LMFDB - Embedded newform 4014.2.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 14x^{6} + 28x^{5} + 43x^{4} - 90x^{3} - 23x^{2} + 82x - 28 $ x^8 - 2x^7 - 14x^6 + 28x^5 + 43x^4 - 90x^3 - 23x^2 + 82*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.16537$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.09191 q^{5} +1.60541 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09191 q^{5} +1.60541 q^{7} +1.00000 q^{8} -1.09191 q^{10} -0.449399 q^{11} +2.24619 q^{13} +1.60541 q^{14} +1.00000 q^{16} -7.95615 q^{17} -3.17102 q^{19} -1.09191 q^{20} -0.449399 q^{22} -6.84524 q^{23} -3.80772 q^{25} +2.24619 q^{26} +1.60541 q^{28} -2.86723 q^{29} -5.51730 q^{31} +1.00000 q^{32} -7.95615 q^{34} -1.75297 q^{35} +6.70619 q^{37} -3.17102 q^{38} -1.09191 q^{40} +0.500847 q^{41} +4.03871 q^{43} -0.449399 q^{44} -6.84524 q^{46} -4.16558 q^{47} -4.42265 q^{49} -3.80772 q^{50} +2.24619 q^{52} -0.688633 q^{53} +0.490705 q^{55} +1.60541 q^{56} -2.86723 q^{58} -0.561562 q^{59} +3.56206 q^{61} -5.51730 q^{62} +1.00000 q^{64} -2.45264 q^{65} -4.84401 q^{67} -7.95615 q^{68} -1.75297 q^{70} +1.34991 q^{71} -4.50443 q^{73} +6.70619 q^{74} -3.17102 q^{76} -0.721472 q^{77} -7.27055 q^{79} -1.09191 q^{80} +0.500847 q^{82} +11.1092 q^{83} +8.68743 q^{85} +4.03871 q^{86} -0.449399 q^{88} +10.1459 q^{89} +3.60606 q^{91} -6.84524 q^{92} -4.16558 q^{94} +3.46248 q^{95} +0.701838 q^{97} -4.42265 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} + 8 q^{8} - 6 q^{10} - 11 q^{11} - q^{13} - 6 q^{14} + 8 q^{16} - 16 q^{17} - q^{19} - 6 q^{20} - 11 q^{22} - 14 q^{23} + 10 q^{25} - q^{26} - 6 q^{28} - 21 q^{29} - 6 q^{31} + 8 q^{32} - 16 q^{34} - 8 q^{35} - 14 q^{37} - q^{38} - 6 q^{40} - 16 q^{41} - 29 q^{43} - 11 q^{44} - 14 q^{46} - 9 q^{47} - 2 q^{49} + 10 q^{50} - q^{52} - 11 q^{53} - 22 q^{55} - 6 q^{56} - 21 q^{58} - 21 q^{59} + 3 q^{61} - 6 q^{62} + 8 q^{64} - 24 q^{65} - 20 q^{67} - 16 q^{68} - 8 q^{70} - 32 q^{71} + 13 q^{73} - 14 q^{74} - q^{76} - 4 q^{77} + 21 q^{79} - 6 q^{80} - 16 q^{82} - 28 q^{83} - 14 q^{85} - 29 q^{86} - 11 q^{88} - 54 q^{89} - 36 q^{91} - 14 q^{92} - 9 q^{94} - 30 q^{95} + 10 q^{97} - 2 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 + 8 * q^8 - 6 * q^10 - 11 * q^11 - q^13 - 6 * q^14 + 8 * q^16 - 16 * q^17 - q^19 - 6 * q^20 - 11 * q^22 - 14 * q^23 + 10 * q^25 - q^26 - 6 * q^28 - 21 * q^29 - 6 * q^31 + 8 * q^32 - 16 * q^34 - 8 * q^35 - 14 * q^37 - q^38 - 6 * q^40 - 16 * q^41 - 29 * q^43 - 11 * q^44 - 14 * q^46 - 9 * q^47 - 2 * q^49 + 10 * q^50 - q^52 - 11 * q^53 - 22 * q^55 - 6 * q^56 - 21 * q^58 - 21 * q^59 + 3 * q^61 - 6 * q^62 + 8 * q^64 - 24 * q^65 - 20 * q^67 - 16 * q^68 - 8 * q^70 - 32 * q^71 + 13 * q^73 - 14 * q^74 - q^76 - 4 * q^77 + 21 * q^79 - 6 * q^80 - 16 * q^82 - 28 * q^83 - 14 * q^85 - 29 * q^86 - 11 * q^88 - 54 * q^89 - 36 * q^91 - 14 * q^92 - 9 * q^94 - 30 * q^95 + 10 * q^97 - 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.09191	−0.488319	−0.244159	−	0.969735i	$-0.578512\pi$
$5$	−1.09191	−0.488319	−0.244159	+	0.969735i	$0.578512\pi$
$6$	0	0
$7$	1.60541	0.606789	0.303395	−	0.952865i	$-0.401880\pi$
$7$	1.60541	0.606789	0.303395	+	0.952865i	$0.401880\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.09191	−0.345294
$11$	−0.449399	−0.135499	−0.0677495	−	0.997702i	$-0.521582\pi$
$11$	−0.449399	−0.135499	−0.0677495	+	0.997702i	$0.521582\pi$
$12$	0	0
$13$	2.24619	0.622980	0.311490	−	0.950249i	$-0.399172\pi$
$13$	2.24619	0.622980	0.311490	+	0.950249i	$0.399172\pi$
$14$	1.60541	0.429065
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.95615	−1.92965	−0.964825	−	0.262895i	$-0.915323\pi$
$17$	−7.95615	−1.92965	−0.964825	+	0.262895i	$0.915323\pi$
$18$	0	0
$19$	−3.17102	−0.727481	−0.363741	−	0.931500i	$-0.618501\pi$
$19$	−3.17102	−0.727481	−0.363741	+	0.931500i	$0.618501\pi$
$20$	−1.09191	−0.244159
$21$	0	0
$22$	−0.449399	−0.0958122
$23$	−6.84524	−1.42733	−0.713665	−	0.700487i	$-0.752966\pi$
$23$	−6.84524	−1.42733	−0.713665	+	0.700487i	$0.752966\pi$
$24$	0	0
$25$	−3.80772	−0.761545
$26$	2.24619	0.440513
$27$	0	0
$28$	1.60541	0.303395
$29$	−2.86723	−0.532430	−0.266215	−	0.963914i	$-0.585773\pi$
$29$	−2.86723	−0.532430	−0.266215	+	0.963914i	$0.585773\pi$
$30$	0	0
$31$	−5.51730	−0.990937	−0.495468	−	0.868626i	$-0.665004\pi$
$31$	−5.51730	−0.990937	−0.495468	+	0.868626i	$0.665004\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.95615	−1.36447
$35$	−1.75297	−0.296307
$36$	0	0
$37$	6.70619	1.10249	0.551245	−	0.834343i	$-0.314153\pi$
$37$	6.70619	1.10249	0.551245	+	0.834343i	$0.314153\pi$
$38$	−3.17102	−0.514407
$39$	0	0
$40$	−1.09191	−0.172647
$41$	0.500847	0.0782191	0.0391096	−	0.999235i	$-0.487548\pi$
$41$	0.500847	0.0782191	0.0391096	+	0.999235i	$0.487548\pi$
$42$	0	0
$43$	4.03871	0.615898	0.307949	−	0.951403i	$-0.400357\pi$
$43$	4.03871	0.615898	0.307949	+	0.951403i	$0.400357\pi$
$44$	−0.449399	−0.0677495
$45$	0	0
$46$	−6.84524	−1.00928
$47$	−4.16558	−0.607613	−0.303806	−	0.952734i	$-0.598258\pi$
$47$	−4.16558	−0.607613	−0.303806	+	0.952734i	$0.598258\pi$
$48$	0	0
$49$	−4.42265	−0.631807
$50$	−3.80772	−0.538493
$51$	0	0
$52$	2.24619	0.311490
$53$	−0.688633	−0.0945910	−0.0472955	−	0.998881i	$-0.515060\pi$
