LMFDB - Embedded newform 4056.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.71914$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.408195 q^{5} -1.71914 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.408195 q^{5} -1.71914 q^{7} +1.00000 q^{9} +4.20771 q^{11} +0.408195 q^{15} +5.92820 q^{17} -0.457112 q^{19} +1.71914 q^{21} +2.66784 q^{23} -4.83338 q^{25} -1.00000 q^{27} -0.393743 q^{29} +5.77779 q^{31} -4.20771 q^{33} +0.701746 q^{35} -6.31197 q^{37} -5.39314 q^{41} +3.71622 q^{43} -0.408195 q^{45} -6.07169 q^{47} -4.04454 q^{49} -5.92820 q^{51} -1.69629 q^{53} -1.71756 q^{55} +0.457112 q^{57} +6.69391 q^{59} +9.84116 q^{61} -1.71914 q^{63} -10.2270 q^{67} -2.66784 q^{69} +10.2118 q^{71} -11.8793 q^{73} +4.83338 q^{75} -7.23365 q^{77} +4.58878 q^{79} +1.00000 q^{81} +10.6340 q^{83} -2.41986 q^{85} +0.393743 q^{87} +3.97450 q^{89} -5.77779 q^{93} +0.186591 q^{95} -0.852514 q^{97} +4.20771 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9	$ 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9} + 8 q^{11} - q^{15} - 5 q^{17} + 19 q^{19} - 7 q^{21} - 6 q^{23} + 17 q^{25} - 6 q^{27} + 3 q^{29} + 9 q^{31} - 8 q^{33} + 4 q^{35} + 6 q^{37} - 15 q^{41} + 11 q^{43} + q^{45} + 5 q^{47} + 5 q^{49} + 5 q^{51} + 12 q^{53} + 7 q^{55} - 19 q^{57} - 5 q^{59} + 17 q^{61} + 7 q^{63} + 13 q^{67} + 6 q^{69} + 26 q^{71} - 49 q^{73} - 17 q^{75} - 25 q^{77} + 14 q^{79} + 6 q^{81} - 3 q^{83} - 19 q^{85} - 3 q^{87} + 13 q^{89} - 9 q^{93} + 43 q^{95} - 14 q^{97} + 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9 + 8 * q^11 - q^15 - 5 * q^17 + 19 * q^19 - 7 * q^21 - 6 * q^23 + 17 * q^25 - 6 * q^27 + 3 * q^29 + 9 * q^31 - 8 * q^33 + 4 * q^35 + 6 * q^37 - 15 * q^41 + 11 * q^43 + q^45 + 5 * q^47 + 5 * q^49 + 5 * q^51 + 12 * q^53 + 7 * q^55 - 19 * q^57 - 5 * q^59 + 17 * q^61 + 7 * q^63 + 13 * q^67 + 6 * q^69 + 26 * q^71 - 49 * q^73 - 17 * q^75 - 25 * q^77 + 14 * q^79 + 6 * q^81 - 3 * q^83 - 19 * q^85 - 3 * q^87 + 13 * q^89 - 9 * q^93 + 43 * q^95 - 14 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.408195	−0.182550	−0.0912752	−	0.995826i	$-0.529094\pi$
$5$	−0.408195	−0.182550	−0.0912752	+	0.995826i	$0.529094\pi$
$6$	0	0
$7$	−1.71914	−0.649775	−0.324888	−	0.945753i	$-0.605326\pi$
$7$	−1.71914	−0.649775	−0.324888	+	0.945753i	$0.605326\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.20771	1.26867	0.634336	−	0.773058i	$-0.281274\pi$
$11$	4.20771	1.26867	0.634336	+	0.773058i	$0.281274\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.408195	0.105396
$16$	0	0
$17$	5.92820	1.43780	0.718899	−	0.695114i	$-0.244646\pi$
$17$	5.92820	1.43780	0.718899	+	0.695114i	$0.244646\pi$
$18$	0	0
$19$	−0.457112	−0.104869	−0.0524344	−	0.998624i	$-0.516698\pi$
$19$	−0.457112	−0.104869	−0.0524344	+	0.998624i	$0.516698\pi$
$20$	0	0
$21$	1.71914	0.375148
$22$	0	0
$23$	2.66784	0.556284	0.278142	−	0.960540i	$-0.410281\pi$
$23$	2.66784	0.556284	0.278142	+	0.960540i	$0.410281\pi$
$24$	0	0
$25$	−4.83338	−0.966675
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.393743	−0.0731162	−0.0365581	−	0.999332i	$-0.511639\pi$
$29$	−0.393743	−0.0731162	−0.0365581	+	0.999332i	$0.511639\pi$
$30$	0	0
$31$	5.77779	1.03772	0.518860	−	0.854859i	$-0.326356\pi$
$31$	5.77779	1.03772	0.518860	+	0.854859i	$0.326356\pi$
$32$	0	0
$33$	−4.20771	−0.732468
$34$	0	0
$35$	0.701746	0.118617
$36$	0	0
$37$	−6.31197	−1.03768	−0.518841	−	0.854871i	$-0.673636\pi$
$37$	−6.31197	−1.03768	−0.518841	+	0.854871i	$0.673636\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.39314	−0.842267	−0.421134	−	0.906999i	$-0.638368\pi$
$41$	−5.39314	−0.842267	−0.421134	+	0.906999i	$0.638368\pi$
$42$	0	0
$43$	3.71622	0.566718	0.283359	−	0.959014i	$-0.408551\pi$
$43$	3.71622	0.566718	0.283359	+	0.959014i	$0.408551\pi$
$44$	0	0
$45$	−0.408195	−0.0608501
