LMFDB - Embedded newform 4014.2.a.w.1.2

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 12x^{4} + 50x^{3} - 36x^{2} - 38x + 18 $ x^7 - x^6 - 14x^5 + 12x^4 + 50x^3 - 36x^2 - 38*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 446)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.26499$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.88711 q^{5} -3.39932 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.88711 q^{5} -3.39932 q^{7} -1.00000 q^{8} +1.88711 q^{10} -5.81384 q^{11} +2.85805 q^{13} +3.39932 q^{14} +1.00000 q^{16} +1.23918 q^{17} -2.52998 q^{19} -1.88711 q^{20} +5.81384 q^{22} -5.67798 q^{23} -1.43880 q^{25} -2.85805 q^{26} -3.39932 q^{28} +1.34432 q^{29} -5.94253 q^{31} -1.00000 q^{32} -1.23918 q^{34} +6.41490 q^{35} -10.2539 q^{37} +2.52998 q^{38} +1.88711 q^{40} +11.0370 q^{41} +4.19631 q^{43} -5.81384 q^{44} +5.67798 q^{46} -4.02616 q^{47} +4.55535 q^{49} +1.43880 q^{50} +2.85805 q^{52} -5.78588 q^{53} +10.9714 q^{55} +3.39932 q^{56} -1.34432 q^{58} -6.53884 q^{59} -3.49279 q^{61} +5.94253 q^{62} +1.00000 q^{64} -5.39347 q^{65} -9.86561 q^{67} +1.23918 q^{68} -6.41490 q^{70} -13.8230 q^{71} -5.52291 q^{73} +10.2539 q^{74} -2.52998 q^{76} +19.7631 q^{77} -4.38133 q^{79} -1.88711 q^{80} -11.0370 q^{82} +9.42926 q^{83} -2.33848 q^{85} -4.19631 q^{86} +5.81384 q^{88} -1.73150 q^{89} -9.71542 q^{91} -5.67798 q^{92} +4.02616 q^{94} +4.77437 q^{95} +13.6359 q^{97} -4.55535 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8	$ 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8} + 2 q^{10} - 9 q^{11} - 2 q^{13} - 6 q^{14} + 7 q^{16} + 7 q^{17} - 2 q^{19} - 2 q^{20} + 9 q^{22} - 15 q^{23} + 13 q^{25} + 2 q^{26} + 6 q^{28} - 9 q^{29} - 2 q^{31} - 7 q^{32} - 7 q^{34} + 4 q^{35} + 5 q^{37} + 2 q^{38} + 2 q^{40} + 33 q^{41} + 20 q^{43} - 9 q^{44} + 15 q^{46} + 2 q^{47} + 3 q^{49} - 13 q^{50} - 2 q^{52} + 13 q^{53} - 18 q^{55} - 6 q^{56} + 9 q^{58} - 9 q^{59} + 8 q^{61} + 2 q^{62} + 7 q^{64} + 44 q^{65} + 29 q^{67} + 7 q^{68} - 4 q^{70} - 37 q^{73} - 5 q^{74} - 2 q^{76} + 18 q^{77} + 32 q^{79} - 2 q^{80} - 33 q^{82} + 6 q^{83} - 4 q^{85} - 20 q^{86} + 9 q^{88} + 17 q^{89} - 4 q^{91} - 15 q^{92} - 2 q^{94} + 12 q^{95} + 12 q^{97} - 3 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8 + 2 * q^10 - 9 * q^11 - 2 * q^13 - 6 * q^14 + 7 * q^16 + 7 * q^17 - 2 * q^19 - 2 * q^20 + 9 * q^22 - 15 * q^23 + 13 * q^25 + 2 * q^26 + 6 * q^28 - 9 * q^29 - 2 * q^31 - 7 * q^32 - 7 * q^34 + 4 * q^35 + 5 * q^37 + 2 * q^38 + 2 * q^40 + 33 * q^41 + 20 * q^43 - 9 * q^44 + 15 * q^46 + 2 * q^47 + 3 * q^49 - 13 * q^50 - 2 * q^52 + 13 * q^53 - 18 * q^55 - 6 * q^56 + 9 * q^58 - 9 * q^59 + 8 * q^61 + 2 * q^62 + 7 * q^64 + 44 * q^65 + 29 * q^67 + 7 * q^68 - 4 * q^70 - 37 * q^73 - 5 * q^74 - 2 * q^76 + 18 * q^77 + 32 * q^79 - 2 * q^80 - 33 * q^82 + 6 * q^83 - 4 * q^85 - 20 * q^86 + 9 * q^88 + 17 * q^89 - 4 * q^91 - 15 * q^92 - 2 * q^94 + 12 * q^95 + 12 * q^97 - 3 * 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
$5$	−1.88711	−0.843943	−0.421971	−	0.906609i	$-0.638662\pi$
$5$	−1.88711	−0.843943	−0.421971	+	0.906609i	$0.638662\pi$
$6$	0	0
$7$	−3.39932	−1.28482	−0.642410	−	0.766361i	$-0.722065\pi$
$7$	−3.39932	−1.28482	−0.642410	+	0.766361i	$0.722065\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.88711	0.596758
$11$	−5.81384	−1.75294	−0.876469	−	0.481457i	$-0.840108\pi$
$11$	−5.81384	−1.75294	−0.876469	+	0.481457i	$0.840108\pi$
$12$	0	0
$13$	2.85805	0.792681	0.396340	−	0.918104i	$-0.370280\pi$
$13$	2.85805	0.792681	0.396340	+	0.918104i	$0.370280\pi$
$14$	3.39932	0.908505
$15$	0	0
$16$	1.00000	0.250000
$17$	1.23918	0.300546	0.150273	−	0.988645i	$-0.451985\pi$
$17$	1.23918	0.300546	0.150273	+	0.988645i	$0.451985\pi$
$18$	0	0
$19$	−2.52998	−0.580418	−0.290209	−	0.956963i	$-0.593725\pi$
$19$	−2.52998	−0.580418	−0.290209	+	0.956963i	$0.593725\pi$
$20$	−1.88711	−0.421971
$21$	0	0
$22$	5.81384	1.23952
$23$	−5.67798	−1.18394	−0.591971	−	0.805959i	$-0.701650\pi$
$23$	−5.67798	−1.18394	−0.591971	+	0.805959i	$0.701650\pi$
$24$	0	0
$25$	−1.43880	−0.287760
$26$	−2.85805	−0.560510
$27$	0	0
$28$	−3.39932	−0.642410
$29$	1.34432	0.249633	0.124817	−	0.992180i	$-0.460166\pi$
$29$	1.34432	0.249633	0.124817	+	0.992180i	$0.460166\pi$
$30$	0	0
$31$	−5.94253	−1.06731	−0.533655	−	0.845702i	$-0.679182\pi$
$31$	−5.94253	−1.06731	−0.533655	+	0.845702i	$0.679182\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.23918	−0.212518
$35$	6.41490	1.08432
$36$	0	0
$37$	−10.2539	−1.68573	−0.842866	−	0.538123i	$-0.819133\pi$
$37$	−10.2539	−1.68573	−0.842866	+	0.538123i	$0.819133\pi$
$38$	2.52998	0.410417
$39$	0	0
$40$	1.88711	0.298379
$41$	11.0370	1.72369	0.861846	−	0.507170i	$-0.169308\pi$
$41$	11.0370	1.72369	0.861846	+	0.507170i	$0.169308\pi$
$42$	0	0
$43$	4.19631	0.639932	0.319966	−	0.947429i	$-0.396329\pi$
$43$	4.19631	0.639932	0.319966	+	0.947429i	$0.396329\pi$
$44$	−5.81384	−0.876469
$45$	0	0
$46$	5.67798	0.837173
$47$	−4.02616	−0.587276	−0.293638	−	0.955917i	$-0.594866\pi$
$47$	−4.02616	−0.587276	−0.293638	+	0.955917i	$0.594866\pi$
$48$	0	0
$49$	4.55535	0.650764
$50$	1.43880	0.203477
$51$	0	0
$52$	2.85805	0.396340