$53$	−0.688633	−0.0945910	−0.0472955	+	0.998881i	$0.515060\pi$
$54$	0	0
$55$	0.490705	0.0661667
$56$	1.60541	0.214532
$57$	0	0
$58$	−2.86723	−0.376485
$59$	−0.561562	−0.0731091	−0.0365546	−	0.999332i	$-0.511638\pi$
$59$	−0.561562	−0.0731091	−0.0365546	+	0.999332i	$0.511638\pi$
$60$	0	0
$61$	3.56206	0.456075	0.228037	−	0.973652i	$-0.426769\pi$
$61$	3.56206	0.456075	0.228037	+	0.973652i	$0.426769\pi$
$62$	−5.51730	−0.700698
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.45264	−0.304213
$66$	0	0
$67$	−4.84401	−0.591790	−0.295895	−	0.955221i	$-0.595618\pi$
$67$	−4.84401	−0.591790	−0.295895	+	0.955221i	$0.595618\pi$
$68$	−7.95615	−0.964825
$69$	0	0
$70$	−1.75297	−0.209520
$71$	1.34991	0.160205	0.0801026	−	0.996787i	$-0.474475\pi$
$71$	1.34991	0.160205	0.0801026	+	0.996787i	$0.474475\pi$
$72$	0	0
$73$	−4.50443	−0.527204	−0.263602	−	0.964632i	$-0.584911\pi$
$73$	−4.50443	−0.527204	−0.263602	+	0.964632i	$0.584911\pi$
$74$	6.70619	0.779579
$75$	0	0
$76$	−3.17102	−0.363741
$77$	−0.721472	−0.0822193
$78$	0	0
$79$	−7.27055	−0.818001	−0.409000	−	0.912534i	$-0.634122\pi$
$79$	−7.27055	−0.818001	−0.409000	+	0.912534i	$0.634122\pi$
$80$	−1.09191	−0.122080
$81$	0	0
$82$	0.500847	0.0553093
$83$	11.1092	1.21939	0.609695	−	0.792636i	$-0.291292\pi$
$83$	11.1092	1.21939	0.609695	+	0.792636i	$0.291292\pi$
$84$	0	0
$85$	8.68743	0.942284
$86$	4.03871	0.435505
$87$	0	0
$88$	−0.449399	−0.0479061
$89$	10.1459	1.07546	0.537730	−	0.843117i	$-0.319282\pi$
$89$	10.1459	1.07546	0.537730	+	0.843117i	$0.319282\pi$
$90$	0	0
$91$	3.60606	0.378018
$92$	−6.84524	−0.713665
$93$	0	0
$94$	−4.16558	−0.429647
$95$	3.46248	0.355243
$96$	0	0
$97$	0.701838	0.0712608	0.0356304	−	0.999365i	$-0.488656\pi$
$97$	0.701838	0.0712608	0.0356304	+	0.999365i	$0.488656\pi$
$98$	−4.42265	−0.446755
$99$	0	0
$100$	−3.80772	−0.380772
$101$	−15.3690	−1.52927	−0.764634	−	0.644465i	$-0.777080\pi$
$101$	−15.3690	−1.52927	−0.764634	+	0.644465i	$0.777080\pi$
$102$	0	0
$103$	17.3502	1.70956	0.854781	−	0.518989i	$-0.173691\pi$
$103$	17.3502	1.70956	0.854781	+	0.518989i	$0.173691\pi$
$104$	2.24619	0.220257
$105$	0	0
$106$	−0.688633	−0.0668860
$107$	−11.0927	−1.07237	−0.536184	−	0.844101i	$-0.680135\pi$
$107$	−11.0927	−1.07237	−0.536184	+	0.844101i	$0.680135\pi$
$108$	0	0
$109$	18.4837	1.77041	0.885207	−	0.465198i	$-0.154017\pi$
$109$	18.4837	1.77041	0.885207	+	0.465198i	$0.154017\pi$
$110$	0.490705	0.0467869
$111$	0	0
$112$	1.60541	0.151697
$113$	−14.4163	−1.35617	−0.678085	−	0.734983i	$-0.737190\pi$
$113$	−14.4163	−1.35617	−0.678085	+	0.734983i	$0.737190\pi$
$114$	0	0
$115$	7.47441	0.696993
$116$	−2.86723	−0.266215
$117$	0	0
$118$	−0.561562	−0.0516959
$119$	−12.7729	−1.17089
$120$	0	0
$121$	−10.7980	−0.981640
$122$	3.56206	0.322494
$123$	0	0
$124$	−5.51730	−0.495468
$125$	9.61728	0.860195
$126$	0	0
$127$	−2.33654	−0.207334	−0.103667	−	0.994612i	$-0.533058\pi$
$127$	−2.33654	−0.207334	−0.103667	+	0.994612i	$0.533058\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.45264	−0.215111
$131$	13.1744	1.15106	0.575528	−	0.817782i	$-0.304797\pi$
$131$	13.1744	1.15106	0.575528	+	0.817782i	$0.304797\pi$
$132$	0	0
$133$	−5.09080	−0.441428
$134$	−4.84401	−0.418459
$135$	0	0
$136$	−7.95615	−0.682234
$137$	−14.9941	−1.28103	−0.640517	−	0.767944i	$-0.721280\pi$
$137$	−14.9941	−1.28103	−0.640517	+	0.767944i	$0.721280\pi$
$138$	0	0
$139$	−8.56880	−0.726796	−0.363398	−	0.931634i	$-0.618384\pi$
$139$	−8.56880	−0.726796	−0.363398	+	0.931634i	$0.618384\pi$
$140$	−1.75297	−0.148153
$141$	0	0
$142$	1.34991	0.113282
$143$	−1.00943	−0.0844131
$144$	0	0
$145$	3.13076	0.259996
$146$	−4.50443	−0.372789
$147$	0	0
$148$	6.70619	0.551245
$149$	9.04419	0.740929	0.370464	−	0.928847i	$-0.379199\pi$
$149$	9.04419	0.740929	0.370464	+	0.928847i	$0.379199\pi$
$150$	0	0
$151$	−11.2276	−0.913689	−0.456845	−	0.889547i	$-0.651020\pi$
$151$	−11.2276	−0.913689	−0.456845	+	0.889547i	$0.651020\pi$
$152$	−3.17102	−0.257203
$153$	0	0
$154$	−0.721472	−0.0581378
$155$	6.02442	0.483893
$156$	0	0
$157$	−6.05283	−0.483069	−0.241534	−	0.970392i	$-0.577651\pi$
$157$	−6.05283	−0.483069	−0.241534	+	0.970392i	$0.577651\pi$
$158$	−7.27055	−0.578414
$159$	0	0
$160$	−1.09191	−0.0863234
$161$	−10.9894	−0.866089
$162$	0	0
$163$	−17.5320	−1.37321	−0.686606	−	0.727030i	$-0.740900\pi$
$163$	−17.5320	−1.37321	−0.686606	+	0.727030i	$0.740900\pi$
$164$	0.500847	0.0391096
$165$	0	0
$166$	11.1092	0.862239
$167$	4.65235	0.360009	0.180005	−	0.983666i	$-0.442389\pi$
$167$	4.65235	0.360009	0.180005	+	0.983666i	$0.442389\pi$
$168$	0	0
$169$	−7.95465	−0.611896
$170$	8.68743	0.666295
$171$	0	0
$172$	4.03871	0.307949
$173$	−7.97464	−0.606301	−0.303150	−	0.952943i	$-0.598038\pi$
$173$	−7.97464	−0.606301	−0.303150	+	0.952943i	$0.598038\pi$
$174$	0	0
$175$	−6.11297	−0.462097
$176$	−0.449399	−0.0338747
$177$	0	0
$178$	10.1459	0.760465
$179$	−5.42490	−0.405476	−0.202738	−	0.979233i	$-0.564984\pi$
$179$	−5.42490	−0.405476	−0.202738	+	0.979233i	$0.564984\pi$
$180$	0	0
$181$	0.400729	0.0297860	0.0148930	−	0.999889i	$-0.495259\pi$
$181$	0.400729	0.0297860	0.0148930	+	0.999889i	$0.495259\pi$
$182$	3.60606	0.267299
$183$	0	0
$184$	−6.84524	−0.504638
