333	−	0.999751i	$-0.492891\pi$
$307$	0.782621	0.0446666	0.0223333	+	0.999751i	$0.492891\pi$
$308$	−1.44651	−0.0824223
$309$	0	0
$310$	−16.6323	−0.944650
$311$	−27.4489	−1.55649	−0.778243	−	0.627963i	$-0.783889\pi$
$311$	−27.4489	−1.55649	−0.778243	+	0.627963i	$0.783889\pi$
$312$	0	0
$313$	−17.6863	−0.999690	−0.499845	−	0.866115i	$-0.666610\pi$
$313$	−17.6863	−0.999690	−0.499845	+	0.866115i	$0.666610\pi$
$314$	35.1517	1.98373
$315$	0	0
$316$	5.12422	0.288260
$317$	−5.84866	−0.328493	−0.164247	−	0.986419i	$-0.552519\pi$
$317$	−5.84866	−0.328493	−0.164247	+	0.986419i	$0.552519\pi$
$318$	0	0
$319$	6.35004	0.355534
$320$	−1.91333	−0.106958
$321$	0	0
$322$	12.5539	0.699604
$323$	−16.3799	−0.911404
$324$	0	0
$325$	21.2091	1.17647
$326$	−23.2939	−1.29013
$327$	0	0
$328$	−11.8695	−0.655386
$329$	2.82123	0.155540
$330$	0	0
$331$	−6.76050	−0.371591	−0.185795	−	0.982588i	$-0.559486\pi$
$331$	−6.76050	−0.371591	−0.185795	+	0.982588i	$0.559486\pi$
$332$	5.66775	0.311058
$333$	0	0
$334$	−42.6288	−2.33254
$335$	15.7016	0.857869
$336$	0	0
$337$	21.7933	1.18715	0.593577	−	0.804777i	$-0.297715\pi$
$337$	21.7933	1.18715	0.593577	+	0.804777i	$0.297715\pi$
$338$	−30.4935	−1.65863
$339$	0	0
$340$	−3.36441	−0.182461
$341$	14.1695	0.767319
$342$	0	0
$343$	1.00000	0.0539949
$344$	−15.4184	−0.831302
$345$	0	0
$346$	−13.3315	−0.716707
$347$	−26.0834	−1.40023	−0.700115	−	0.714030i	$-0.746868\pi$
$347$	−26.0834	−1.40023	−0.700115	+	0.714030i	$0.746868\pi$
$348$	0	0
$349$	−33.5729	−1.79711	−0.898557	−	0.438856i	$-0.855384\pi$
$349$	−33.5729	−1.79711	−0.898557	+	0.438856i	$0.855384\pi$
$350$	6.51800	0.348402
$351$	0	0
$352$	−7.61928	−0.406109
$353$	27.4331	1.46011	0.730057	−	0.683386i	$-0.239493\pi$
$353$	27.4331	1.46011	0.730057	+	0.683386i	$0.239493\pi$
$354$	0	0
$355$	5.00171	0.265463
$356$	13.5023	0.715619
$357$	0	0
$358$	−34.9145	−1.84529
$359$	−6.58084	−0.347324	−0.173662	−	0.984805i	$-0.555560\pi$
$359$	−6.58084	−0.347324	−0.173662	+	0.984805i	$0.555560\pi$
$360$	0	0
$361$	4.44491	0.233943
$362$	7.38010	0.387889
$363$	0	0
$364$	−5.08145	−0.266341
$365$	12.4161	0.649887
$366$	0	0
$367$	23.1790	1.20994	0.604968	−	0.796250i	$-0.293186\pi$
$367$	23.1790	1.20994	0.604968	+	0.796250i	$0.293186\pi$
$368$	36.7131	1.91380
$369$	0	0
$370$	−7.31375	−0.380224
$371$	2.86232	0.148604
$372$	0	0
$373$	14.4055	0.745886	0.372943	−	0.927854i	$-0.378349\pi$
$373$	14.4055	0.745886	0.372943	+	0.927854i	$0.378349\pi$
$374$	9.13321	0.472267
$375$	0	0
$376$	5.22737	0.269581
$377$	22.3072	1.14888
$378$	0	0
$379$	19.3549	0.994194	0.497097	−	0.867695i	$-0.334399\pi$
$379$	19.3549	0.994194	0.497097	+	0.867695i	$0.334399\pi$
$380$	4.81555	0.247032
$381$	0	0
$382$	32.1356	1.64420
$383$	−21.4947	−1.09833	−0.549163	−	0.835715i	$-0.685054\pi$
$383$	−21.4947	−1.09833	−0.549163	+	0.835715i	$0.685054\pi$
$384$	0	0
$385$	1.71941	0.0876290
$386$	−6.88301	−0.350336
$387$	0	0
$388$	−4.56888	−0.231950
$389$	−24.0187	−1.21779	−0.608897	−	0.793249i	$-0.708388\pi$
$389$	−24.0187	−1.21779	−0.608897	+	0.793249i	$0.708388\pi$
$390$	0	0
$391$	−24.8754	−1.25800
$392$	1.85287	0.0935840
$393$	0	0
$394$	14.9779	0.754578