$53$	−5.78588	−0.794752	−0.397376	−	0.917656i	$-0.630079\pi$
$53$	−5.78588	−0.794752	−0.397376	+	0.917656i	$0.630079\pi$
$54$	0	0
$55$	10.9714	1.47938
$56$	3.39932	0.454253
$57$	0	0
$58$	−1.34432	−0.176517
$59$	−6.53884	−0.851285	−0.425642	−	0.904891i	$-0.639952\pi$
$59$	−6.53884	−0.851285	−0.425642	+	0.904891i	$0.639952\pi$
$60$	0	0
$61$	−3.49279	−0.447205	−0.223603	−	0.974680i	$-0.571782\pi$
$61$	−3.49279	−0.447205	−0.223603	+	0.974680i	$0.571782\pi$
$62$	5.94253	0.754702
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.39347	−0.668977
$66$	0	0
$67$	−9.86561	−1.20528	−0.602638	−	0.798015i	$-0.705884\pi$
$67$	−9.86561	−1.20528	−0.602638	+	0.798015i	$0.705884\pi$
$68$	1.23918	0.150273
$69$	0	0
$70$	−6.41490	−0.766727
$71$	−13.8230	−1.64049	−0.820246	−	0.572011i	$-0.806164\pi$
$71$	−13.8230	−1.64049	−0.820246	+	0.572011i	$0.806164\pi$
$72$	0	0
$73$	−5.52291	−0.646408	−0.323204	−	0.946329i	$-0.604760\pi$
$73$	−5.52291	−0.646408	−0.323204	+	0.946329i	$0.604760\pi$
$74$	10.2539	1.19199
$75$	0	0
$76$	−2.52998	−0.290209
$77$	19.7631	2.25221
$78$	0	0
$79$	−4.38133	−0.492939	−0.246469	−	0.969151i	$-0.579270\pi$
$79$	−4.38133	−0.492939	−0.246469	+	0.969151i	$0.579270\pi$
$80$	−1.88711	−0.210986
$81$	0	0
$82$	−11.0370	−1.21883
$83$	9.42926	1.03500	0.517498	−	0.855684i	$-0.326863\pi$
$83$	9.42926	1.03500	0.517498	+	0.855684i	$0.326863\pi$
$84$	0	0
$85$	−2.33848	−0.253644
$86$	−4.19631	−0.452500
$87$	0	0
$88$	5.81384	0.619758
$89$	−1.73150	−0.183539	−0.0917693	−	0.995780i	$-0.529252\pi$
$89$	−1.73150	−0.183539	−0.0917693	+	0.995780i	$0.529252\pi$
$90$	0	0
$91$	−9.71542	−1.01845
$92$	−5.67798	−0.591971
$93$	0	0
$94$	4.02616	0.415267
$95$	4.77437	0.489840
$96$	0	0
$97$	13.6359	1.38451	0.692257	−	0.721651i	$-0.256616\pi$
$97$	13.6359	1.38451	0.692257	+	0.721651i	$0.256616\pi$
$98$	−4.55535	−0.460160
$99$	0	0
$100$	−1.43880	−0.143880
$101$	0.246233	0.0245011	0.0122505	−	0.999925i	$-0.496100\pi$
$101$	0.246233	0.0245011	0.0122505	+	0.999925i	$0.496100\pi$
$102$	0	0
$103$	12.1187	1.19409	0.597043	−	0.802209i	$-0.296342\pi$
$103$	12.1187	1.19409	0.597043	+	0.802209i	$0.296342\pi$
$104$	−2.85805	−0.280255
$105$	0	0
$106$	5.78588	0.561974
$107$	5.08978	0.492047	0.246024	−	0.969264i	$-0.420876\pi$
$107$	5.08978	0.492047	0.246024	+	0.969264i	$0.420876\pi$
$108$	0	0
$109$	13.7404	1.31609	0.658044	−	0.752979i	$-0.271384\pi$
$109$	13.7404	1.31609	0.658044	+	0.752979i	$0.271384\pi$
$110$	−10.9714	−1.04608
$111$	0	0
$112$	−3.39932	−0.321205
$113$	21.0447	1.97972	0.989860	−	0.142045i	$-0.0453678\pi$
$113$	21.0447	1.97972	0.989860	+	0.142045i	$0.0453678\pi$
$114$	0	0
$115$	10.7150	0.999179
$116$	1.34432	0.124817
$117$	0	0
$118$	6.53884	0.601949
$119$	−4.21237	−0.386147
$120$	0	0
$121$	22.8007	2.07279
$122$	3.49279	0.316222
$123$	0	0
$124$	−5.94253	−0.533655
$125$	12.1508	1.08680
$126$	0	0
$127$	−5.96135	−0.528984	−0.264492	−	0.964388i	$-0.585204\pi$
$127$	−5.96135	−0.528984	−0.264492	+	0.964388i	$0.585204\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	5.39347	0.473038
$131$	−1.29586	−0.113220	−0.0566100	−	0.998396i	$-0.518029\pi$
$131$	−1.29586	−0.113220	−0.0566100	+	0.998396i	$0.518029\pi$
$132$	0	0
$133$	8.60021	0.745733
$134$	9.86561	0.852259
$135$	0	0
$136$	−1.23918	−0.106259
$137$	−15.3980	−1.31554	−0.657768	−	0.753221i	$-0.728499\pi$
$137$	−15.3980	−1.31554	−0.657768	+	0.753221i	$0.728499\pi$
$138$	0	0
$139$	10.1313	0.859325	0.429663	−	0.902990i	$-0.358632\pi$
$139$	10.1313	0.859325	0.429663	+	0.902990i	$0.358632\pi$
$140$	6.41490	0.542158
$141$	0	0
$142$	13.8230	1.16000
$143$	−16.6163	−1.38952
$144$	0	0
$145$	−2.53688	−0.210676
$146$	5.52291	0.457079
$147$	0	0
$148$	−10.2539	−0.842866
$149$	10.7649	0.881892	0.440946	−	0.897534i	$-0.354643\pi$
$149$	10.7649	0.881892	0.440946	+	0.897534i	$0.354643\pi$
$150$	0	0
$151$	19.9679	1.62497	0.812483	−	0.582986i	$-0.198116\pi$
$151$	19.9679	1.62497	0.812483	+	0.582986i	$0.198116\pi$
$152$	2.52998	0.205209
$153$	0	0
$154$	−19.7631	−1.59255
$155$	11.2142	0.900749
$156$	0	0
$157$	3.46782	0.276762	0.138381	−	0.990379i	$-0.455810\pi$
$157$	3.46782	0.276762	0.138381	+	0.990379i	$0.455810\pi$
$158$	4.38133	0.348560
$159$	0	0
$160$	1.88711	0.149189
$161$	19.3013	1.52115
$162$	0	0
$163$	−18.4716	−1.44681	−0.723405	−	0.690423i	$-0.757424\pi$
$163$	−18.4716	−1.44681	−0.723405	+	0.690423i	$0.757424\pi$
$164$	11.0370	0.861846
$165$	0	0
$166$	−9.42926	−0.731853
$167$	−6.06238	−0.469121	−0.234561	−	0.972101i	$-0.575365\pi$
$167$	−6.06238	−0.469121	−0.234561	+	0.972101i	$0.575365\pi$
$168$	0	0
$169$	−4.83155	−0.371658
$170$	2.33848	0.179353
$171$	0	0
$172$	4.19631	0.319966
$173$	11.4872	0.873355	0.436677	−	0.899618i	$-0.356155\pi$
$173$	11.4872	0.873355	0.436677	+	0.899618i	$0.356155\pi$
$174$	0	0
$175$	4.89094	0.369720
$176$	−5.81384	−0.438235
$177$	0	0
$178$	1.73150	0.129781
$179$	−8.63233	−0.645211	−0.322605	−	0.946534i	$-0.604559\pi$
$179$	−8.63233	−0.645211	−0.322605	+	0.946534i	$0.604559\pi$
$180$	0	0
$181$	−3.83970	−0.285402	−0.142701	−	0.989766i	$-0.545579\pi$
$181$	−3.83970	−0.285402	−0.142701	+	0.989766i	$0.545579\pi$
$182$	9.71542	0.720155
$183$	0	0
$184$	5.67798	0.418587
$185$	19.3503	1.42266
$186$	0	0