$185$	−7.32258	−0.538367
$186$	0	0
$187$	3.57549	0.261465
$188$	−4.16558	−0.303806
$189$	0	0
$190$	3.46248	0.251195
$191$	−8.85349	−0.640616	−0.320308	−	0.947313i	$-0.603786\pi$
$191$	−8.85349	−0.640616	−0.320308	+	0.947313i	$0.603786\pi$
$192$	0	0
$193$	4.35343	0.313367	0.156684	−	0.987649i	$-0.449920\pi$
$193$	4.35343	0.313367	0.156684	+	0.987649i	$0.449920\pi$
$194$	0.701838	0.0503890
$195$	0	0
$196$	−4.42265	−0.315903
$197$	−9.04254	−0.644254	−0.322127	−	0.946696i	$-0.604398\pi$
$197$	−9.04254	−0.644254	−0.322127	+	0.946696i	$0.604398\pi$
$198$	0	0
$199$	5.43241	0.385093	0.192547	−	0.981288i	$-0.438325\pi$
$199$	5.43241	0.385093	0.192547	+	0.981288i	$0.438325\pi$
$200$	−3.80772	−0.269247
$201$	0	0
$202$	−15.3690	−1.08136
$203$	−4.60308	−0.323073
$204$	0	0
$205$	−0.546881	−0.0381959
$206$	17.3502	1.20884
$207$	0	0
$208$	2.24619	0.155745
$209$	1.42505	0.0985729
$210$	0	0
$211$	6.89567	0.474718	0.237359	−	0.971422i	$-0.423718\pi$
$211$	6.89567	0.474718	0.237359	+	0.971422i	$0.423718\pi$
$212$	−0.688633	−0.0472955
$213$	0	0
$214$	−11.0927	−0.758279
$215$	−4.40992	−0.300754
$216$	0	0
$217$	−8.85756	−0.601290
$218$	18.4837	1.25187
$219$	0	0
$220$	0.490705	0.0330833
$221$	−17.8710	−1.20213
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	1.60541	0.107266
$225$	0	0
$226$	−14.4163	−0.958957
$227$	−13.7181	−0.910505	−0.455252	−	0.890362i	$-0.650451\pi$
$227$	−13.7181	−0.910505	−0.455252	+	0.890362i	$0.650451\pi$
$228$	0	0
$229$	−12.6145	−0.833587	−0.416794	−	0.909001i	$-0.636846\pi$
$229$	−12.6145	−0.833587	−0.416794	+	0.909001i	$0.636846\pi$
$230$	7.47441	0.492848
$231$	0	0
$232$	−2.86723	−0.188243
$233$	11.0552	0.724248	0.362124	−	0.932130i	$-0.382052\pi$
$233$	11.0552	0.724248	0.362124	+	0.932130i	$0.382052\pi$
$234$	0	0
$235$	4.54846	0.296709
$236$	−0.561562	−0.0365546
$237$	0	0
$238$	−12.7729	−0.827945
$239$	12.1340	0.784886	0.392443	−	0.919776i	$-0.371630\pi$
$239$	12.1340	0.784886	0.392443	+	0.919776i	$0.371630\pi$
$240$	0	0
$241$	−5.31832	−0.342583	−0.171291	−	0.985220i	$-0.554794\pi$
$241$	−5.31832	−0.342583	−0.171291	+	0.985220i	$0.554794\pi$
$242$	−10.7980	−0.694124
$243$	0	0
$244$	3.56206	0.228037
$245$	4.82915	0.308523
$246$	0	0
$247$	−7.12269	−0.453206
$248$	−5.51730	−0.350349
$249$	0	0
$250$	9.61728	0.608250
$251$	12.2148	0.770991	0.385495	−	0.922710i	$-0.374031\pi$
$251$	12.2148	0.770991	0.385495	+	0.922710i	$0.374031\pi$
$252$	0	0
$253$	3.07624	0.193402
$254$	−2.33654	−0.146608
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.8331	0.738127	0.369064	−	0.929404i	$-0.379678\pi$
$257$	11.8331	0.738127	0.369064	+	0.929404i	$0.379678\pi$
$258$	0	0
$259$	10.7662	0.668980
$260$	−2.45264	−0.152106
$261$	0	0
$262$	13.1744	0.813919
$263$	−7.60976	−0.469238	−0.234619	−	0.972087i	$-0.575384\pi$
$263$	−7.60976	−0.469238	−0.234619	+	0.972087i	$0.575384\pi$
$264$	0	0
$265$	0.751928	0.0461906
$266$	−5.09080	−0.312137
$267$	0	0
$268$	−4.84401	−0.295895
$269$	−2.11682	−0.129065	−0.0645323	−	0.997916i	$-0.520556\pi$
$269$	−2.11682	−0.129065	−0.0645323	+	0.997916i	$0.520556\pi$
$270$	0	0
$271$	17.9785	1.09211	0.546057	−	0.837748i	$-0.316128\pi$
$271$	17.9785	1.09211	0.546057	+	0.837748i	$0.316128\pi$
$272$	−7.95615	−0.482412
$273$	0	0
$274$	−14.9941	−0.905828
$275$	1.71119	0.103188
$276$	0	0
$277$	18.3721	1.10387	0.551937	−	0.833886i	$-0.313889\pi$
$277$	18.3721	1.10387	0.551937	+	0.833886i	$0.313889\pi$
$278$	−8.56880	−0.513923
$279$	0	0
$280$	−1.75297	−0.104760
$281$	−12.2253	−0.729299	−0.364650	−	0.931145i	$-0.618811\pi$
$281$	−12.2253	−0.729299	−0.364650	+	0.931145i	$0.618811\pi$
$282$	0	0
$283$	−3.36534	−0.200049	−0.100024	−	0.994985i	$-0.531892\pi$
$283$	−3.36534	−0.200049	−0.100024	+	0.994985i	$0.531892\pi$
$284$	1.34991	0.0801026
$285$	0	0
$286$	−1.00943	−0.0596891
$287$	0.804066	0.0474625
$288$	0	0
$289$	46.3003	2.72355
$290$	3.13076	0.183845
$291$	0	0
$292$	−4.50443	−0.263602
$293$	−10.5218	−0.614691	−0.307345	−	0.951598i	$-0.599441\pi$
$293$	−10.5218	−0.614691	−0.307345	+	0.951598i	$0.599441\pi$
$294$	0	0
$295$	0.613177	0.0357006
$296$	6.70619	0.389789
$297$	0	0
$298$	9.04419	0.523916
$299$	−15.3757	−0.889198
$300$	0	0
$301$	6.48380	0.373720
$302$	−11.2276	−0.646076
$303$	0	0
$304$	−3.17102	−0.181870
$305$	−3.88946	−0.222710
$306$	0	0
$307$	2.68123	0.153026	0.0765129	−	0.997069i	$-0.475621\pi$
$307$	2.68123	0.153026	0.0765129	+	0.997069i	$0.475621\pi$
$308$	−0.721472	−0.0411097
$309$	0	0
$310$	6.02442	0.342164
$311$	10.3840	0.588824	0.294412	−	0.955679i	$-0.404876\pi$
$311$	10.3840	0.588824	0.294412	+	0.955679i	$0.404876\pi$
$312$	0	0
$313$	25.6770	1.45135	0.725675	−	0.688037i	$-0.241528\pi$
$313$	25.6770	1.45135	0.725675	+	0.688037i	$0.241528\pi$
$314$	−6.05283	−0.341581
$315$	0	0
$316$	−7.27055	−0.409000
$317$	11.5721	0.649954	0.324977	−	0.945722i	$-0.394643\pi$
$317$	11.5721	0.649954	0.324977	+	0.945722i	$0.394643\pi$
$318$	0	0
$319$	1.28853	0.0721437
$320$	−1.09191	−0.0610399
$321$	0	0
$322$	−10.9894	−0.612418
$323$	25.2291	1.40378
$324$	0	0
$325$	−8.55286	−0.474427
$326$	−17.5320	−0.971008
$327$	0	0
$328$	0.500847	0.0276546
$329$	−6.68749	−0.368693
$330$	0	0
$331$	3.32446	0.182729	0.0913644	−	0.995818i	$-0.470877\pi$