$395$	−6.09096	−0.306469
$396$	0	0
$397$	−22.9033	−1.14948	−0.574741	−	0.818336i	$-0.694897\pi$
$397$	−22.9033	−1.14948	−0.574741	+	0.818336i	$0.694897\pi$
$398$	27.6565	1.38629
$399$	0	0
$400$	19.0614	0.953069
$401$	36.2294	1.80921	0.904606	−	0.426249i	$-0.140165\pi$
$401$	36.2294	1.80921	0.904606	+	0.426249i	$0.140165\pi$
$402$	0	0
$403$	49.7761	2.47953
$404$	−14.0675	−0.699886
$405$	0	0
$406$	6.85544	0.340230
$407$	6.23077	0.308848
$408$	0	0
$409$	10.3032	0.509461	0.254731	−	0.967012i	$-0.418013\pi$
$409$	10.3032	0.509461	0.254731	+	0.967012i	$0.418013\pi$
$410$	−11.8912	−0.587265
$411$	0	0
$412$	16.8830	0.831768
$413$	3.19939	0.157432
$414$	0	0
$415$	−6.73704	−0.330708
$416$	−26.7659	−1.31231
$417$	0	0
$418$	−13.0725	−0.639399
$419$	−34.4188	−1.68147	−0.840735	−	0.541446i	$-0.817877\pi$
$419$	−34.4188	−1.68147	−0.840735	+	0.541446i	$0.817877\pi$
$420$	0	0
$421$	5.58303	0.272100	0.136050	−	0.990702i	$-0.456559\pi$
$421$	5.58303	0.272100	0.136050	+	0.990702i	$0.456559\pi$
$422$	11.3382	0.551936
$423$	0	0
$424$	5.30351	0.257561
$425$	−12.9153	−0.626484
$426$	0	0
$427$	15.1992	0.735540
$428$	3.05934	0.147879
$429$	0	0
$430$	−15.4465	−0.744898
$431$	6.32096	0.304470	0.152235	−	0.988344i	$-0.451353\pi$
$431$	6.32096	0.304470	0.152235	+	0.988344i	$0.451353\pi$
$432$	0	0
$433$	2.42420	0.116500	0.0582498	−	0.998302i	$-0.481448\pi$
$433$	2.42420	0.116500	0.0582498	+	0.998302i	$0.481448\pi$
$434$	15.2972	0.734289
$435$	0	0
$436$	−2.61807	−0.125383
$437$	35.6047	1.70320
$438$	0	0
$439$	20.2577	0.966845	0.483423	−	0.875387i	$-0.339393\pi$
$439$	20.2577	0.966845	0.483423	+	0.875387i	$0.339393\pi$
$440$	3.18583	0.151879
$441$	0	0
$442$	32.0842	1.52609
$443$	22.8915	1.08761	0.543804	−	0.839212i	$-0.316983\pi$
$443$	22.8915	1.08761	0.543804	+	0.839212i	$0.316983\pi$
$444$	0	0
$445$	−16.0496	−0.760826
$446$	−32.4271	−1.53547
$447$	0	0
$448$	1.75975	0.0831402
$449$	6.07459	0.286677	0.143339	−	0.989674i	$-0.454216\pi$
$449$	6.07459	0.286677	0.143339	+	0.989674i	$0.454216\pi$
$450$	0	0
$451$	10.1304	0.477023
$452$	6.07431	0.285712
$453$	0	0
$454$	15.6889	0.736318
$455$	6.04013	0.283166
$456$	0	0
$457$	12.0326	0.562862	0.281431	−	0.959582i	$-0.409191\pi$
$457$	12.0326	0.562862	0.281431	+	0.959582i	$0.409191\pi$
$458$	−1.20825	−0.0564578
$459$	0	0
$460$	7.31315	0.340977
$461$	35.2714	1.64275	0.821377	−	0.570386i	$-0.193206\pi$
$461$	35.2714	1.64275	0.821377	+	0.570386i	$0.193206\pi$
$462$	0	0
$463$	41.2830	1.91858	0.959291	−	0.282420i	$-0.0911372\pi$
$463$	41.2830	1.91858	0.959291	+	0.282420i	$0.0911372\pi$
$464$	20.0482	0.930714
$465$	0	0
$466$	−22.9287	−1.06215
$467$	−3.10679	−0.143765	−0.0718825	−	0.997413i	$-0.522901\pi$
$467$	−3.10679	−0.143765	−0.0718825	+	0.997413i	$0.522901\pi$
$468$	0	0
$469$	−14.4412	−0.666833
$470$	5.23692	0.241561
$471$	0	0
$472$	5.92805	0.272861
$473$	13.1593	0.605064
$474$	0	0
$475$	18.4859	0.848192
$476$	3.09435	0.141829
$477$	0	0
$478$	−1.06379	−0.0486567
$479$	15.5577	0.710848	0.355424	−	0.934705i	$-0.384336\pi$
$479$	15.5577	0.710848	0.355424	+	0.934705i	$0.384336\pi$
$480$	0	0
$481$	21.8882	0.998014
$482$	−1.57919	−0.0719302
$483$	0	0