$187$	−7.20441	−0.526838
$188$	−4.02616	−0.293638
$189$	0	0
$190$	−4.77437	−0.346369
$191$	−25.7228	−1.86124	−0.930619	−	0.365990i	$-0.880730\pi$
$191$	−25.7228	−1.86124	−0.930619	+	0.365990i	$0.880730\pi$
$192$	0	0
$193$	−0.697244	−0.0501887	−0.0250943	−	0.999685i	$-0.507989\pi$
$193$	−0.697244	−0.0501887	−0.0250943	+	0.999685i	$0.507989\pi$
$194$	−13.6359	−0.979000
$195$	0	0
$196$	4.55535	0.325382
$197$	−17.8963	−1.27506	−0.637530	−	0.770425i	$-0.720044\pi$
$197$	−17.8963	−1.27506	−0.637530	+	0.770425i	$0.720044\pi$
$198$	0	0
$199$	23.1451	1.64071	0.820355	−	0.571855i	$-0.193776\pi$
$199$	23.1451	1.64071	0.820355	+	0.571855i	$0.193776\pi$
$200$	1.43880	0.101739
$201$	0	0
$202$	−0.246233	−0.0173249
$203$	−4.56975	−0.320734
$204$	0	0
$205$	−20.8281	−1.45470
$206$	−12.1187	−0.844347
$207$	0	0
$208$	2.85805	0.198170
$209$	14.7089	1.01744
$210$	0	0
$211$	19.6554	1.35313	0.676566	−	0.736382i	$-0.263467\pi$
$211$	19.6554	1.35313	0.676566	+	0.736382i	$0.263467\pi$
$212$	−5.78588	−0.397376
$213$	0	0
$214$	−5.08978	−0.347930
$215$	−7.91892	−0.540066
$216$	0	0
$217$	20.2005	1.37130
$218$	−13.7404	−0.930615
$219$	0	0
$220$	10.9714	0.739690
$221$	3.54165	0.238237
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	3.39932	0.227126
$225$	0	0
$226$	−21.0447	−1.39987
$227$	6.51371	0.432330	0.216165	−	0.976357i	$-0.430645\pi$
$227$	6.51371	0.432330	0.216165	+	0.976357i	$0.430645\pi$
$228$	0	0
$229$	3.61411	0.238827	0.119413	−	0.992845i	$-0.461899\pi$
$229$	3.61411	0.238827	0.119413	+	0.992845i	$0.461899\pi$
$230$	−10.7150	−0.706526
$231$	0	0
$232$	−1.34432	−0.0882587
$233$	−16.7699	−1.09863	−0.549317	−	0.835614i	$-0.685112\pi$
$233$	−16.7699	−1.09863	−0.549317	+	0.835614i	$0.685112\pi$
$234$	0	0
$235$	7.59782	0.495627
$236$	−6.53884	−0.425642
$237$	0	0
$238$	4.21237	0.273047
$239$	8.90587	0.576073	0.288036	−	0.957619i	$-0.406998\pi$
$239$	8.90587	0.576073	0.288036	+	0.957619i	$0.406998\pi$
$240$	0	0
$241$	−5.77767	−0.372173	−0.186086	−	0.982533i	$-0.559580\pi$
$241$	−5.77767	−0.372173	−0.186086	+	0.982533i	$0.559580\pi$
$242$	−22.8007	−1.46569
$243$	0	0
$244$	−3.49279	−0.223603
$245$	−8.59646	−0.549208
$246$	0	0
$247$	−7.23082	−0.460086
$248$	5.94253	0.377351
$249$	0	0
$250$	−12.1508	−0.768481
$251$	−11.3277	−0.714995	−0.357498	−	0.933914i	$-0.616370\pi$
$251$	−11.3277	−0.714995	−0.357498	+	0.933914i	$0.616370\pi$
$252$	0	0
$253$	33.0109	2.07538
$254$	5.96135	0.374048
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.5197	0.905716	0.452858	−	0.891583i	$-0.350404\pi$
$257$	14.5197	0.905716	0.452858	+	0.891583i	$0.350404\pi$
$258$	0	0
$259$	34.8563	2.16586
$260$	−5.39347	−0.334489
$261$	0	0
$262$	1.29586	0.0800587
$263$	20.5460	1.26692	0.633459	−	0.773776i	$-0.281635\pi$
$263$	20.5460	1.26692	0.633459	+	0.773776i	$0.281635\pi$
$264$	0	0
$265$	10.9186	0.670725
$266$	−8.60021	−0.527313
$267$	0	0
$268$	−9.86561	−0.602638
$269$	−14.4476	−0.880884	−0.440442	−	0.897781i	$-0.645178\pi$
$269$	−14.4476	−0.880884	−0.440442	+	0.897781i	$0.645178\pi$
$270$	0	0
$271$	3.59831	0.218582	0.109291	−	0.994010i	$-0.465142\pi$
$271$	3.59831	0.218582	0.109291	+	0.994010i	$0.465142\pi$
$272$	1.23918	0.0751365
$273$	0	0
$274$	15.3980	0.930224
$275$	8.36497	0.504426
$276$	0	0
$277$	−7.80848	−0.469166	−0.234583	−	0.972096i	$-0.575372\pi$
$277$	−7.80848	−0.469166	−0.234583	+	0.972096i	$0.575372\pi$
$278$	−10.1313	−0.607635
$279$	0	0
$280$	−6.41490	−0.383363
$281$	20.7989	1.24076	0.620381	−	0.784301i	$-0.286978\pi$
$281$	20.7989	1.24076	0.620381	+	0.784301i	$0.286978\pi$
$282$	0	0
$283$	−15.0612	−0.895295	−0.447648	−	0.894210i	$-0.647738\pi$
$283$	−15.0612	−0.895295	−0.447648	+	0.894210i	$0.647738\pi$
$284$	−13.8230	−0.820246
$285$	0	0
$286$	16.6163	0.982539
$287$	−37.5183	−2.21464
$288$	0	0
$289$	−15.4644	−0.909672
$290$	2.53688	0.148971
$291$	0	0
$292$	−5.52291	−0.323204
$293$	2.65346	0.155017	0.0775084	−	0.996992i	$-0.475304\pi$
$293$	2.65346	0.155017	0.0775084	+	0.996992i	$0.475304\pi$
$294$	0	0
$295$	12.3395	0.718436
$296$	10.2539	0.595996
$297$	0	0
$298$	−10.7649	−0.623592
$299$	−16.2280	−0.938487
$300$	0	0
$301$	−14.2646	−0.822198
$302$	−19.9679	−1.14902
$303$	0	0
$304$	−2.52998	−0.145104
$305$	6.59128	0.377416
$306$	0	0
$307$	17.3596	0.990767	0.495383	−	0.868674i	$-0.335028\pi$
$307$	17.3596	0.990767	0.495383	+	0.868674i	$0.335028\pi$
$308$	19.7631	1.12611
$309$	0	0
$310$	−11.2142	−0.636926
$311$	18.2101	1.03260	0.516299	−	0.856408i	$-0.327309\pi$
$311$	18.2101	1.03260	0.516299	+	0.856408i	$0.327309\pi$
$312$	0	0
$313$	14.3258	0.809743	0.404871	−	0.914374i	$-0.367316\pi$
$313$	14.3258	0.809743	0.404871	+	0.914374i	$0.367316\pi$
$314$	−3.46782	−0.195700
$315$	0	0
$316$	−4.38133	−0.246469
$317$	−15.1146	−0.848920	−0.424460	−	0.905447i	$-0.639536\pi$
$317$	−15.1146	−0.848920	−0.424460	+	0.905447i	$0.639536\pi$
$318$	0	0
$319$	−7.81564	−0.437592
$320$	−1.88711	−0.105493
$321$	0	0
$322$	−19.3013	−1.07562
$323$	−3.13511	−0.174442
$324$	0	0
$325$	−4.11217	−0.228102
$326$	18.4716	1.02305
$327$	0	0
$328$	−11.0370	−0.609417
$329$	13.6862	0.754544
$330$	0	0
$331$	4.72682	0.259809	0.129905	−	0.991526i	$-0.458533\pi$
$331$	4.72682	0.259809	0.129905	+	0.991526i	$0.458533\pi$