$331$	3.32446	0.182729	0.0913644	+	0.995818i	$0.470877\pi$
$332$	11.1092	0.609695
$333$	0	0
$334$	4.65235	0.254565
$335$	5.28924	0.288982
$336$	0	0
$337$	5.12244	0.279037	0.139518	−	0.990219i	$-0.455445\pi$
$337$	5.12244	0.279037	0.139518	+	0.990219i	$0.455445\pi$
$338$	−7.95465	−0.432676
$339$	0	0
$340$	8.68743	0.471142
$341$	2.47947	0.134271
$342$	0	0
$343$	−18.3381	−0.990163
$344$	4.03871	0.217753
$345$	0	0
$346$	−7.97464	−0.428720
$347$	−35.3904	−1.89986	−0.949929	−	0.312465i	$-0.898845\pi$
$347$	−35.3904	−1.89986	−0.949929	+	0.312465i	$0.898845\pi$
$348$	0	0
$349$	1.36609	0.0731253	0.0365626	−	0.999331i	$-0.488359\pi$
$349$	1.36609	0.0731253	0.0365626	+	0.999331i	$0.488359\pi$
$350$	−6.11297	−0.326752
$351$	0	0
$352$	−0.449399	−0.0239531
$353$	7.39739	0.393723	0.196862	−	0.980431i	$-0.436925\pi$
$353$	7.39739	0.393723	0.196862	+	0.980431i	$0.436925\pi$
$354$	0	0
$355$	−1.47399	−0.0782312
$356$	10.1459	0.537730
$357$	0	0
$358$	−5.42490	−0.286715
$359$	−19.4435	−1.02619	−0.513093	−	0.858333i	$-0.671500\pi$
$359$	−19.4435	−1.02619	−0.513093	+	0.858333i	$0.671500\pi$
$360$	0	0
$361$	−8.94465	−0.470771
$362$	0.400729	0.0210619
$363$	0	0
$364$	3.60606	0.189009
$365$	4.91845	0.257444
$366$	0	0
$367$	30.9241	1.61422	0.807112	−	0.590398i	$-0.201029\pi$
$367$	30.9241	1.61422	0.807112	+	0.590398i	$0.201029\pi$
$368$	−6.84524	−0.356833
$369$	0	0
$370$	−7.32258	−0.380683
$371$	−1.10554	−0.0573968
$372$	0	0
$373$	13.1506	0.680910	0.340455	−	0.940261i	$-0.389419\pi$
$373$	13.1506	0.680910	0.340455	+	0.940261i	$0.389419\pi$
$374$	3.57549	0.184884
$375$	0	0
$376$	−4.16558	−0.214824
$377$	−6.44032	−0.331693
$378$	0	0
$379$	16.3105	0.837812	0.418906	−	0.908030i	$-0.362414\pi$
$379$	16.3105	0.837812	0.418906	+	0.908030i	$0.362414\pi$
$380$	3.46248	0.177621
$381$	0	0
$382$	−8.85349	−0.452984
$383$	−9.84463	−0.503037	−0.251519	−	0.967852i	$-0.580930\pi$
$383$	−9.84463	−0.503037	−0.251519	+	0.967852i	$0.580930\pi$
$384$	0	0
$385$	0.787785	0.0401492
$386$	4.35343	0.221584
$387$	0	0
$388$	0.701838	0.0356304
$389$	27.6443	1.40162	0.700811	−	0.713347i	$-0.252822\pi$
$389$	27.6443	1.40162	0.700811	+	0.713347i	$0.252822\pi$
$390$	0	0
$391$	54.4617	2.75425
$392$	−4.42265	−0.223377
$393$	0	0
$394$	−9.04254	−0.455557
$395$	7.93881	0.399445
$396$	0	0
$397$	17.5979	0.883215	0.441607	−	0.897208i	$-0.354408\pi$
$397$	17.5979	0.883215	0.441607	+	0.897208i	$0.354408\pi$
$398$	5.43241	0.272302
$399$	0	0
$400$	−3.80772	−0.190386
$401$	−29.6039	−1.47835	−0.739175	−	0.673514i	$-0.764784\pi$
$401$	−29.6039	−1.47835	−0.739175	+	0.673514i	$0.764784\pi$
$402$	0	0
$403$	−12.3929	−0.617334
$404$	−15.3690	−0.764634
$405$	0	0
$406$	−4.60308	−0.228447
$407$	−3.01376	−0.149386
$408$	0	0
$409$	−22.3741	−1.10633	−0.553164	−	0.833072i	$-0.686580\pi$
$409$	−22.3741	−1.10633	−0.553164	+	0.833072i	$0.686580\pi$
$410$	−0.546881	−0.0270086
$411$	0	0
$412$	17.3502	0.854781
$413$	−0.901539	−0.0443618
$414$	0	0
$415$	−12.1303	−0.595451
$416$	2.24619	0.110128
$417$	0	0
$418$	1.42505	0.0697016
$419$	18.8267	0.919746	0.459873	−	0.887985i	$-0.347895\pi$
$419$	18.8267	0.919746	0.459873	+	0.887985i	$0.347895\pi$
$420$	0	0
$421$	−8.21829	−0.400535	−0.200267	−	0.979741i	$-0.564181\pi$
$421$	−8.21829	−0.400535	−0.200267	+	0.979741i	$0.564181\pi$
$422$	6.89567	0.335676
$423$	0	0
$424$	−0.688633	−0.0334430
$425$	30.2948	1.46951
$426$	0	0
$427$	5.71858	0.276741
$428$	−11.0927	−0.536184
$429$	0	0
$430$	−4.40992	−0.212665
$431$	−3.68113	−0.177314	−0.0886570	−	0.996062i	$-0.528257\pi$
$431$	−3.68113	−0.177314	−0.0886570	+	0.996062i	$0.528257\pi$
$432$	0	0
$433$	19.5290	0.938506	0.469253	−	0.883064i	$-0.344523\pi$
$433$	19.5290	0.938506	0.469253	+	0.883064i	$0.344523\pi$
$434$	−8.85756	−0.425176
$435$	0	0
$436$	18.4837	0.885207
$437$	21.7064	1.03836
$438$	0	0
$439$	18.0868	0.863238	0.431619	−	0.902056i	$-0.357943\pi$
$439$	18.0868	0.863238	0.431619	+	0.902056i	$0.357943\pi$
$440$	0.490705	0.0233935
$441$	0	0
$442$	−17.8710	−0.850036
$443$	10.5885	0.503073	0.251537	−	0.967848i	$-0.419064\pi$
$443$	10.5885	0.503073	0.251537	+	0.967848i	$0.419064\pi$
$444$	0	0
$445$	−11.0784	−0.525168
$446$	1.00000	0.0473514
$447$	0	0
$448$	1.60541	0.0758487
$449$	−29.8328	−1.40790	−0.703949	−	0.710251i	$-0.748582\pi$
$449$	−29.8328	−1.40790	−0.703949	+	0.710251i	$0.748582\pi$
$450$	0	0
$451$	−0.225080	−0.0105986
$452$	−14.4163	−0.678085
$453$	0	0
$454$	−13.7181	−0.643824
$455$	−3.93751	−0.184593
$456$	0	0
$457$	−9.46149	−0.442590	−0.221295	−	0.975207i	$-0.571028\pi$
$457$	−9.46149	−0.442590	−0.221295	+	0.975207i	$0.571028\pi$
$458$	−12.6145	−0.589435
$459$	0	0
$460$	7.47441	0.348496
$461$	30.5173	1.42133	0.710667	−	0.703528i	$-0.248393\pi$
$461$	30.5173	1.42133	0.710667	+	0.703528i	$0.248393\pi$
$462$	0	0
$463$	−1.24315	−0.0577739	−0.0288869	−	0.999583i	$-0.509196\pi$
$463$	−1.24315	−0.0577739	−0.0288869	+	0.999583i	$0.509196\pi$
$464$	−2.86723	−0.133108
$465$	0	0
$466$	11.0552	0.512121
$467$	17.2797	0.799608	0.399804	−	0.916601i	$-0.369078\pi$
$467$	17.2797	0.799608	0.399804	+	0.916601i	$0.369078\pi$
$468$	0	0
$469$	−7.77664	−0.359092
$470$	4.54846	0.209805
$471$	0	0
$472$	−0.561562	−0.0258480