$484$	−7.77428	−0.353377
$485$	5.43085	0.246602
$486$	0	0
$487$	−29.6687	−1.34442	−0.672210	−	0.740361i	$-0.734655\pi$
$487$	−29.6687	−1.34442	−0.672210	+	0.740361i	$0.734655\pi$
$488$	28.1621	1.27484
$489$	0	0
$490$	1.85625	0.0838570
$491$	−0.760535	−0.0343225	−0.0171612	−	0.999853i	$-0.505463\pi$
$491$	−0.760535	−0.0343225	−0.0171612	+	0.999853i	$0.505463\pi$
$492$	0	0
$493$	−13.5839	−0.611790
$494$	−45.9228	−2.06616
$495$	0	0
$496$	44.7355	2.00868
$497$	−4.60022	−0.206348
$498$	0	0
$499$	−16.4337	−0.735673	−0.367837	−	0.929890i	$-0.619901\pi$
$499$	−16.4337	−0.735673	−0.367837	+	0.929890i	$0.619901\pi$
$500$	8.76967	0.392192
$501$	0	0
$502$	19.3162	0.862123
$503$	−38.5672	−1.71963	−0.859813	−	0.510610i	$-0.829420\pi$
$503$	−38.5672	−1.71963	−0.859813	+	0.510610i	$0.829420\pi$
$504$	0	0
$505$	16.7215	0.744098
$506$	−19.8527	−0.882559
$507$	0	0
$508$	0.914706	0.0405835
$509$	26.3389	1.16745	0.583726	−	0.811951i	$-0.301594\pi$
$509$	26.3389	1.16745	0.583726	+	0.811951i	$0.301594\pi$
$510$	0	0
$511$	−11.4194	−0.505166
$512$	−5.55371	−0.245442
$513$	0	0
$514$	38.3852	1.69310
$515$	−20.0682	−0.884312
$516$	0	0
$517$	−4.46146	−0.196215
$518$	6.72667	0.295553
$519$	0	0
$520$	11.1916	0.490783
$521$	−12.5229	−0.548640	−0.274320	−	0.961638i	$-0.588453\pi$
$521$	−12.5229	−0.548640	−0.274320	+	0.961638i	$0.588453\pi$
$522$	0	0
$523$	−29.0101	−1.26852	−0.634261	−	0.773119i	$-0.718696\pi$
$523$	−29.0101	−1.26852	−0.634261	+	0.773119i	$0.718696\pi$
$524$	5.69271	0.248687
$525$	0	0
$526$	12.3935	0.540382
$527$	−30.3112	−1.32037
$528$	0	0
$529$	31.0712	1.35092
$530$	5.31320	0.230790
$531$	0	0
$532$	−4.42900	−0.192022
$533$	35.5873	1.54146
$534$	0	0
$535$	−3.63652	−0.157220
$536$	−26.7577	−1.15576
$537$	0	0
$538$	−9.58812	−0.413373
$539$	−1.58139	−0.0681152
$540$	0	0
$541$	−19.1020	−0.821260	−0.410630	−	0.911802i	$-0.634691\pi$
$541$	−19.1020	−0.821260	−0.410630	+	0.911802i	$0.634691\pi$
$542$	−26.3550	−1.13204
$543$	0	0
$544$	16.2991	0.698818
$545$	3.11199	0.133303
$546$	0	0
$547$	−38.1266	−1.63018	−0.815089	−	0.579336i	$-0.803312\pi$
$547$	−38.1266	−1.63018	−0.815089	+	0.579336i	$0.803312\pi$
$548$	−0.395797	−0.0169076
$549$	0	0
$550$	−10.3075	−0.439513
$551$	19.4429	0.828297
$552$	0	0
$553$	5.60203	0.238223
$554$	−36.5882	−1.55448
$555$	0	0
$556$	−5.65194	−0.239696
$557$	15.8515	0.671649	0.335825	−	0.941925i	$-0.390985\pi$
$557$	15.8515	0.671649	0.335825	+	0.941925i	$0.390985\pi$
$558$	0	0
$559$	46.2275	1.95521
$560$	5.42847	0.229395
$561$	0	0
$562$	31.0533	1.30991
$563$	−12.5290	−0.528035	−0.264017	−	0.964518i	$-0.585048\pi$
$563$	−12.5290	−0.528035	−0.264017	+	0.964518i	$0.585048\pi$
$564$	0	0
$565$	−7.22030	−0.303760
$566$	11.9276	0.501352
$567$	0	0
$568$	−8.52360	−0.357642
$569$	26.3672	1.10537	0.552685	−	0.833390i	$-0.313603\pi$
$569$	26.3672	1.10537	0.552685	+	0.833390i	$0.313603\pi$
$570$	0	0
$571$	24.5409	1.02700	0.513502	−	0.858089i	$-0.328348\pi$
$571$	24.5409	1.02700	0.513502	+	0.858089i	$0.328348\pi$
$572$	8.03575	0.335992
$573$	0	0
$574$	10.9367	0.456489
$575$	28.0737	1.17075
$576$	0	0
$577$	−28.3026	−1.17825	−0.589126	−	0.808042i	$-0.700528\pi$