$332$	9.42926	0.517498
$333$	0	0
$334$	6.06238	0.331719
$335$	18.6175	1.01718
$336$	0	0
$337$	−12.6704	−0.690202	−0.345101	−	0.938566i	$-0.612155\pi$
$337$	−12.6704	−0.690202	−0.345101	+	0.938566i	$0.612155\pi$
$338$	4.83155	0.262802
$339$	0	0
$340$	−2.33848	−0.126822
$341$	34.5489	1.87093
$342$	0	0
$343$	8.31014	0.448706
$344$	−4.19631	−0.226250
$345$	0	0
$346$	−11.4872	−0.617555
$347$	−26.4289	−1.41878	−0.709389	−	0.704817i	$-0.751029\pi$
$347$	−26.4289	−1.41878	−0.709389	+	0.704817i	$0.751029\pi$
$348$	0	0
$349$	−25.9704	−1.39016	−0.695081	−	0.718932i	$-0.744631\pi$
$349$	−25.9704	−1.39016	−0.695081	+	0.718932i	$0.744631\pi$
$350$	−4.89094	−0.261432
$351$	0	0
$352$	5.81384	0.309879
$353$	31.4804	1.67553	0.837767	−	0.546029i	$-0.183861\pi$
$353$	31.4804	1.67553	0.837767	+	0.546029i	$0.183861\pi$
$354$	0	0
$355$	26.0856	1.38448
$356$	−1.73150	−0.0917693
$357$	0	0
$358$	8.63233	0.456233
$359$	6.75528	0.356530	0.178265	−	0.983982i	$-0.442952\pi$
$359$	6.75528	0.356530	0.178265	+	0.983982i	$0.442952\pi$
$360$	0	0
$361$	−12.5992	−0.663115
$362$	3.83970	0.201810
$363$	0	0
$364$	−9.71542	−0.509226
$365$	10.4224	0.545531
$366$	0	0
$367$	16.2821	0.849920	0.424960	−	0.905212i	$-0.360288\pi$
$367$	16.2821	0.849920	0.424960	+	0.905212i	$0.360288\pi$
$368$	−5.67798	−0.295985
$369$	0	0
$370$	−19.3503	−1.00597
$371$	19.6680	1.02111
$372$	0	0
$373$	15.5643	0.805889	0.402945	−	0.915224i	$-0.367987\pi$
$373$	15.5643	0.805889	0.402945	+	0.915224i	$0.367987\pi$
$374$	7.20441	0.372531
$375$	0	0
$376$	4.02616	0.207633
$377$	3.84212	0.197879
$378$	0	0
$379$	15.4256	0.792362	0.396181	−	0.918172i	$-0.370335\pi$
$379$	15.4256	0.792362	0.396181	+	0.918172i	$0.370335\pi$
$380$	4.77437	0.244920
$381$	0	0
$382$	25.7228	1.31609
$383$	−22.1437	−1.13149	−0.565745	−	0.824580i	$-0.691411\pi$
$383$	−22.1437	−1.13149	−0.565745	+	0.824580i	$0.691411\pi$
$384$	0	0
$385$	−37.2952	−1.90074
$386$	0.697244	0.0354888
$387$	0	0
$388$	13.6359	0.692257
$389$	−15.6749	−0.794749	−0.397374	−	0.917657i	$-0.630079\pi$
$389$	−15.6749	−0.794749	−0.397374	+	0.917657i	$0.630079\pi$
$390$	0	0
$391$	−7.03606	−0.355829
$392$	−4.55535	−0.230080
$393$	0	0
$394$	17.8963	0.901604
$395$	8.26807	0.416012
$396$	0	0
$397$	−35.0234	−1.75777	−0.878886	−	0.477031i	$-0.841713\pi$
$397$	−35.0234	−1.75777	−0.878886	+	0.477031i	$0.841713\pi$
$398$	−23.1451	−1.16016
$399$	0	0
$400$	−1.43880	−0.0719401
$401$	−28.4561	−1.42103	−0.710515	−	0.703682i	$-0.751538\pi$
$401$	−28.4561	−1.42103	−0.710515	+	0.703682i	$0.751538\pi$
$402$	0	0
$403$	−16.9841	−0.846036
$404$	0.246233	0.0122505
$405$	0	0
$406$	4.56975	0.226793
$407$	59.6146	2.95499
$408$	0	0
$409$	27.0225	1.33618	0.668088	−	0.744082i	$-0.267113\pi$
$409$	27.0225	1.33618	0.668088	+	0.744082i	$0.267113\pi$
$410$	20.8281	1.02863
$411$	0	0
$412$	12.1187	0.597043
$413$	22.2276	1.09375
$414$	0	0
$415$	−17.7941	−0.873478
$416$	−2.85805	−0.140127
$417$	0	0
$418$	−14.7089	−0.719437
$419$	28.3236	1.38370	0.691848	−	0.722043i	$-0.256797\pi$
$419$	28.3236	1.38370	0.691848	+	0.722043i	$0.256797\pi$
$420$	0	0
$421$	17.8659	0.870732	0.435366	−	0.900254i	$-0.356619\pi$
$421$	17.8659	0.870732	0.435366	+	0.900254i	$0.356619\pi$
$422$	−19.6554	−0.956809
$423$	0	0
$424$	5.78588	0.280987
$425$	−1.78294	−0.0864852
$426$	0	0
$427$	11.8731	0.574579
$428$	5.08978	0.246024
$429$	0	0
$430$	7.91892	0.381884
$431$	−16.5255	−0.796008	−0.398004	−	0.917384i	$-0.630297\pi$
$431$	−16.5255	−0.796008	−0.398004	+	0.917384i	$0.630297\pi$
$432$	0	0
$433$	20.0981	0.965852	0.482926	−	0.875661i	$-0.339574\pi$
$433$	20.0981	0.965852	0.482926	+	0.875661i	$0.339574\pi$
$434$	−20.2005	−0.969657
$435$	0	0
$436$	13.7404	0.658044
$437$	14.3652	0.687181
$438$	0	0
$439$	−7.59975	−0.362716	−0.181358	−	0.983417i	$-0.558049\pi$
$439$	−7.59975	−0.362716	−0.181358	+	0.983417i	$0.558049\pi$
$440$	−10.9714	−0.523040
$441$	0	0
$442$	−3.54165	−0.168459
$443$	−2.56789	−0.122004	−0.0610020	−	0.998138i	$-0.519430\pi$
$443$	−2.56789	−0.122004	−0.0610020	+	0.998138i	$0.519430\pi$
$444$	0	0
$445$	3.26754	0.154896
$446$	1.00000	0.0473514
$447$	0	0
$448$	−3.39932	−0.160603
$449$	−15.0053	−0.708145	−0.354073	−	0.935218i	$-0.615203\pi$
$449$	−15.0053	−0.708145	−0.354073	+	0.935218i	$0.615203\pi$
$450$	0	0
$451$	−64.1675	−3.02153
$452$	21.0447	0.989860
$453$	0	0
$454$	−6.51371	−0.305704
$455$	18.3341	0.859516
$456$	0	0
$457$	−32.7811	−1.53344	−0.766718	−	0.641984i	$-0.778112\pi$
$457$	−32.7811	−1.53344	−0.766718	+	0.641984i	$0.778112\pi$
$458$	−3.61411	−0.168876
$459$	0	0
$460$	10.7150	0.499590
$461$	−28.4766	−1.32629	−0.663144	−	0.748492i	$-0.730778\pi$
$461$	−28.4766	−1.32629	−0.663144	+	0.748492i	$0.730778\pi$
$462$	0	0
$463$	−40.9330	−1.90232	−0.951160	−	0.308700i	$-0.900106\pi$
$463$	−40.9330	−1.90232	−0.951160	+	0.308700i	$0.900106\pi$
$464$	1.34432	0.0624083
$465$	0	0
$466$	16.7699	0.776851
$467$	38.6853	1.79014	0.895071	−	0.445923i	$-0.147124\pi$
$467$	38.6853	1.79014	0.895071	+	0.445923i	$0.147124\pi$
$468$	0	0
$469$	33.5363	1.54856
$470$	−7.59782	−0.350461
$471$	0	0
$472$	6.53884	0.300975
$473$	−24.3967	−1.12176
$474$	0	0
$475$	3.64015	0.167021
$476$	−4.21237	−0.193074
$477$	0	0