$473$	−1.81499	−0.0834535
$474$	0	0
$475$	12.0744	0.554010
$476$	−12.7729	−0.585445
$477$	0	0
$478$	12.1340	0.554998
$479$	−24.7125	−1.12914	−0.564571	−	0.825385i	$-0.690958\pi$
$479$	−24.7125	−1.12914	−0.564571	+	0.825385i	$0.690958\pi$
$480$	0	0
$481$	15.0633	0.686830
$482$	−5.31832	−0.242243
$483$	0	0
$484$	−10.7980	−0.490820
$485$	−0.766346	−0.0347980
$486$	0	0
$487$	−31.1571	−1.41186	−0.705932	−	0.708279i	$-0.749472\pi$
$487$	−31.1571	−1.41186	−0.705932	+	0.708279i	$0.749472\pi$
$488$	3.56206	0.161247
$489$	0	0
$490$	4.82915	0.218159
$491$	−39.6086	−1.78751	−0.893756	−	0.448553i	$-0.851940\pi$
$491$	−39.6086	−1.78751	−0.893756	+	0.448553i	$0.851940\pi$
$492$	0	0
$493$	22.8121	1.02740
$494$	−7.12269	−0.320465
$495$	0	0
$496$	−5.51730	−0.247734
$497$	2.16717	0.0972108
$498$	0	0
$499$	−14.1849	−0.635001	−0.317501	−	0.948258i	$-0.602844\pi$
$499$	−14.1849	−0.635001	−0.317501	+	0.948258i	$0.602844\pi$
$500$	9.61728	0.430098
$501$	0	0
$502$	12.2148	0.545173
$503$	−22.7136	−1.01275	−0.506374	−	0.862314i	$-0.669014\pi$
$503$	−22.7136	−1.01275	−0.506374	+	0.862314i	$0.669014\pi$
$504$	0	0
$505$	16.7816	0.746770
$506$	3.07624	0.136756
$507$	0	0
$508$	−2.33654	−0.103667
$509$	−8.98179	−0.398111	−0.199055	−	0.979988i	$-0.563787\pi$
$509$	−8.98179	−0.398111	−0.199055	+	0.979988i	$0.563787\pi$
$510$	0	0
$511$	−7.23148	−0.319902
$512$	1.00000	0.0441942
$513$	0	0
$514$	11.8331	0.521935
$515$	−18.9449	−0.834811
$516$	0	0
$517$	1.87201	0.0823309
$518$	10.7662	0.473040
$519$	0	0
$520$	−2.45264	−0.107555
$521$	−14.0077	−0.613688	−0.306844	−	0.951760i	$-0.599273\pi$
$521$	−14.0077	−0.613688	−0.306844	+	0.951760i	$0.599273\pi$
$522$	0	0
$523$	−36.4788	−1.59511	−0.797553	−	0.603249i	$-0.793872\pi$
$523$	−36.4788	−1.59511	−0.797553	+	0.603249i	$0.793872\pi$
$524$	13.1744	0.575528
$525$	0	0
$526$	−7.60976	−0.331801
$527$	43.8965	1.91216
$528$	0	0
$529$	23.8573	1.03727
$530$	0.751928	0.0326617
$531$	0	0
$532$	−5.09080	−0.220714
$533$	1.12499	0.0487289
$534$	0	0
$535$	12.1122	0.523658
$536$	−4.84401	−0.209229
$537$	0	0
$538$	−2.11682	−0.0912624
$539$	1.98753	0.0856091
$540$	0	0
$541$	11.4950	0.494211	0.247105	−	0.968989i	$-0.420521\pi$
$541$	11.4950	0.494211	0.247105	+	0.968989i	$0.420521\pi$
$542$	17.9785	0.772241
$543$	0	0
$544$	−7.95615	−0.341117
$545$	−20.1826	−0.864526
$546$	0	0
$547$	8.27961	0.354011	0.177005	−	0.984210i	$-0.443359\pi$
$547$	8.27961	0.354011	0.177005	+	0.984210i	$0.443359\pi$
$548$	−14.9941	−0.640517
$549$	0	0
$550$	1.71119	0.0729653
$551$	9.09202	0.387333
$552$	0	0
$553$	−11.6722	−0.496354
$554$	18.3721	0.780557
$555$	0	0
$556$	−8.56880	−0.363398
$557$	11.1722	0.473380	0.236690	−	0.971585i	$-0.423937\pi$
$557$	11.1722	0.473380	0.236690	+	0.971585i	$0.423937\pi$
$558$	0	0
$559$	9.07169	0.383692
$560$	−1.75297	−0.0740767
$561$	0	0
$562$	−12.2253	−0.515693
$563$	44.6884	1.88339	0.941696	−	0.336465i	$-0.109231\pi$
$563$	44.6884	1.88339	0.941696	+	0.336465i	$0.109231\pi$
$564$	0	0
$565$	15.7413	0.662243
$566$	−3.36534	−0.141456
$567$	0	0
$568$	1.34991	0.0566411
$569$	14.5066	0.608148	0.304074	−	0.952648i	$-0.401653\pi$
$569$	14.5066	0.608148	0.304074	+	0.952648i	$0.401653\pi$
$570$	0	0
$571$	−14.0799	−0.589228	−0.294614	−	0.955616i	$-0.595191\pi$
$571$	−14.0799	−0.589228	−0.294614	+	0.955616i	$0.595191\pi$
$572$	−1.00943	−0.0422066
$573$	0	0
$574$	0.804066	0.0335611
$575$	26.0648	1.08698
$576$	0	0
$577$	9.96168	0.414710	0.207355	−	0.978266i	$-0.433514\pi$
$577$	9.96168	0.414710	0.207355	+	0.978266i	$0.433514\pi$
$578$	46.3003	1.92584
$579$	0	0
$580$	3.13076	0.129998
$581$	17.8348	0.739913
$582$	0	0
$583$	0.309471	0.0128170
$584$	−4.50443	−0.186395
$585$	0	0
$586$	−10.5218	−0.434652
$587$	12.4924	0.515617	0.257809	−	0.966196i	$-0.417000\pi$
$587$	12.4924	0.515617	0.257809	+	0.966196i	$0.417000\pi$
$588$	0	0
$589$	17.4955	0.720888
$590$	0.613177	0.0252441
$591$	0	0
$592$	6.70619	0.275623
$593$	−1.57247	−0.0645736	−0.0322868	−	0.999479i	$-0.510279\pi$
$593$	−1.57247	−0.0645736	−0.0322868	+	0.999479i	$0.510279\pi$
$594$	0	0
$595$	13.9469	0.571768
$596$	9.04419	0.370464
$597$	0	0
$598$	−15.3757	−0.628758
$599$	−5.00404	−0.204459	−0.102230	−	0.994761i	$-0.532598\pi$
$599$	−5.00404	−0.204459	−0.102230	+	0.994761i	$0.532598\pi$
$600$	0	0
$601$	32.5890	1.32933	0.664667	−	0.747140i	$-0.268573\pi$
$601$	32.5890	1.32933	0.664667	+	0.747140i	$0.268573\pi$
$602$	6.48380	0.264260
$603$	0	0
$604$	−11.2276	−0.456845
$605$	11.7905	0.479353
$606$	0	0
$607$	27.5971	1.12013	0.560066	−	0.828448i	$-0.310776\pi$
$607$	27.5971	1.12013	0.560066	+	0.828448i	$0.310776\pi$
$608$	−3.17102	−0.128602
$609$	0	0
$610$	−3.88946	−0.157480
$611$	−9.35667	−0.378531
$612$	0	0
$613$	6.13182	0.247662	0.123831	−	0.992303i	$-0.460482\pi$
$613$	6.13182	0.247662	0.123831	+	0.992303i	$0.460482\pi$
$614$	2.68123	0.108206
$615$	0	0
$616$	−0.721472	−0.0290689
$617$	13.8855	0.559007	0.279504	−	0.960145i	$-0.409830\pi$
$617$	13.8855	0.559007	0.279504	+	0.960145i	$0.409830\pi$
$618$	0	0
$619$	−41.2416	−1.65764	−0.828820	−	0.559516i	$-0.810987\pi$
$619$	−41.2416	−1.65764	−0.828820	+	0.559516i	$0.810987\pi$
$620$	6.02442	0.241947
$621$	0	0
$622$	10.3840	0.416362
$623$	16.2883	0.652578