$577$	−28.3026	−1.17825	−0.589126	+	0.808042i	$0.700528\pi$
$578$	9.48559	0.394549
$579$	0	0
$580$	3.99355	0.165823
$581$	6.19625	0.257064
$582$	0	0
$583$	−4.52644	−0.187466
$584$	−21.1587	−0.875554
$585$	0	0
$586$	6.26769	0.258916
$587$	7.70006	0.317815	0.158908	−	0.987293i	$-0.449203\pi$
$587$	7.70006	0.317815	0.158908	+	0.987293i	$0.449203\pi$
$588$	0	0
$589$	43.3849	1.78765
$590$	5.93888	0.244500
$591$	0	0
$592$	19.6716	0.808499
$593$	−25.5437	−1.04895	−0.524477	−	0.851424i	$-0.675739\pi$
$593$	−25.5437	−1.04895	−0.524477	+	0.851424i	$0.675739\pi$
$594$	0	0
$595$	−3.67813	−0.150789
$596$	−12.2639	−0.502348
$597$	0	0
$598$	−69.7408	−2.85191
$599$	25.6812	1.04930	0.524652	−	0.851317i	$-0.324196\pi$
$599$	25.6812	1.04930	0.524652	+	0.851317i	$0.324196\pi$
$600$	0	0
$601$	−33.8437	−1.38051	−0.690256	−	0.723565i	$-0.742502\pi$
$601$	−33.8437	−1.38051	−0.690256	+	0.723565i	$0.742502\pi$
$602$	14.2066	0.579019
$603$	0	0
$604$	10.3151	0.419717
$605$	9.24099	0.375700
$606$	0	0
$607$	−15.9169	−0.646046	−0.323023	−	0.946391i	$-0.604699\pi$
$607$	−15.9169	−0.646046	−0.323023	+	0.946391i	$0.604699\pi$
$608$	−23.3292	−0.946124
$609$	0	0
$610$	28.2135	1.14233
$611$	−15.6727	−0.634051
$612$	0	0
$613$	−26.9742	−1.08948	−0.544740	−	0.838605i	$-0.683372\pi$
$613$	−26.9742	−1.08948	−0.544740	+	0.838605i	$0.683372\pi$
$614$	−1.33613	−0.0539219
$615$	0	0
$616$	−2.93010	−0.118057
$617$	45.1074	1.81596	0.907979	−	0.419016i	$-0.137625\pi$
$617$	45.1074	1.81596	0.907979	+	0.419016i	$0.137625\pi$
$618$	0	0
$619$	−24.4678	−0.983443	−0.491722	−	0.870752i	$-0.663632\pi$
$619$	−24.4678	−0.983443	−0.491722	+	0.870752i	$0.663632\pi$
$620$	8.91120	0.357882
$621$	0	0
$622$	46.8622	1.87900
$623$	14.7613	0.591400
$624$	0	0
$625$	8.66499	0.346600
$626$	30.1950	1.20683
$627$	0	0
$628$	−18.8335	−0.751538
$629$	−13.3288	−0.531454
$630$	0	0
$631$	4.21378	0.167748	0.0838740	−	0.996476i	$-0.473271\pi$
$631$	4.21378	0.167748	0.0838740	+	0.996476i	$0.473271\pi$
$632$	10.3798	0.412888
$633$	0	0
$634$	9.98513	0.396560
$635$	−1.08728	−0.0431472
$636$	0	0
$637$	−5.55528	−0.220108
$638$	−10.8411	−0.429204
$639$	0	0
$640$	13.7437	0.543269
$641$	−42.2656	−1.66939	−0.834696	−	0.550711i	$-0.814357\pi$
$641$	−42.2656	−1.66939	−0.834696	+	0.550711i	$0.814357\pi$
$642$	0	0
$643$	18.5428	0.731257	0.365628	−	0.930761i	$-0.380854\pi$
$643$	18.5428	0.731257	0.365628	+	0.930761i	$0.380854\pi$
$644$	−6.72612	−0.265046
$645$	0	0
$646$	27.9647	1.10025
$647$	−22.4540	−0.882760	−0.441380	−	0.897320i	$-0.645511\pi$
$647$	−22.4540	−0.882760	−0.441380	+	0.897320i	$0.645511\pi$
$648$	0	0
$649$	−5.05948	−0.198602
$650$	−36.2093	−1.42025
$651$	0	0
$652$	12.4803	0.488767
$653$	36.2822	1.41983	0.709917	−	0.704285i	$-0.248732\pi$
$653$	36.2822	1.41983	0.709917	+	0.704285i	$0.248732\pi$
$654$	0	0
$655$	−6.76670	−0.264397
$656$	31.9836	1.24875
$657$	0	0
$658$	−4.81655	−0.187769
$659$	−15.1601	−0.590555	−0.295278	−	0.955412i	$-0.595412\pi$
$659$	−15.1601	−0.590555	−0.295278	+	0.955412i	$0.595412\pi$
$660$	0	0
$661$	2.70883	0.105361	0.0526805	−	0.998611i	$-0.483223\pi$
$661$	2.70883	0.105361	0.0526805	+	0.998611i	$0.483223\pi$
$662$	11.5419	0.448587