$478$	−8.90587	−0.407345
$479$	34.2534	1.56508	0.782538	−	0.622602i	$-0.213925\pi$
$479$	34.2534	1.56508	0.782538	+	0.622602i	$0.213925\pi$
$480$	0	0
$481$	−29.3062	−1.33625
$482$	5.77767	0.263166
$483$	0	0
$484$	22.8007	1.03640
$485$	−25.7325	−1.16845
$486$	0	0
$487$	−1.31777	−0.0597138	−0.0298569	−	0.999554i	$-0.509505\pi$
$487$	−1.31777	−0.0597138	−0.0298569	+	0.999554i	$0.509505\pi$
$488$	3.49279	0.158111
$489$	0	0
$490$	8.59646	0.388348
$491$	42.0221	1.89643	0.948216	−	0.317626i	$-0.102886\pi$
$491$	42.0221	1.89643	0.948216	+	0.317626i	$0.102886\pi$
$492$	0	0
$493$	1.66585	0.0750262
$494$	7.23082	0.325330
$495$	0	0
$496$	−5.94253	−0.266828
$497$	46.9889	2.10774
$498$	0	0
$499$	34.4470	1.54206	0.771029	−	0.636800i	$-0.219742\pi$
$499$	34.4470	1.54206	0.771029	+	0.636800i	$0.219742\pi$
$500$	12.1508	0.543398
$501$	0	0
$502$	11.3277	0.505578
$503$	−2.16167	−0.0963842	−0.0481921	−	0.998838i	$-0.515346\pi$
$503$	−2.16167	−0.0963842	−0.0481921	+	0.998838i	$0.515346\pi$
$504$	0	0
$505$	−0.464669	−0.0206775
$506$	−33.0109	−1.46751
$507$	0	0
$508$	−5.96135	−0.264492
$509$	26.3722	1.16893	0.584463	−	0.811420i	$-0.301305\pi$
$509$	26.3722	1.16893	0.584463	+	0.811420i	$0.301305\pi$
$510$	0	0
$511$	18.7741	0.830518
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−14.5197	−0.640438
$515$	−22.8693	−1.00774
$516$	0	0
$517$	23.4075	1.02946
$518$	−34.8563	−1.53150
$519$	0	0
$520$	5.39347	0.236519
$521$	14.7140	0.644633	0.322316	−	0.946632i	$-0.395539\pi$
$521$	14.7140	0.644633	0.322316	+	0.946632i	$0.395539\pi$
$522$	0	0
$523$	15.3128	0.669583	0.334792	−	0.942292i	$-0.391334\pi$
$523$	15.3128	0.669583	0.334792	+	0.942292i	$0.391334\pi$
$524$	−1.29586	−0.0566100
$525$	0	0
$526$	−20.5460	−0.895846
$527$	−7.36388	−0.320776
$528$	0	0
$529$	9.23950	0.401718
$530$	−10.9186	−0.474274
$531$	0	0
$532$	8.60021	0.372866
$533$	31.5444	1.36634
$534$	0	0
$535$	−9.60499	−0.415260
$536$	9.86561	0.426129
$537$	0	0
$538$	14.4476	0.622879
$539$	−26.4841	−1.14075
$540$	0	0
$541$	8.02388	0.344974	0.172487	−	0.985012i	$-0.444820\pi$
$541$	8.02388	0.344974	0.172487	+	0.985012i	$0.444820\pi$
$542$	−3.59831	−0.154561
$543$	0	0
$544$	−1.23918	−0.0531295
$545$	−25.9296	−1.11070
$546$	0	0
$547$	−38.7712	−1.65773	−0.828867	−	0.559445i	$-0.811014\pi$
$547$	−38.7712	−1.65773	−0.828867	+	0.559445i	$0.811014\pi$
$548$	−15.3980	−0.657768
$549$	0	0
$550$	−8.36497	−0.356683
$551$	−3.40110	−0.144892
$552$	0	0
$553$	14.8935	0.633338
$554$	7.80848	0.331750
$555$	0	0
$556$	10.1313	0.429663
$557$	1.58029	0.0669589	0.0334795	−	0.999439i	$-0.489341\pi$
$557$	1.58029	0.0669589	0.0334795	+	0.999439i	$0.489341\pi$
$558$	0	0
$559$	11.9933	0.507262
$560$	6.41490	0.271079
$561$	0	0
$562$	−20.7989	−0.877351
$563$	14.8469	0.625723	0.312862	−	0.949799i	$-0.398712\pi$
$563$	14.8469	0.625723	0.312862	+	0.949799i	$0.398712\pi$
$564$	0	0
$565$	−39.7138	−1.67077
$566$	15.0612	0.633069
$567$	0	0
$568$	13.8230	0.580002
$569$	−8.00600	−0.335629	−0.167815	−	0.985819i	$-0.553671\pi$
$569$	−8.00600	−0.335629	−0.167815	+	0.985819i	$0.553671\pi$
$570$	0	0
$571$	−13.0603	−0.546558	−0.273279	−	0.961935i	$-0.588108\pi$
$571$	−13.0603	−0.546558	−0.273279	+	0.961935i	$0.588108\pi$
$572$	−16.6163	−0.694760
$573$	0	0
$574$	37.5183	1.56598
$575$	8.16950	0.340691
$576$	0	0
$577$	−19.0951	−0.794939	−0.397469	−	0.917615i	$-0.630112\pi$
$577$	−19.0951	−0.794939	−0.397469	+	0.917615i	$0.630112\pi$
$578$	15.4644	0.643235
$579$	0	0
$580$	−2.53688	−0.105338
$581$	−32.0530	−1.32978
$582$	0	0
$583$	33.6382	1.39315
$584$	5.52291	0.228540
$585$	0	0
$586$	−2.65346	−0.109613
$587$	−21.0818	−0.870140	−0.435070	−	0.900397i	$-0.643276\pi$
$587$	−21.0818	−0.870140	−0.435070	+	0.900397i	$0.643276\pi$
$588$	0	0
$589$	15.0345	0.619486
$590$	−12.3395	−0.508011
$591$	0	0
$592$	−10.2539	−0.421433
$593$	37.2607	1.53011	0.765057	−	0.643962i	$-0.222711\pi$
$593$	37.2607	1.53011	0.765057	+	0.643962i	$0.222711\pi$
$594$	0	0
$595$	7.94922	0.325886
$596$	10.7649	0.440946
$597$	0	0
$598$	16.2280	0.663611
$599$	−36.8967	−1.50756	−0.753780	−	0.657127i	$-0.771771\pi$
$599$	−36.8967	−1.50756	−0.753780	+	0.657127i	$0.771771\pi$
$600$	0	0
$601$	−0.861589	−0.0351450	−0.0175725	−	0.999846i	$-0.505594\pi$
$601$	−0.861589	−0.0351450	−0.0175725	+	0.999846i	$0.505594\pi$
$602$	14.2646	0.581382
$603$	0	0
$604$	19.9679	0.812483
$605$	−43.0276	−1.74932
$606$	0	0
$607$	−5.14970	−0.209020	−0.104510	−	0.994524i	$-0.533327\pi$
$607$	−5.14970	−0.209020	−0.104510	+	0.994524i	$0.533327\pi$
$608$	2.52998	0.102604
$609$	0	0
$610$	−6.59128	−0.266873
$611$	−11.5070	−0.465522
$612$	0	0
$613$	20.9728	0.847084	0.423542	−	0.905876i	$-0.360787\pi$
$613$	20.9728	0.847084	0.423542	+	0.905876i	$0.360787\pi$
$614$	−17.3596	−0.700578
$615$	0	0
$616$	−19.7631	−0.796277
$617$	−24.1716	−0.973112	−0.486556	−	0.873649i	$-0.661747\pi$
$617$	−24.1716	−0.973112	−0.486556	+	0.873649i	$0.661747\pi$
$618$	0	0
$619$	−39.9939	−1.60749	−0.803745	−	0.594974i	$-0.797162\pi$
$619$	−39.9939	−1.60749	−0.803745	+	0.594974i	$0.797162\pi$
$620$	11.2142	0.450374
$621$	0	0
$622$	−18.2101	−0.730157
$623$	5.88591	0.235814
$624$	0	0
$625$	−15.7358	−0.629434
$626$	−14.3258	−0.572575
$627$	0	0