$624$	0	0
$625$	8.53738	0.341495
$626$	25.6770	1.02626
$627$	0	0
$628$	−6.05283	−0.241534
$629$	−53.3554	−2.12742
$630$	0	0
$631$	29.5640	1.17692	0.588461	−	0.808525i	$-0.299734\pi$
$631$	29.5640	1.17692	0.588461	+	0.808525i	$0.299734\pi$
$632$	−7.27055	−0.289207
$633$	0	0
$634$	11.5721	0.459587
$635$	2.55130	0.101245
$636$	0	0
$637$	−9.93408	−0.393603
$638$	1.28853	0.0510133
$639$	0	0
$640$	−1.09191	−0.0431617
$641$	36.4633	1.44021	0.720106	−	0.693864i	$-0.244093\pi$
$641$	36.4633	1.44021	0.720106	+	0.693864i	$0.244093\pi$
$642$	0	0
$643$	−44.0408	−1.73680	−0.868400	−	0.495864i	$-0.834852\pi$
$643$	−44.0408	−1.73680	−0.868400	+	0.495864i	$0.834852\pi$
$644$	−10.9894	−0.433045
$645$	0	0
$646$	25.2291	0.992625
$647$	−23.3653	−0.918586	−0.459293	−	0.888285i	$-0.651897\pi$
$647$	−23.3653	−0.918586	−0.459293	+	0.888285i	$0.651897\pi$
$648$	0	0
$649$	0.252365	0.00990621
$650$	−8.55286	−0.335471
$651$	0	0
$652$	−17.5320	−0.686606
$653$	31.8819	1.24763	0.623817	−	0.781571i	$-0.285581\pi$
$653$	31.8819	1.24763	0.623817	+	0.781571i	$0.285581\pi$
$654$	0	0
$655$	−14.3854	−0.562082
$656$	0.500847	0.0195548
$657$	0	0
$658$	−6.68749	−0.260705
$659$	24.1989	0.942654	0.471327	−	0.881958i	$-0.343775\pi$
$659$	24.1989	0.942654	0.471327	+	0.881958i	$0.343775\pi$
$660$	0	0
$661$	−10.5341	−0.409729	−0.204865	−	0.978790i	$-0.565675\pi$
$661$	−10.5341	−0.409729	−0.204865	+	0.978790i	$0.565675\pi$
$662$	3.32446	0.129209
$663$	0	0
$664$	11.1092	0.431119
$665$	5.55871	0.215558
$666$	0	0
$667$	19.6268	0.759954
$668$	4.65235	0.180005
$669$	0	0
$670$	5.28924	0.204341
$671$	−1.60079	−0.0617977
$672$	0	0
$673$	35.3536	1.36278	0.681390	−	0.731920i	$-0.261376\pi$
$673$	35.3536	1.36278	0.681390	+	0.731920i	$0.261376\pi$
$674$	5.12244	0.197309
$675$	0	0
$676$	−7.95465	−0.305948
$677$	30.5037	1.17235	0.586177	−	0.810183i	$-0.300632\pi$
$677$	30.5037	1.17235	0.586177	+	0.810183i	$0.300632\pi$
$678$	0	0
$679$	1.12674	0.0432403
$680$	8.68743	0.333148
$681$	0	0
$682$	2.47947	0.0949439
$683$	1.19189	0.0456066	0.0228033	−	0.999740i	$-0.492741\pi$
$683$	1.19189	0.0456066	0.0228033	+	0.999740i	$0.492741\pi$
$684$	0	0
$685$	16.3723	0.625553
$686$	−18.3381	−0.700151
$687$	0	0
$688$	4.03871	0.153974
$689$	−1.54680	−0.0589283
$690$	0	0
$691$	2.64383	0.100576	0.0502881	−	0.998735i	$-0.483986\pi$
$691$	2.64383	0.100576	0.0502881	+	0.998735i	$0.483986\pi$
$692$	−7.97464	−0.303150
$693$	0	0
$694$	−35.3904	−1.34340
$695$	9.35639	0.354908
$696$	0	0
$697$	−3.98481	−0.150935
$698$	1.36609	0.0517074
$699$	0	0
$700$	−6.11297	−0.231049
$701$	−38.7891	−1.46504	−0.732522	−	0.680743i	$-0.761657\pi$
$701$	−38.7891	−1.46504	−0.732522	+	0.680743i	$0.761657\pi$
$702$	0	0
$703$	−21.2654	−0.802041
$704$	−0.449399	−0.0169374
$705$	0	0
$706$	7.39739	0.278405
$707$	−24.6735	−0.927944
$708$	0	0
$709$	−15.4217	−0.579174	−0.289587	−	0.957152i	$-0.593518\pi$
$709$	−15.4217	−0.579174	−0.289587	+	0.957152i	$0.593518\pi$
$710$	−1.47399	−0.0553178
$711$	0	0
$712$	10.1459	0.380233
$713$	37.7673	1.41439
$714$	0	0
$715$	1.10222	0.0412205
$716$	−5.42490	−0.202738
$717$	0	0
$718$	−19.4435	−0.725623
$719$	16.5764	0.618195	0.309097	−	0.951030i	$-0.399973\pi$
$719$	16.5764	0.618195	0.309097	+	0.951030i	$0.399973\pi$
$720$	0	0
$721$	27.8542	1.03734
$722$	−8.94465	−0.332885
$723$	0	0
$724$	0.400729	0.0148930
$725$	10.9176	0.405470
$726$	0	0
$727$	−46.6075	−1.72858	−0.864288	−	0.502998i	$-0.832230\pi$
$727$	−46.6075	−1.72858	−0.864288	+	0.502998i	$0.832230\pi$
$728$	3.60606	0.133649
$729$	0	0
$730$	4.91845	0.182040
$731$	−32.1326	−1.18847
$732$	0	0
$733$	−53.5695	−1.97863	−0.989317	−	0.145780i	$-0.953431\pi$
$733$	−53.5695	−1.97863	−0.989317	+	0.145780i	$0.953431\pi$
$734$	30.9241	1.14143
$735$	0	0
$736$	−6.84524	−0.252319
$737$	2.17689	0.0801869
$738$	0	0
$739$	2.51791	0.0926228	0.0463114	−	0.998927i	$-0.485253\pi$
$739$	2.51791	0.0926228	0.0463114	+	0.998927i	$0.485253\pi$
$740$	−7.32258	−0.269183
$741$	0	0
$742$	−1.10554	−0.0405857
$743$	24.9160	0.914080	0.457040	−	0.889446i	$-0.348910\pi$
$743$	24.9160	0.914080	0.457040	+	0.889446i	$0.348910\pi$
$744$	0	0
$745$	−9.87547	−0.361809
$746$	13.1506	0.481476
$747$	0	0
$748$	3.57549	0.130733
$749$	−17.8083	−0.650702
$750$	0	0
$751$	−16.2491	−0.592938	−0.296469	−	0.955042i	$-0.595809\pi$
$751$	−16.2491	−0.592938	−0.296469	+	0.955042i	$0.595809\pi$
$752$	−4.16558	−0.151903
$753$	0	0
$754$	−6.44032	−0.234543
$755$	12.2596	0.446172
$756$	0	0
$757$	33.4182	1.21460	0.607302	−	0.794471i	$-0.292252\pi$
$757$	33.4182	1.21460	0.607302	+	0.794471i	$0.292252\pi$
$758$	16.3105	0.592423
$759$	0	0
$760$	3.46248	0.125597
$761$	−21.4737	−0.778421	−0.389211	−	0.921149i	$-0.627252\pi$
$761$	−21.4737	−0.778421	−0.389211	+	0.921149i	$0.627252\pi$
$762$	0	0
$763$	29.6739	1.07427
$764$	−8.85349	−0.320308
$765$	0	0
$766$	−9.84463	−0.355701
$767$	−1.26137	−0.0455455
$768$	0	0
$769$	−40.9252	−1.47580	−0.737901	−	0.674909i	$-0.764183\pi$
$769$	−40.9252	−1.47580	−0.737901	+	0.674909i	$0.764183\pi$
$770$	0.787785	0.0283898
$771$	0	0
$772$	4.35343	0.156684
$773$	−22.5520	−0.811138	−0.405569	−	0.914065i	$-0.632927\pi$
$773$	−22.5520	−0.811138	−0.405569	+	0.914065i	$0.632927\pi$
$774$	0	0