$663$	0	0
$664$	11.4808	0.445543
$665$	5.26458	0.204152
$666$	0	0
$667$	29.5271	1.14329
$668$	22.8395	0.883688
$669$	0	0
$670$	−26.8065	−1.03563
$671$	−24.0358	−0.927892
$672$	0	0
$673$	12.1551	0.468545	0.234273	−	0.972171i	$-0.424729\pi$
$673$	12.1551	0.468545	0.234273	+	0.972171i	$0.424729\pi$
$674$	−37.2066	−1.43314
$675$	0	0
$676$	16.3377	0.628375
$677$	−18.2524	−0.701498	−0.350749	−	0.936469i	$-0.614073\pi$
$677$	−18.2524	−0.701498	−0.350749	+	0.936469i	$0.614073\pi$
$678$	0	0
$679$	−4.99491	−0.191687
$680$	−6.81510	−0.261347
$681$	0	0
$682$	−24.1908	−0.926314
$683$	14.5033	0.554955	0.277478	−	0.960732i	$-0.410502\pi$
$683$	14.5033	0.554955	0.277478	+	0.960732i	$0.410502\pi$
$684$	0	0
$685$	0.470468	0.0179757
$686$	−1.70725	−0.0651832
$687$	0	0
$688$	41.5462	1.58393
$689$	−15.9010	−0.605781
$690$	0	0
$691$	34.0254	1.29439	0.647193	−	0.762326i	$-0.275943\pi$
$691$	34.0254	1.29439	0.647193	+	0.762326i	$0.275943\pi$
$692$	7.14273	0.271526
$693$	0	0
$694$	44.5309	1.69037
$695$	6.71825	0.254838
$696$	0	0
$697$	−21.6709	−0.820844
$698$	57.3173	2.16949
$699$	0	0
$700$	−3.49219	−0.131993
$701$	20.9705	0.792043	0.396022	−	0.918241i	$-0.370391\pi$
$701$	20.9705	0.792043	0.396022	+	0.918241i	$0.370391\pi$
$702$	0	0
$703$	19.0778	0.719531
$704$	−2.78284	−0.104882
$705$	0	0
$706$	−46.8351	−1.76266
$707$	−15.3793	−0.578398
$708$	0	0
$709$	−12.8542	−0.482749	−0.241375	−	0.970432i	$-0.577598\pi$
$709$	−12.8542	−0.482749	−0.241375	+	0.970432i	$0.577598\pi$
$710$	−8.53917	−0.320469
$711$	0	0
$712$	27.3508	1.02501
$713$	65.8867	2.46748
$714$	0	0
$715$	−9.55179	−0.357217
$716$	18.7064	0.699091
$717$	0	0
$718$	11.2352	0.419292
$719$	−16.2432	−0.605768	−0.302884	−	0.953027i	$-0.597949\pi$
$719$	−16.2432	−0.605768	−0.302884	+	0.953027i	$0.597949\pi$
$720$	0	0
$721$	18.4573	0.687387
$722$	−7.58858	−0.282418
$723$	0	0
$724$	−3.95409	−0.146953
$725$	15.3304	0.569358
$726$	0	0
$727$	−31.2233	−1.15801	−0.579003	−	0.815325i	$-0.696558\pi$
$727$	−31.2233	−1.15801	−0.579003	+	0.815325i	$0.696558\pi$
$728$	−10.2932	−0.381492
$729$	0	0
$730$	−21.1974	−0.784549
$731$	−28.1502	−1.04117
$732$	0	0
$733$	−9.59601	−0.354437	−0.177218	−	0.984172i	$-0.556710\pi$
$733$	−9.59601	−0.354437	−0.177218	+	0.984172i	$0.556710\pi$
$734$	−39.5724	−1.46065
$735$	0	0
$736$	−35.4290	−1.30593
$737$	22.8372	0.841218
$738$	0	0
$739$	16.5843	0.610065	0.305032	−	0.952342i	$-0.401333\pi$
$739$	16.5843	0.610065	0.305032	+	0.952342i	$0.401333\pi$
$740$	3.91854	0.144048
$741$	0	0
$742$	−4.88670	−0.179397
$743$	44.5444	1.63418	0.817088	−	0.576513i	$-0.195587\pi$
$743$	44.5444	1.63418	0.817088	+	0.576513i	$0.195587\pi$
$744$	0	0
$745$	14.5776	0.534082
$746$	−24.5937	−0.900440
$747$	0	0
$748$	−4.89337	−0.178919
$749$	3.34462	0.122210
$750$	0	0
$751$	−6.32997	−0.230984	−0.115492	−	0.993308i	$-0.536844\pi$
$751$	−6.32997	−0.230984	−0.115492	+	0.993308i	$0.536844\pi$
$752$	−14.0856	−0.513650
$753$	0	0
$754$	−38.0839	−1.38693
$755$	−12.2612	−0.446230
$756$	0	0
$757$	46.6198	1.69442	0.847212	−	0.531255i	$-0.178279\pi$
$757$	46.6198	1.69442	0.847212	+	0.531255i	$0.178279\pi$
$758$	−33.0436	−1.20020
$759$	0	0