$628$	3.46782	0.138381
$629$	−12.7065	−0.506640
$630$	0	0
$631$	12.7275	0.506674	0.253337	−	0.967378i	$-0.418472\pi$
$631$	12.7275	0.506674	0.253337	+	0.967378i	$0.418472\pi$
$632$	4.38133	0.174280
$633$	0	0
$634$	15.1146	0.600277
$635$	11.2497	0.446433
$636$	0	0
$637$	13.0194	0.515848
$638$	7.81564	0.309424
$639$	0	0
$640$	1.88711	0.0745947
$641$	−41.5097	−1.63954	−0.819768	−	0.572696i	$-0.805897\pi$
$641$	−41.5097	−1.63954	−0.819768	+	0.572696i	$0.805897\pi$
$642$	0	0
$643$	−27.9379	−1.10176	−0.550882	−	0.834583i	$-0.685708\pi$
$643$	−27.9379	−1.10176	−0.550882	+	0.834583i	$0.685708\pi$
$644$	19.3013	0.760576
$645$	0	0
$646$	3.13511	0.123349
$647$	−10.3614	−0.407350	−0.203675	−	0.979039i	$-0.565289\pi$
$647$	−10.3614	−0.407350	−0.203675	+	0.979039i	$0.565289\pi$
$648$	0	0
$649$	38.0158	1.49225
$650$	4.11217	0.161293
$651$	0	0
$652$	−18.4716	−0.723405
$653$	−18.0622	−0.706829	−0.353415	−	0.935467i	$-0.614980\pi$
$653$	−18.0622	−0.706829	−0.353415	+	0.935467i	$0.614980\pi$
$654$	0	0
$655$	2.44544	0.0955513
$656$	11.0370	0.430923
$657$	0	0
$658$	−13.6862	−0.533543
$659$	−0.597557	−0.0232775	−0.0116388	−	0.999932i	$-0.503705\pi$
$659$	−0.597557	−0.0232775	−0.0116388	+	0.999932i	$0.503705\pi$
$660$	0	0
$661$	−23.5518	−0.916059	−0.458029	−	0.888937i	$-0.651445\pi$
$661$	−23.5518	−0.916059	−0.458029	+	0.888937i	$0.651445\pi$
$662$	−4.72682	−0.183713
$663$	0	0
$664$	−9.42926	−0.365926
$665$	−16.2296	−0.629356
$666$	0	0
$667$	−7.63300	−0.295551
$668$	−6.06238	−0.234561
$669$	0	0
$670$	−18.6175	−0.719258
$671$	20.3065	0.783924
$672$	0	0
$673$	10.7866	0.415794	0.207897	−	0.978151i	$-0.433338\pi$
$673$	10.7866	0.415794	0.207897	+	0.978151i	$0.433338\pi$
$674$	12.6704	0.488046
$675$	0	0
$676$	−4.83155	−0.185829
$677$	−13.2892	−0.510746	−0.255373	−	0.966843i	$-0.582198\pi$
$677$	−13.2892	−0.510746	−0.255373	+	0.966843i	$0.582198\pi$
$678$	0	0
$679$	−46.3527	−1.77885
$680$	2.33848	0.0896765
$681$	0	0
$682$	−34.5489	−1.32295
$683$	8.22597	0.314758	0.157379	−	0.987538i	$-0.449696\pi$
$683$	8.22597	0.314758	0.157379	+	0.987538i	$0.449696\pi$
$684$	0	0
$685$	29.0577	1.11024
$686$	−8.31014	−0.317283
$687$	0	0
$688$	4.19631	0.159983
$689$	−16.5363	−0.629984
$690$	0	0
$691$	27.8761	1.06046	0.530229	−	0.847855i	$-0.322106\pi$
$691$	27.8761	1.06046	0.530229	+	0.847855i	$0.322106\pi$
$692$	11.4872	0.436677
$693$	0	0
$694$	26.4289	1.00323
$695$	−19.1189	−0.725221
$696$	0	0
$697$	13.6769	0.518049
$698$	25.9704	0.982993
$699$	0	0
$700$	4.89094	0.184860
$701$	16.1833	0.611236	0.305618	−	0.952154i	$-0.401137\pi$
$701$	16.1833	0.611236	0.305618	+	0.952154i	$0.401137\pi$
$702$	0	0
$703$	25.9422	0.978429
$704$	−5.81384	−0.219117
$705$	0	0
$706$	−31.4804	−1.18478
$707$	−0.837022	−0.0314795
$708$	0	0
$709$	27.2076	1.02180	0.510902	−	0.859639i	$-0.329312\pi$
$709$	27.2076	1.02180	0.510902	+	0.859639i	$0.329312\pi$
$710$	−26.0856	−0.978977
$711$	0	0
$712$	1.73150	0.0648907
$713$	33.7416	1.26363
$714$	0	0
$715$	31.3568	1.17268
$716$	−8.63233	−0.322605
$717$	0	0
$718$	−6.75528	−0.252105
$719$	16.5745	0.618123	0.309062	−	0.951042i	$-0.399985\pi$
$719$	16.5745	0.618123	0.309062	+	0.951042i	$0.399985\pi$
$720$	0	0
$721$	−41.1951	−1.53419
$722$	12.5992	0.468893
$723$	0	0
$724$	−3.83970	−0.142701
$725$	−1.93420	−0.0718345
$726$	0	0
$727$	8.96490	0.332490	0.166245	−	0.986085i	$-0.446836\pi$
$727$	8.96490	0.332490	0.166245	+	0.986085i	$0.446836\pi$
$728$	9.71542	0.360077
$729$	0	0
$730$	−10.4224	−0.385749
$731$	5.20000	0.192329
$732$	0	0
$733$	27.4498	1.01388	0.506941	−	0.861981i	$-0.330776\pi$
$733$	27.4498	1.01388	0.506941	+	0.861981i	$0.330776\pi$
$734$	−16.2821	−0.600984
$735$	0	0
$736$	5.67798	0.209293
$737$	57.3571	2.11278
$738$	0	0
$739$	25.8299	0.950167	0.475084	−	0.879941i	$-0.342418\pi$
$739$	25.8299	0.950167	0.475084	+	0.879941i	$0.342418\pi$
$740$	19.3503	0.711331
$741$	0	0
$742$	−19.6680	−0.722036
$743$	9.37139	0.343803	0.171901	−	0.985114i	$-0.445009\pi$
$743$	9.37139	0.343803	0.171901	+	0.985114i	$0.445009\pi$
$744$	0	0
$745$	−20.3145	−0.744267
$746$	−15.5643	−0.569850
$747$	0	0
$748$	−7.20441	−0.263419
$749$	−17.3018	−0.632192
$750$	0	0
$751$	34.4033	1.25539	0.627697	−	0.778458i	$-0.283998\pi$
$751$	34.4033	1.25539	0.627697	+	0.778458i	$0.283998\pi$
$752$	−4.02616	−0.146819
$753$	0	0
$754$	−3.84212	−0.139922
$755$	−37.6817	−1.37138
$756$	0	0
$757$	−22.5044	−0.817936	−0.408968	−	0.912549i	$-0.634111\pi$
$757$	−22.5044	−0.817936	−0.408968	+	0.912549i	$0.634111\pi$
$758$	−15.4256	−0.560285
$759$	0	0
$760$	−4.77437	−0.173184
$761$	−25.8770	−0.938040	−0.469020	−	0.883188i	$-0.655393\pi$
$761$	−25.8770	−0.938040	−0.469020	+	0.883188i	$0.655393\pi$
$762$	0	0
$763$	−46.7078	−1.69094
$764$	−25.7228	−0.930619
$765$	0	0
$766$	22.1437	0.800084
$767$	−18.6883	−0.674797
$768$	0	0
$769$	−0.0570330	−0.00205666	−0.00102833	−	0.999999i	$-0.500327\pi$
$769$	−0.0570330	−0.00205666	−0.00102833	+	0.999999i	$0.500327\pi$
$770$	37.2952	1.34403
$771$	0	0
$772$	−0.697244	−0.0250943
$773$	5.29591	0.190481	0.0952404	−	0.995454i	$-0.469638\pi$
$773$	5.29591	0.190481	0.0952404	+	0.995454i	$0.469638\pi$
$774$	0	0
$775$	8.55013	0.307130
$776$	−13.6359	−0.489500
$777$	0	0