$775$	21.0084	0.754643
$776$	0.701838	0.0251945
$777$	0	0
$778$	27.6443	0.991096
$779$	−1.58819	−0.0569029
$780$	0	0
$781$	−0.606650	−0.0217076
$782$	54.4617	1.94755
$783$	0	0
$784$	−4.42265	−0.157952
$785$	6.60917	0.235891
$786$	0	0
$787$	−38.9567	−1.38866	−0.694328	−	0.719659i	$-0.744298\pi$
$787$	−38.9567	−1.38866	−0.694328	+	0.719659i	$0.744298\pi$
$788$	−9.04254	−0.322127
$789$	0	0
$790$	7.93881	0.282450
$791$	−23.1441	−0.822910
$792$	0	0
$793$	8.00105	0.284125
$794$	17.5979	0.624527
$795$	0	0
$796$	5.43241	0.192547
$797$	−37.4911	−1.32800	−0.664001	−	0.747732i	$-0.731143\pi$
$797$	−37.4911	−1.32800	−0.664001	+	0.747732i	$0.731143\pi$
$798$	0	0
$799$	33.1420	1.17248
$800$	−3.80772	−0.134623
$801$	0	0
$802$	−29.6039	−1.04535
$803$	2.02429	0.0714356
$804$	0	0
$805$	11.9995	0.422928
$806$	−12.3929	−0.436521
$807$	0	0
$808$	−15.3690	−0.540678
$809$	−16.1108	−0.566424	−0.283212	−	0.959057i	$-0.591400\pi$
$809$	−16.1108	−0.566424	−0.283212	+	0.959057i	$0.591400\pi$
$810$	0	0
$811$	−17.9197	−0.629246	−0.314623	−	0.949217i	$-0.601878\pi$
$811$	−17.9197	−0.629246	−0.314623	+	0.949217i	$0.601878\pi$
$812$	−4.60308	−0.161537
$813$	0	0
$814$	−3.01376	−0.105632
$815$	19.1434	0.670565
$816$	0	0
$817$	−12.8068	−0.448054
$818$	−22.3741	−0.782292
$819$	0	0
$820$	−0.546881	−0.0190979
$821$	−5.55467	−0.193859	−0.0969297	−	0.995291i	$-0.530902\pi$
$821$	−5.55467	−0.193859	−0.0969297	+	0.995291i	$0.530902\pi$
$822$	0	0
$823$	−31.4419	−1.09599	−0.547997	−	0.836480i	$-0.684610\pi$
$823$	−31.4419	−1.09599	−0.547997	+	0.836480i	$0.684610\pi$
$824$	17.3502	0.604421
$825$	0	0
$826$	−0.901539	−0.0313686
$827$	13.5673	0.471780	0.235890	−	0.971780i	$-0.424200\pi$
$827$	13.5673	0.471780	0.235890	+	0.971780i	$0.424200\pi$
$828$	0	0
$829$	24.6751	0.857003	0.428502	−	0.903541i	$-0.359042\pi$
$829$	24.6751	0.857003	0.428502	+	0.903541i	$0.359042\pi$
$830$	−12.1303	−0.421047
$831$	0	0
$832$	2.24619	0.0778725
$833$	35.1872	1.21917
$834$	0	0
$835$	−5.07996	−0.175799
$836$	1.42505	0.0492865
$837$	0	0
$838$	18.8267	0.650359
$839$	19.1149	0.659919	0.329960	−	0.943995i	$-0.392965\pi$
$839$	19.1149	0.659919	0.329960	+	0.943995i	$0.392965\pi$
$840$	0	0
$841$	−20.7790	−0.716518
$842$	−8.21829	−0.283221
$843$	0	0
$844$	6.89567	0.237359
$845$	8.68579	0.298800
$846$	0	0
$847$	−17.3353	−0.595649
$848$	−0.688633	−0.0236478
$849$	0	0
$850$	30.2948	1.03910
$851$	−45.9055	−1.57362
$852$	0	0
$853$	0.106985	0.00366310	0.00183155	−	0.999998i	$-0.499417\pi$
$853$	0.106985	0.00366310	0.00183155	+	0.999998i	$0.499417\pi$
$854$	5.71858	0.195686
$855$	0	0
$856$	−11.0927	−0.379140
$857$	24.9825	0.853387	0.426693	−	0.904396i	$-0.359678\pi$
$857$	24.9825	0.853387	0.426693	+	0.904396i	$0.359678\pi$
$858$	0	0
$859$	−27.9667	−0.954212	−0.477106	−	0.878846i	$-0.658314\pi$
$859$	−27.9667	−0.954212	−0.477106	+	0.878846i	$0.658314\pi$
$860$	−4.40992	−0.150377
$861$	0	0
$862$	−3.68113	−0.125380
$863$	−20.6883	−0.704237	−0.352118	−	0.935956i	$-0.614539\pi$
$863$	−20.6883	−0.704237	−0.352118	+	0.935956i	$0.614539\pi$
$864$	0	0
$865$	8.70763	0.296068
$866$	19.5290	0.663624
$867$	0	0
$868$	−8.85756	−0.300645
$869$	3.26738	0.110838
$870$	0	0
$871$	−10.8805	−0.368673
$872$	18.4837	0.625936
$873$	0	0
$874$	21.7064	0.734229
$875$	15.4397	0.521957
$876$	0	0
$877$	−39.5113	−1.33420	−0.667101	−	0.744968i	$-0.732465\pi$
$877$	−39.5113	−1.33420	−0.667101	+	0.744968i	$0.732465\pi$
$878$	18.0868	0.610401
$879$	0	0
$880$	0.490705	0.0165417
$881$	−2.82668	−0.0952334	−0.0476167	−	0.998866i	$-0.515163\pi$
$881$	−2.82668	−0.0952334	−0.0476167	+	0.998866i	$0.515163\pi$
$882$	0	0
$883$	−45.9656	−1.54686	−0.773432	−	0.633879i	$-0.781462\pi$
$883$	−45.9656	−1.54686	−0.773432	+	0.633879i	$0.781462\pi$
$884$	−17.8710	−0.601066
$885$	0	0
$886$	10.5885	0.355727
$887$	21.7293	0.729600	0.364800	−	0.931086i	$-0.381137\pi$
$887$	21.7293	0.729600	0.364800	+	0.931086i	$0.381137\pi$
$888$	0	0
$889$	−3.75112	−0.125808
$890$	−11.0784	−0.371350
$891$	0	0
$892$	1.00000	0.0334825
$893$	13.2091	0.442027
$894$	0	0
$895$	5.92352	0.198002
$896$	1.60541	0.0536331
$897$	0	0
$898$	−29.8328	−0.995534
$899$	15.8194	0.527605
$900$	0	0
$901$	5.47887	0.182527
$902$	−0.225080	−0.00749435
$903$	0	0
$904$	−14.4163	−0.479479
$905$	−0.437562	−0.0145451
$906$	0	0
$907$	13.9338	0.462663	0.231332	−	0.972875i	$-0.425692\pi$
$907$	13.9338	0.462663	0.231332	+	0.972875i	$0.425692\pi$
$908$	−13.7181	−0.455252
$909$	0	0
$910$	−3.93751	−0.130527
$911$	51.8658	1.71839	0.859196	−	0.511646i	$-0.170964\pi$
$911$	51.8658	1.71839	0.859196	+	0.511646i	$0.170964\pi$
$912$	0	0
$913$	−4.99245	−0.165226
$914$	−9.46149	−0.312958
$915$	0	0
$916$	−12.6145	−0.416794
$917$	21.1504	0.698448
$918$	0	0
$919$	20.2667	0.668536	0.334268	−	0.942478i	$-0.391511\pi$
$919$	20.2667	0.668536	0.334268	+	0.942478i	$0.391511\pi$
$920$	7.47441	0.246424
$921$	0	0
$922$	30.5173	1.00504
$923$	3.03216	0.0998046
$924$	0	0
$925$	−25.5353	−0.839596
$926$	−1.24315	−0.0408523
$927$	0	0
$928$	−2.86723	−0.0941213
$929$	47.9314	1.57258	0.786289	−	0.617859i	$-0.212000\pi$
$929$	47.9314	1.57258	0.786289	+	0.617859i	$0.212000\pi$
$930$	0	0
$931$	14.0243	0.459627
$932$	11.0552	0.362124
$933$	0	0