$760$	9.75458	0.353836
$761$	−22.2626	−0.807017	−0.403509	−	0.914976i	$-0.632209\pi$
$761$	−22.2626	−0.807017	−0.403509	+	0.914976i	$0.632209\pi$
$762$	0	0
$763$	−2.86219	−0.103618
$764$	−17.2175	−0.622908
$765$	0	0
$766$	36.6968	1.32591
$767$	−17.7735	−0.641765
$768$	0	0
$769$	−45.6713	−1.64695	−0.823475	−	0.567352i	$-0.807968\pi$
$769$	−45.6713	−1.64695	−0.823475	+	0.567352i	$0.807968\pi$
$770$	−2.93546	−0.105787
$771$	0	0
$772$	3.68776	0.132725
$773$	18.8556	0.678189	0.339094	−	0.940752i	$-0.389879\pi$
$773$	18.8556	0.678189	0.339094	+	0.940752i	$0.389879\pi$
$774$	0	0
$775$	34.2083	1.22880
$776$	−9.25492	−0.332232
$777$	0	0
$778$	41.0059	1.47013
$779$	31.0180	1.11133
$780$	0	0
$781$	7.27473	0.260310
$782$	42.4686	1.51867
$783$	0	0
$784$	−4.99273	−0.178312
$785$	22.3866	0.799014
$786$	0	0
$787$	−12.6245	−0.450015	−0.225008	−	0.974357i	$-0.572241\pi$
$787$	−12.6245	−0.450015	−0.225008	+	0.974357i	$0.572241\pi$
$788$	−8.02484	−0.285873
$789$	0	0
$790$	10.3988	0.369972
$791$	6.64072	0.236117
$792$	0	0
$793$	−84.4358	−2.99840
$794$	39.1016	1.38766
$795$	0	0
$796$	−14.8177	−0.525200
$797$	30.7074	1.08771	0.543856	−	0.839178i	$-0.316964\pi$
$797$	30.7074	1.08771	0.543856	+	0.839178i	$0.316964\pi$
$798$	0	0
$799$	9.54391	0.337639
$800$	−18.3947	−0.650350
$801$	0	0
$802$	−61.8527	−2.18410
$803$	18.0586	0.637273
$804$	0	0
$805$	7.99508	0.281790
$806$	−84.9803	−2.99330
$807$	0	0
$808$	−28.4958	−1.00248
$809$	18.4310	0.648001	0.324001	−	0.946057i	$-0.394972\pi$
$809$	18.4310	0.648001	0.324001	+	0.946057i	$0.394972\pi$
$810$	0	0
$811$	7.64652	0.268506	0.134253	−	0.990947i	$-0.457137\pi$
$811$	7.64652	0.268506	0.134253	+	0.990947i	$0.457137\pi$
$812$	−3.67299	−0.128897
$813$	0	0
$814$	−10.6375	−0.372844
$815$	−14.8349	−0.519643
$816$	0	0
$817$	40.2919	1.40964
$818$	−17.5902	−0.615026
$819$	0	0
$820$	6.37104	0.222487
$821$	−9.83737	−0.343326	−0.171663	−	0.985156i	$-0.554914\pi$
$821$	−9.83737	−0.343326	−0.171663	+	0.985156i	$0.554914\pi$
$822$	0	0
$823$	−9.32310	−0.324983	−0.162491	−	0.986710i	$-0.551953\pi$
$823$	−9.32310	−0.324983	−0.162491	+	0.986710i	$0.551953\pi$
$824$	34.1990	1.19138
$825$	0	0
$826$	−5.46217	−0.190053
$827$	4.19232	0.145781	0.0728906	−	0.997340i	$-0.476778\pi$
$827$	4.19232	0.145781	0.0728906	+	0.997340i	$0.476778\pi$
$828$	0	0
$829$	3.57413	0.124135	0.0620673	−	0.998072i	$-0.480231\pi$
$829$	3.57413	0.124135	0.0620673	+	0.998072i	$0.480231\pi$
$830$	11.5018	0.399234
$831$	0	0
$832$	−9.77590	−0.338918
$833$	3.38289	0.117210
$834$	0	0
$835$	−27.1485	−0.939511
$836$	7.00397	0.242238
$837$	0	0
$838$	58.7616	2.02989
$839$	46.2021	1.59507	0.797537	−	0.603270i	$-0.206136\pi$
$839$	46.2021	1.59507	0.797537	+	0.603270i	$0.206136\pi$
$840$	0	0
$841$	−12.8759	−0.443996
$842$	−9.53164	−0.328482
$843$	0	0
$844$	−6.07477	−0.209102
$845$	−19.4200	−0.668070
$846$	0	0
$847$	−8.49921	−0.292036
$848$	−14.2908	−0.490748
$849$	0	0
$850$	22.0497	0.756297
$851$	28.9725	0.993164
$852$	0	0
$853$	−41.5480	−1.42257	−0.711287	−	0.702901i	$-0.751888\pi$
$853$	−41.5480	−1.42257	−0.711287	+	0.702901i	$0.751888\pi$
$854$	−25.9488	−0.887950
$855$	0	0
$856$	6.19713	0.211814