$778$	15.6749	0.561972
$779$	−27.9235	−1.00046
$780$	0	0
$781$	80.3649	2.87568
$782$	7.03606	0.251609
$783$	0	0
$784$	4.55535	0.162691
$785$	−6.54416	−0.233571
$786$	0	0
$787$	−38.0732	−1.35716	−0.678582	−	0.734524i	$-0.737405\pi$
$787$	−38.0732	−1.35716	−0.678582	+	0.734524i	$0.737405\pi$
$788$	−17.8963	−0.637530
$789$	0	0
$790$	−8.26807	−0.294165
$791$	−71.5376	−2.54359
$792$	0	0
$793$	−9.98256	−0.354491
$794$	35.0234	1.24293
$795$	0	0
$796$	23.1451	0.820355
$797$	−22.8614	−0.809793	−0.404896	−	0.914363i	$-0.632692\pi$
$797$	−22.8614	−0.809793	−0.404896	+	0.914363i	$0.632692\pi$
$798$	0	0
$799$	−4.98915	−0.176503
$800$	1.43880	0.0508693
$801$	0	0
$802$	28.4561	1.00482
$803$	32.1093	1.13311
$804$	0	0
$805$	−36.4237	−1.28377
$806$	16.9841	0.598238
$807$	0	0
$808$	−0.246233	−0.00866243
$809$	27.8988	0.980869	0.490435	−	0.871478i	$-0.336838\pi$
$809$	27.8988	0.980869	0.490435	+	0.871478i	$0.336838\pi$
$810$	0	0
$811$	−36.2827	−1.27406	−0.637029	−	0.770840i	$-0.719837\pi$
$811$	−36.2827	−1.27406	−0.637029	+	0.770840i	$0.719837\pi$
$812$	−4.56975	−0.160367
$813$	0	0
$814$	−59.6146	−2.08949
$815$	34.8581	1.22103
$816$	0	0
$817$	−10.6166	−0.371428
$818$	−27.0225	−0.944820
$819$	0	0
$820$	−20.8281	−0.727349
$821$	15.3995	0.537445	0.268722	−	0.963218i	$-0.413399\pi$
$821$	15.3995	0.537445	0.268722	+	0.963218i	$0.413399\pi$
$822$	0	0
$823$	−23.5271	−0.820103	−0.410052	−	0.912062i	$-0.634489\pi$
$823$	−23.5271	−0.820103	−0.410052	+	0.912062i	$0.634489\pi$
$824$	−12.1187	−0.422173
$825$	0	0
$826$	−22.2276	−0.773397
$827$	6.99278	0.243163	0.121581	−	0.992581i	$-0.461203\pi$
$827$	6.99278	0.243163	0.121581	+	0.992581i	$0.461203\pi$
$828$	0	0
$829$	−5.07365	−0.176215	−0.0881076	−	0.996111i	$-0.528082\pi$
$829$	−5.07365	−0.176215	−0.0881076	+	0.996111i	$0.528082\pi$
$830$	17.7941	0.617642
$831$	0	0
$832$	2.85805	0.0990851
$833$	5.64491	0.195584
$834$	0	0
$835$	11.4404	0.395912
$836$	14.7089	0.508719
$837$	0	0
$838$	−28.3236	−0.978421
$839$	57.6955	1.99187	0.995935	−	0.0900751i	$-0.0287107\pi$
$839$	57.6955	1.99187	0.995935	+	0.0900751i	$0.0287107\pi$
$840$	0	0
$841$	−27.1928	−0.937683
$842$	−17.8659	−0.615701
$843$	0	0
$844$	19.6554	0.676566
$845$	9.11768	0.313658
$846$	0	0
$847$	−77.5069	−2.66317
$848$	−5.78588	−0.198688
$849$	0	0
$850$	1.78294	0.0611543
$851$	58.2215	1.99581
$852$	0	0
$853$	−56.5123	−1.93494	−0.967472	−	0.252978i	$-0.918590\pi$
$853$	−56.5123	−1.93494	−0.967472	+	0.252978i	$0.918590\pi$
$854$	−11.8731	−0.406288
$855$	0	0
$856$	−5.08978	−0.173965
$857$	51.7967	1.76934	0.884672	−	0.466215i	$-0.154383\pi$
$857$	51.7967	1.76934	0.884672	+	0.466215i	$0.154383\pi$
$858$	0	0
$859$	21.9803	0.749958	0.374979	−	0.927033i	$-0.377650\pi$
$859$	21.9803	0.749958	0.374979	+	0.927033i	$0.377650\pi$
$860$	−7.91892	−0.270033
$861$	0	0
$862$	16.5255	0.562862
$863$	−46.5616	−1.58498	−0.792488	−	0.609888i	$-0.791215\pi$
$863$	−46.5616	−1.58498	−0.792488	+	0.609888i	$0.791215\pi$
$864$	0	0
$865$	−21.6776	−0.737061
$866$	−20.0981	−0.682960
$867$	0	0
$868$	20.2005	0.685651
$869$	25.4724	0.864091
$870$	0	0
$871$	−28.1964	−0.955399
$872$	−13.7404	−0.465308
$873$	0	0
$874$	−14.3652	−0.485910
$875$	−41.3042	−1.39634
$876$	0	0
$877$	9.29819	0.313978	0.156989	−	0.987600i	$-0.449821\pi$
$877$	9.29819	0.313978	0.156989	+	0.987600i	$0.449821\pi$
$878$	7.59975	0.256479
$879$	0	0
$880$	10.9714	0.369845
$881$	−27.4806	−0.925843	−0.462922	−	0.886399i	$-0.653199\pi$
$881$	−27.4806	−0.925843	−0.462922	+	0.886399i	$0.653199\pi$
$882$	0	0
$883$	−53.7772	−1.80975	−0.904874	−	0.425679i	$-0.860035\pi$
$883$	−53.7772	−1.80975	−0.904874	+	0.425679i	$0.860035\pi$
$884$	3.54165	0.119118
$885$	0	0
$886$	2.56789	0.0862698
$887$	−2.10352	−0.0706291	−0.0353146	−	0.999376i	$-0.511243\pi$
$887$	−2.10352	−0.0706291	−0.0353146	+	0.999376i	$0.511243\pi$
$888$	0	0
$889$	20.2645	0.679650
$890$	−3.26754	−0.109528
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	10.1861	0.340866
$894$	0	0
$895$	16.2902	0.544521
$896$	3.39932	0.113563
$897$	0	0
$898$	15.0053	0.500734
$899$	−7.98864	−0.266436
$900$	0	0
$901$	−7.16976	−0.238859
$902$	64.1675	2.13654
$903$	0	0
$904$	−21.0447	−0.699937
$905$	7.24595	0.240863
$906$	0	0
$907$	−12.0881	−0.401379	−0.200690	−	0.979655i	$-0.564318\pi$
$907$	−12.0881	−0.401379	−0.200690	+	0.979655i	$0.564318\pi$
$908$	6.51371	0.216165
$909$	0	0
$910$	−18.3341	−0.607769
$911$	9.61744	0.318640	0.159320	−	0.987227i	$-0.449070\pi$
$911$	9.61744	0.318640	0.159320	+	0.987227i	$0.449070\pi$
$912$	0	0
$913$	−54.8202	−1.81428
$914$	32.7811	1.08430
$915$	0	0
$916$	3.61411	0.119413
$917$	4.40505	0.145467
$918$	0	0
$919$	47.2053	1.55716	0.778580	−	0.627545i	$-0.215940\pi$
$919$	47.2053	1.55716	0.778580	+	0.627545i	$0.215940\pi$
$920$	−10.7150	−0.353263
$921$	0	0
$922$	28.4766	0.937827
$923$	−39.5069	−1.30039
$924$	0	0
$925$	14.7533	0.485087
$926$	40.9330	1.34514
$927$	0	0
$928$	−1.34432	−0.0441293
$929$	−5.75600	−0.188848	−0.0944241	−	0.995532i	$-0.530101\pi$
$929$	−5.75600	−0.188848	−0.0944241	+	0.995532i	$0.530101\pi$
$930$	0	0
$931$	−11.5250	−0.377715
$932$	−16.7699	−0.549317
$933$	0	0
$934$	−38.6853	−1.26582
$935$	13.5955	0.444622
$936$	0	0