$934$	17.2797	0.565408
$935$	−3.90412	−0.127678
$936$	0	0
$937$	50.2746	1.64240	0.821200	−	0.570640i	$-0.193305\pi$
$937$	50.2746	1.64240	0.821200	+	0.570640i	$0.193305\pi$
$938$	−7.77664	−0.253916
$939$	0	0
$940$	4.54846	0.148354
$941$	−8.33097	−0.271582	−0.135791	−	0.990738i	$-0.543358\pi$
$941$	−8.33097	−0.271582	−0.135791	+	0.990738i	$0.543358\pi$
$942$	0	0
$943$	−3.42841	−0.111645
$944$	−0.561562	−0.0182773
$945$	0	0
$946$	−1.81499	−0.0590105
$947$	4.95575	0.161040	0.0805201	−	0.996753i	$-0.474342\pi$
$947$	4.95575	0.161040	0.0805201	+	0.996753i	$0.474342\pi$
$948$	0	0
$949$	−10.1178	−0.328437
$950$	12.0744	0.391744
$951$	0	0
$952$	−12.7729	−0.413972
$953$	28.1641	0.912326	0.456163	−	0.889896i	$-0.349223\pi$
$953$	28.1641	0.912326	0.456163	+	0.889896i	$0.349223\pi$
$954$	0	0
$955$	9.66725	0.312825
$956$	12.1340	0.392443
$957$	0	0
$958$	−24.7125	−0.798424
$959$	−24.0718	−0.777318
$960$	0	0
$961$	−0.559369	−0.0180441
$962$	15.0633	0.485662
$963$	0	0
$964$	−5.31832	−0.171291
$965$	−4.75358	−0.153023
$966$	0	0
$967$	−30.1875	−0.970763	−0.485382	−	0.874302i	$-0.661319\pi$
$967$	−30.1875	−0.970763	−0.485382	+	0.874302i	$0.661319\pi$
$968$	−10.7980	−0.347062
$969$	0	0
$970$	−0.766346	−0.0246059
$971$	22.1428	0.710597	0.355299	−	0.934753i	$-0.384379\pi$
$971$	22.1428	0.710597	0.355299	+	0.934753i	$0.384379\pi$
$972$	0	0
$973$	−13.7565	−0.441012
$974$	−31.1571	−0.998339
$975$	0	0
$976$	3.56206	0.114019
$977$	28.1588	0.900881	0.450441	−	0.892806i	$-0.351267\pi$
$977$	28.1588	0.900881	0.450441	+	0.892806i	$0.351267\pi$
$978$	0	0
$979$	−4.55955	−0.145724
$980$	4.82915	0.154262
$981$	0	0
$982$	−39.6086	−1.26396
$983$	−0.925692	−0.0295250	−0.0147625	−	0.999891i	$-0.504699\pi$
$983$	−0.925692	−0.0295250	−0.0147625	+	0.999891i	$0.504699\pi$
$984$	0	0
$985$	9.87367	0.314601
$986$	22.8121	0.726484
$987$	0	0
$988$	−7.12269	−0.226603
$989$	−27.6459	−0.879090
$990$	0	0
$991$	−25.1080	−0.797582	−0.398791	−	0.917042i	$-0.630570\pi$
$991$	−25.1080	−0.797582	−0.398791	+	0.917042i	$0.630570\pi$
$992$	−5.51730	−0.175175
$993$	0	0
$994$	2.16717	0.0687384
$995$	−5.93173	−0.188048
$996$	0	0
$997$	−23.0338	−0.729488	−0.364744	−	0.931108i	$-0.618844\pi$
$997$	−23.0338	−0.729488	−0.364744	+	0.931108i	$0.618844\pi$
$998$	−14.1849	−0.449014
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.y.1.4	yes	8
3.2	odd	2		4014.2.a.x.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.x.1.5	✓	8	3.2	odd	2
4014.2.a.y.1.4	yes	8	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09191 q^{5} +1.60541 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09191 q^{5} +1.60541 q^{7} +1.00000 q^{8} -1.09191 q^{10} -0.449399 q^{11} +2.24619 q^{13} +1.60541 q^{14} +1.00000 q^{16} -7.95615 q^{17} -3.17102 q^{19} -1.09191 q^{20} -0.449399 q^{22} -6.84524 q^{23} -3.80772 q^{25} +2.24619 q^{26} +1.60541 q^{28} -2.86723 q^{29} -5.51730 q^{31} +1.00000 q^{32} -7.95615 q^{34} -1.75297 q^{35} +6.70619 q^{37} -3.17102 q^{38} -1.09191 q^{40} +0.500847 q^{41} +4.03871 q^{43} -0.449399 q^{44} -6.84524 q^{46} -4.16558 q^{47} -4.42265 q^{49} -3.80772 q^{50} +2.24619 q^{52} -0.688633 q^{53} +0.490705 q^{55} +1.60541 q^{56} -2.86723 q^{58} -0.561562 q^{59} +3.56206 q^{61} -5.51730 q^{62} +1.00000 q^{64} -2.45264 q^{65} -4.84401 q^{67} -7.95615 q^{68} -1.75297 q^{70} +1.34991 q^{71} -4.50443 q^{73} +6.70619 q^{74} -3.17102 q^{76} -0.721472 q^{77} -7.27055 q^{79} -1.09191 q^{80} +0.500847 q^{82} +11.1092 q^{83} +8.68743 q^{85} +4.03871 q^{86} -0.449399 q^{88} +10.1459 q^{89} +3.60606 q^{91} -6.84524 q^{92} -4.16558 q^{94} +3.46248 q^{95} +0.701838 q^{97} -4.42265 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} + 8 q^{8} - 6 q^{10} - 11 q^{11} - q^{13} - 6 q^{14} + 8 q^{16} - 16 q^{17} - q^{19} - 6 q^{20} - 11 q^{22} - 14 q^{23} + 10 q^{25} - q^{26} - 6 q^{28} - 21 q^{29} - 6 q^{31} + 8 q^{32} - 16 q^{34} - 8 q^{35} - 14 q^{37} - q^{38} - 6 q^{40} - 16 q^{41} - 29 q^{43} - 11 q^{44} - 14 q^{46} - 9 q^{47} - 2 q^{49} + 10 q^{50} - q^{52} - 11 q^{53} - 22 q^{55} - 6 q^{56} - 21 q^{58} - 21 q^{59} + 3 q^{61} - 6 q^{62} + 8 q^{64} - 24 q^{65} - 20 q^{67} - 16 q^{68} - 8 q^{70} - 32 q^{71} + 13 q^{73} - 14 q^{74} - q^{76} - 4 q^{77} + 21 q^{79} - 6 q^{80} - 16 q^{82} - 28 q^{83} - 14 q^{85} - 29 q^{86} - 11 q^{88} - 54 q^{89} - 36 q^{91} - 14 q^{92} - 9 q^{94} - 30 q^{95} + 10 q^{97} - 2 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 + 8 * q^8 - 6 * q^10 - 11 * q^11 - q^13 - 6 * q^14 + 8 * q^16 - 16 * q^17 - q^19 - 6 * q^20 - 11 * q^22 - 14 * q^23 + 10 * q^25 - q^26 - 6 * q^28 - 21 * q^29 - 6 * q^31 + 8 * q^32 - 16 * q^34 - 8 * q^35 - 14 * q^37 - q^38 - 6 * q^40 - 16 * q^41 - 29 * q^43 - 11 * q^44 - 14 * q^46 - 9 * q^47 - 2 * q^49 + 10 * q^50 - q^52 - 11 * q^53 - 22 * q^55 - 6 * q^56 - 21 * q^58 - 21 * q^59 + 3 * q^61 - 6 * q^62 + 8 * q^64 - 24 * q^65 - 20 * q^67 - 16 * q^68 - 8 * q^70 - 32 * q^71 + 13 * q^73 - 14 * q^74 - q^76 - 4 * q^77 + 21 * q^79 - 6 * q^80 - 16 * q^82 - 28 * q^83 - 14 * q^85 - 29 * q^86 - 11 * q^88 - 54 * q^89 - 36 * q^91 - 14 * q^92 - 9 * q^94 - 30 * q^95 + 10 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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