$857$	−34.9505	−1.19389	−0.596943	−	0.802284i	$-0.703618\pi$
$857$	−34.9505	−1.19389	−0.596943	+	0.802284i	$0.703618\pi$
$858$	0	0
$859$	14.3297	0.488922	0.244461	−	0.969659i	$-0.421389\pi$
$859$	14.3297	0.488922	0.244461	+	0.969659i	$0.421389\pi$
$860$	8.27590	0.282206
$861$	0	0
$862$	−10.7915	−0.367559
$863$	4.68464	0.159467	0.0797334	−	0.996816i	$-0.474593\pi$
$863$	4.68464	0.159467	0.0797334	+	0.996816i	$0.474593\pi$
$864$	0	0
$865$	−8.49028	−0.288678
$866$	−4.13871	−0.140639
$867$	0	0
$868$	−8.19590	−0.278187
$869$	−8.85899	−0.300521
$870$	0	0
$871$	80.2250	2.71832
$872$	−5.30327	−0.179591
$873$	0	0
$874$	−60.7862	−2.05612
$875$	9.58742	0.324114
$876$	0	0
$877$	−29.4874	−0.995719	−0.497859	−	0.867258i	$-0.665880\pi$
$877$	−29.4874	−0.995719	−0.497859	+	0.867258i	$0.665880\pi$
$878$	−34.5849	−1.16718
$879$	0	0
$880$	−8.58452	−0.289384
$881$	55.1122	1.85678	0.928389	−	0.371610i	$-0.121194\pi$
$881$	55.1122	1.85678	0.928389	+	0.371610i	$0.121194\pi$
$882$	0	0
$883$	−30.1318	−1.01402	−0.507008	−	0.861942i	$-0.669248\pi$
$883$	−30.1318	−1.01402	−0.507008	+	0.861942i	$0.669248\pi$
$884$	−17.1900	−0.578162
$885$	0	0
$886$	−39.0816	−1.31297
$887$	−23.2940	−0.782137	−0.391069	−	0.920362i	$-0.627894\pi$
$887$	−23.2940	−0.782137	−0.391069	+	0.920362i	$0.627894\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	27.4008	0.918476
$891$	0	0
$892$	17.3737	0.581714
$893$	−13.6604	−0.457128
$894$	0	0
$895$	−22.2356	−0.743253
$896$	−12.6405	−0.422290
$897$	0	0
$898$	−10.3708	−0.346080
$899$	35.9793	1.19998
$900$	0	0
$901$	9.68292	0.322585
$902$	−17.2952	−0.575866
$903$	0	0
$904$	12.3044	0.409238
$905$	4.70007	0.156236
$906$	0	0
$907$	−3.20187	−0.106316	−0.0531582	−	0.998586i	$-0.516929\pi$
$907$	−3.20187	−0.106316	−0.0531582	+	0.998586i	$0.516929\pi$
$908$	−8.40577	−0.278955
$909$	0	0
$910$	−10.3120	−0.341840
$911$	1.03289	0.0342212	0.0171106	−	0.999854i	$-0.494553\pi$
$911$	1.03289	0.0342212	0.0171106	+	0.999854i	$0.494553\pi$
$912$	0	0
$913$	−9.79868	−0.324289
$914$	−20.5427	−0.679492
$915$	0	0
$916$	0.647353	0.0213891
$917$	6.22354	0.205519
$918$	0	0
$919$	−3.56770	−0.117688	−0.0588438	−	0.998267i	$-0.518741\pi$
$919$	−3.56770	−0.117688	−0.0588438	+	0.998267i	$0.518741\pi$
$920$	14.8138	0.488398
$921$	0	0
$922$	−60.2172	−1.98315
$923$	25.5555	0.841171
$924$	0	0
$925$	15.0425	0.494594
$926$	−70.4804	−2.31613
$927$	0	0
$928$	−19.3470	−0.635096
$929$	0.360409	0.0118246	0.00591231	−	0.999983i	$-0.498118\pi$
$929$	0.360409	0.0118246	0.00591231	+	0.999983i	$0.498118\pi$
$930$	0	0
$931$	−4.84199	−0.158690
$932$	12.2847	0.402399
$933$	0	0
$934$	5.30407	0.173554
$935$	5.81656	0.190222
$936$	0	0
$937$	−11.7931	−0.385265	−0.192633	−	0.981271i	$-0.561703\pi$
$937$	−11.7931	−0.385265	−0.192633	+	0.981271i	$0.561703\pi$
$938$	24.6548	0.805007
$939$	0	0
$940$	−2.80582	−0.0915159
$941$	14.0874	0.459236	0.229618	−	0.973281i	$-0.426252\pi$
$941$	14.0874	0.459236	0.229618	+	0.973281i	$0.426252\pi$
$942$	0	0
$943$	47.1056	1.53397
$944$	−15.9737	−0.519899
$945$	0	0
$946$	−22.4662	−0.730439
$947$	42.7570	1.38941	0.694707	−	0.719292i	$-0.255534\pi$
$947$	42.7570	1.38941	0.694707	+	0.719292i	$0.255534\pi$