$937$	−12.5790	−0.410937	−0.205469	−	0.978664i	$-0.565872\pi$
$937$	−12.5790	−0.410937	−0.205469	+	0.978664i	$0.565872\pi$
$938$	−33.5363	−1.09500
$939$	0	0
$940$	7.59782	0.247814
$941$	−24.7011	−0.805232	−0.402616	−	0.915369i	$-0.631899\pi$
$941$	−24.7011	−0.805232	−0.402616	+	0.915369i	$0.631899\pi$
$942$	0	0
$943$	−62.6680	−2.04075
$944$	−6.53884	−0.212821
$945$	0	0
$946$	24.3967	0.793205
$947$	5.59397	0.181780	0.0908898	−	0.995861i	$-0.471029\pi$
$947$	5.59397	0.181780	0.0908898	+	0.995861i	$0.471029\pi$
$948$	0	0
$949$	−15.7848	−0.512395
$950$	−3.64015	−0.118102
$951$	0	0
$952$	4.21237	0.136524
$953$	−34.7174	−1.12461	−0.562303	−	0.826931i	$-0.690085\pi$
$953$	−34.7174	−1.12461	−0.562303	+	0.826931i	$0.690085\pi$
$954$	0	0
$955$	48.5419	1.57078
$956$	8.90587	0.288036
$957$	0	0
$958$	−34.2534	−1.10668
$959$	52.3425	1.69023
$960$	0	0
$961$	4.31368	0.139151
$962$	29.3062	0.944869
$963$	0	0
$964$	−5.77767	−0.186086
$965$	1.31578	0.0423564
$966$	0	0
$967$	0.272484	0.00876250	0.00438125	−	0.999990i	$-0.498605\pi$
$967$	0.272484	0.00876250	0.00438125	+	0.999990i	$0.498605\pi$
$968$	−22.8007	−0.732844
$969$	0	0
$970$	25.7325	0.826220
$971$	−34.7968	−1.11668	−0.558342	−	0.829611i	$-0.688562\pi$
$971$	−34.7968	−1.11668	−0.558342	+	0.829611i	$0.688562\pi$
$972$	0	0
$973$	−34.4395	−1.10408
$974$	1.31777	0.0422241
$975$	0	0
$976$	−3.49279	−0.111801
$977$	11.7953	0.377364	0.188682	−	0.982038i	$-0.439578\pi$
$977$	11.7953	0.377364	0.188682	+	0.982038i	$0.439578\pi$
$978$	0	0
$979$	10.0667	0.321732
$980$	−8.59646	−0.274604
$981$	0	0
$982$	−42.0221	−1.34098
$983$	−48.2759	−1.53976	−0.769881	−	0.638188i	$-0.779684\pi$
$983$	−48.2759	−1.53976	−0.769881	+	0.638188i	$0.779684\pi$
$984$	0	0
$985$	33.7724	1.07608
$986$	−1.66585	−0.0530515
$987$	0	0
$988$	−7.23082	−0.230043
$989$	−23.8266	−0.757642
$990$	0	0
$991$	7.38930	0.234729	0.117364	−	0.993089i	$-0.462555\pi$
$991$	7.38930	0.234729	0.117364	+	0.993089i	$0.462555\pi$
$992$	5.94253	0.188676
$993$	0	0
$994$	−46.9889	−1.49040
$995$	−43.6774	−1.38467
$996$	0	0
$997$	−19.8871	−0.629830	−0.314915	−	0.949120i	$-0.601976\pi$
$997$	−19.8871	−0.629830	−0.314915	+	0.949120i	$0.601976\pi$
$998$	−34.4470	−1.09040
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.w.1.2		7
3.2	odd	2		446.2.a.e.1.5	✓	7
12.11	even	2		3568.2.a.l.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
446.2.a.e.1.5	✓	7	3.2	odd	2
3568.2.a.l.1.3		7	12.11	even	2
4014.2.a.w.1.2		7	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.88711 q^{5} -3.39932 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.88711 q^{5} -3.39932 q^{7} -1.00000 q^{8} +1.88711 q^{10} -5.81384 q^{11} +2.85805 q^{13} +3.39932 q^{14} +1.00000 q^{16} +1.23918 q^{17} -2.52998 q^{19} -1.88711 q^{20} +5.81384 q^{22} -5.67798 q^{23} -1.43880 q^{25} -2.85805 q^{26} -3.39932 q^{28} +1.34432 q^{29} -5.94253 q^{31} -1.00000 q^{32} -1.23918 q^{34} +6.41490 q^{35} -10.2539 q^{37} +2.52998 q^{38} +1.88711 q^{40} +11.0370 q^{41} +4.19631 q^{43} -5.81384 q^{44} +5.67798 q^{46} -4.02616 q^{47} +4.55535 q^{49} +1.43880 q^{50} +2.85805 q^{52} -5.78588 q^{53} +10.9714 q^{55} +3.39932 q^{56} -1.34432 q^{58} -6.53884 q^{59} -3.49279 q^{61} +5.94253 q^{62} +1.00000 q^{64} -5.39347 q^{65} -9.86561 q^{67} +1.23918 q^{68} -6.41490 q^{70} -13.8230 q^{71} -5.52291 q^{73} +10.2539 q^{74} -2.52998 q^{76} +19.7631 q^{77} -4.38133 q^{79} -1.88711 q^{80} -11.0370 q^{82} +9.42926 q^{83} -2.33848 q^{85} -4.19631 q^{86} +5.81384 q^{88} -1.73150 q^{89} -9.71542 q^{91} -5.67798 q^{92} +4.02616 q^{94} +4.77437 q^{95} +13.6359 q^{97} -4.55535 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8	\( 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8} + 2 q^{10} - 9 q^{11} - 2 q^{13} - 6 q^{14} + 7 q^{16} + 7 q^{17} - 2 q^{19} - 2 q^{20} + 9 q^{22} - 15 q^{23} + 13 q^{25} + 2 q^{26} + 6 q^{28} - 9 q^{29} - 2 q^{31} - 7 q^{32} - 7 q^{34} + 4 q^{35} + 5 q^{37} + 2 q^{38} + 2 q^{40} + 33 q^{41} + 20 q^{43} - 9 q^{44} + 15 q^{46} + 2 q^{47} + 3 q^{49} - 13 q^{50} - 2 q^{52} + 13 q^{53} - 18 q^{55} - 6 q^{56} + 9 q^{58} - 9 q^{59} + 8 q^{61} + 2 q^{62} + 7 q^{64} + 44 q^{65} + 29 q^{67} + 7 q^{68} - 4 q^{70} - 37 q^{73} - 5 q^{74} - 2 q^{76} + 18 q^{77} + 32 q^{79} - 2 q^{80} - 33 q^{82} + 6 q^{83} - 4 q^{85} - 20 q^{86} + 9 q^{88} + 17 q^{89} - 4 q^{91} - 15 q^{92} - 2 q^{94} + 12 q^{95} + 12 q^{97} - 3 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8 + 2 * q^10 - 9 * q^11 - 2 * q^13 - 6 * q^14 + 7 * q^16 + 7 * q^17 - 2 * q^19 - 2 * q^20 + 9 * q^22 - 15 * q^23 + 13 * q^25 + 2 * q^26 + 6 * q^28 - 9 * q^29 - 2 * q^31 - 7 * q^32 - 7 * q^34 + 4 * q^35 + 5 * q^37 + 2 * q^38 + 2 * q^40 + 33 * q^41 + 20 * q^43 - 9 * q^44 + 15 * q^46 + 2 * q^47 + 3 * q^49 - 13 * q^50 - 2 * q^52 + 13 * q^53 - 18 * q^55 - 6 * q^56 + 9 * q^58 - 9 * q^59 + 8 * q^61 + 2 * q^62 + 7 * q^64 + 44 * q^65 + 29 * q^67 + 7 * q^68 - 4 * q^70 - 37 * q^73 - 5 * q^74 - 2 * q^76 + 18 * q^77 + 32 * q^79 - 2 * q^80 - 33 * q^82 + 6 * q^83 - 4 * q^85 - 20 * q^86 + 9 * q^88 + 17 * q^89 - 4 * q^91 - 15 * q^92 - 2 * q^94 + 12 * q^95 + 12 * q^97 - 3 * q^98

Introduction

L-functions

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