$948$	0	0
$949$	63.4382	2.05929
$950$	−31.5601	−1.02394
$951$	0	0
$952$	6.26805	0.203149
$953$	36.7785	1.19137	0.595686	−	0.803218i	$-0.296880\pi$
$953$	36.7785	1.19137	0.595686	+	0.803218i	$0.296880\pi$
$954$	0	0
$955$	20.4658	0.662257
$956$	0.569956	0.0184337
$957$	0	0
$958$	−26.5608	−0.858142
$959$	−0.432704	−0.0139727
$960$	0	0
$961$	49.2841	1.58981
$962$	−37.3686	−1.20481
$963$	0	0
$964$	0.846095	0.0272509
$965$	−4.38350	−0.141110
$966$	0	0
$967$	25.7951	0.829515	0.414757	−	0.909932i	$-0.363866\pi$
$967$	25.7951	0.829515	0.414757	+	0.909932i	$0.363866\pi$
$968$	−15.7479	−0.506157
$969$	0	0
$970$	−9.27183	−0.297700
$971$	40.2676	1.29225	0.646124	−	0.763232i	$-0.276389\pi$
$971$	40.2676	1.29225	0.646124	+	0.763232i	$0.276389\pi$
$972$	0	0
$973$	−6.17897	−0.198089
$974$	50.6520	1.62299
$975$	0	0
$976$	−75.8853	−2.42903
$977$	23.0942	0.738848	0.369424	−	0.929261i	$-0.379555\pi$
$977$	23.0942	0.738848	0.369424	+	0.929261i	$0.379555\pi$
$978$	0	0
$979$	−23.3434	−0.746058
$980$	−0.994538	−0.0317694
$981$	0	0
$982$	1.29842	0.0414344
$983$	−23.8772	−0.761563	−0.380781	−	0.924665i	$-0.624345\pi$
$983$	−23.8772	−0.761563	−0.380781	+	0.924665i	$0.624345\pi$
$984$	0	0
$985$	9.53882	0.303932
$986$	23.1912	0.738558
$987$	0	0
$988$	24.6044	0.782769
$989$	61.1895	1.94571
$990$	0	0
$991$	13.5689	0.431031	0.215516	−	0.976500i	$-0.430857\pi$
$991$	13.5689	0.431031	0.215516	+	0.976500i	$0.430857\pi$
$992$	−43.1708	−1.37067
$993$	0	0
$994$	7.85373	0.249105
$995$	17.6132	0.558378
$996$	0	0
$997$	53.5594	1.69624	0.848121	−	0.529803i	$-0.177734\pi$
$997$	53.5594	1.69624	0.848121	+	0.529803i	$0.177734\pi$
$998$	28.0564	0.888111
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.4		18
3.2	odd	2		2667.2.a.p.1.15	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.15	✓	18	3.2	odd	2
8001.2.a.u.1.4		18	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70725 q^{2} +0.914706 q^{4} -1.08728 q^{5} +1.00000 q^{7} +1.85287 q^{8} +O(q^{10})\)	\(q-1.70725 q^{2} +0.914706 q^{4} -1.08728 q^{5} +1.00000 q^{7} +1.85287 q^{8} +1.85625 q^{10} -1.58139 q^{11} -5.55528 q^{13} -1.70725 q^{14} -4.99273 q^{16} +3.38289 q^{17} -4.84199 q^{19} -0.994538 q^{20} +2.69983 q^{22} -7.35331 q^{23} -3.81783 q^{25} +9.48427 q^{26} +0.914706 q^{28} -4.01548 q^{29} -8.96014 q^{31} +4.81810 q^{32} -5.77544 q^{34} -1.08728 q^{35} -3.94006 q^{37} +8.26650 q^{38} -2.01458 q^{40} -6.40603 q^{41} -8.32135 q^{43} -1.44651 q^{44} +12.5539 q^{46} +2.82123 q^{47} +1.00000 q^{49} +6.51800 q^{50} -5.08145 q^{52} +2.86232 q^{53} +1.71941 q^{55} +1.85287 q^{56} +6.85544 q^{58} +3.19939 q^{59} +15.1992 q^{61} +15.2972 q^{62} +1.75975 q^{64} +6.04013 q^{65} -14.4412 q^{67} +3.09435 q^{68} +1.85625 q^{70} -4.60022 q^{71} -11.4194 q^{73} +6.72667 q^{74} -4.42900 q^{76} -1.58139 q^{77} +5.60203 q^{79} +5.42847 q^{80} +10.9367 q^{82} +6.19625 q^{83} -3.67813 q^{85} +14.2066 q^{86} -2.93010 q^{88} +14.7613 q^{89} -5.55528 q^{91} -6.72612 q^{92} -4.81655 q^{94} +5.26458 q^{95} -4.99491 q^{97} -1.70725 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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