$46$	0	0
$47$	−6.07169	−0.885647	−0.442824	−	0.896609i	$-0.646023\pi$
$47$	−6.07169	−0.885647	−0.442824	+	0.896609i	$0.646023\pi$
$48$	0	0
$49$	−4.04454	−0.577792
$50$	0	0
$51$	−5.92820	−0.830114
$52$	0	0
$53$	−1.69629	−0.233004	−0.116502	−	0.993190i	$-0.537168\pi$
$53$	−1.69629	−0.233004	−0.116502	+	0.993190i	$0.537168\pi$
$54$	0	0
$55$	−1.71756	−0.231596
$56$	0	0
$57$	0.457112	0.0605460
$58$	0	0
$59$	6.69391	0.871473	0.435737	−	0.900074i	$-0.356488\pi$
$59$	6.69391	0.871473	0.435737	+	0.900074i	$0.356488\pi$
$60$	0	0
$61$	9.84116	1.26003	0.630015	−	0.776583i	$-0.283049\pi$
$61$	9.84116	1.26003	0.630015	+	0.776583i	$0.283049\pi$
$62$	0	0
$63$	−1.71914	−0.216592
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.2270	−1.24943	−0.624715	−	0.780853i	$-0.714785\pi$
$67$	−10.2270	−1.24943	−0.624715	+	0.780853i	$0.714785\pi$
$68$	0	0
$69$	−2.66784	−0.321171
$70$	0	0
$71$	10.2118	1.21192	0.605958	−	0.795497i	$-0.292790\pi$
$71$	10.2118	1.21192	0.605958	+	0.795497i	$0.292790\pi$
$72$	0	0
$73$	−11.8793	−1.39036	−0.695182	−	0.718834i	$-0.744676\pi$
$73$	−11.8793	−1.39036	−0.695182	+	0.718834i	$0.744676\pi$
$74$	0	0
$75$	4.83338	0.558110
$76$	0	0
$77$	−7.23365	−0.824351
$78$	0	0
$79$	4.58878	0.516278	0.258139	−	0.966108i	$-0.416891\pi$
$79$	4.58878	0.516278	0.258139	+	0.966108i	$0.416891\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.6340	1.16723	0.583617	−	0.812029i	$-0.301637\pi$
$83$	10.6340	1.16723	0.583617	+	0.812029i	$0.301637\pi$
$84$	0	0
$85$	−2.41986	−0.262471
$86$	0	0
$87$	0.393743	0.0422136
$88$	0	0
$89$	3.97450	0.421296	0.210648	−	0.977562i	$-0.432443\pi$
$89$	3.97450	0.421296	0.210648	+	0.977562i	$0.432443\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.77779	−0.599128
$94$	0	0
$95$	0.186591	0.0191438
$96$	0	0
$97$	−0.852514	−0.0865597	−0.0432799	−	0.999063i	$-0.513781\pi$
$97$	−0.852514	−0.0865597	−0.0432799	+	0.999063i	$0.513781\pi$
$98$	0	0
$99$	4.20771	0.422890
$100$	0	0
$101$	6.66833	0.663523	0.331762	−	0.943363i	$-0.392357\pi$
$101$	6.66833	0.663523	0.331762	+	0.943363i	$0.392357\pi$
$102$	0	0
$103$	14.3815	1.41705	0.708527	−	0.705683i	$-0.249360\pi$
$103$	14.3815	1.41705	0.708527	+	0.705683i	$0.249360\pi$
$104$	0	0
$105$	−0.701746	−0.0684834
$106$	0	0
$107$	−11.1593	−1.07881	−0.539406	−	0.842046i	$-0.681351\pi$
$107$	−11.1593	−1.07881	−0.539406	+	0.842046i	$0.681351\pi$
$108$	0	0
$109$	13.0163	1.24674	0.623368	−	0.781929i	$-0.285764\pi$
$109$	13.0163	1.24674	0.623368	+	0.781929i	$0.285764\pi$
$110$	0	0
$111$	6.31197	0.599106
$112$	0	0
$113$	14.0137	1.31830	0.659148	−	0.752013i	$-0.270917\pi$
$113$	14.0137	1.31830	0.659148	+	0.752013i	$0.270917\pi$
$114$	0	0
$115$	−1.08900	−0.101550
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.1914	−0.934246
$120$	0	0
$121$	6.70479	0.609527
$122$	0	0
$123$	5.39314	0.486283
$124$	0	0
$125$	4.01394	0.359017
$126$	0	0
$127$	14.4832	1.28517	0.642587	−	0.766213i	$-0.277861\pi$
$127$	14.4832	1.28517	0.642587	+	0.766213i	$0.277861\pi$
$128$	0	0
$129$	−3.71622	−0.327195
$130$	0	0
$131$	−14.1042	−1.23229	−0.616145	−	0.787632i	$-0.711307\pi$
$131$	−14.1042	−1.23229	−0.616145	+	0.787632i	$0.711307\pi$
$132$	0	0
$133$	0.785842	0.0681411
$134$	0	0
$135$	0.408195	0.0351318
$136$	0	0
$137$	−19.8683	−1.69746	−0.848732	−	0.528824i	$-0.822633\pi$
$137$	−19.8683	−1.69746	−0.848732	+	0.528824i	$0.822633\pi$
$138$	0	0
$139$	16.2590	1.37907	0.689534	−	0.724253i	$-0.257815\pi$
$139$	16.2590	1.37907	0.689534	+	0.724253i	$0.257815\pi$
$140$	0	0
$141$	6.07169	0.511329
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.160724	0.0133474
$146$	0	0
$147$	4.04454	0.333588
$148$	0	0
$149$	9.42331	0.771988	0.385994	−	0.922501i	$-0.373859\pi$
$149$	9.42331	0.771988	0.385994	+	0.922501i	$0.373859\pi$
$150$	0	0
$151$	4.91356	0.399860	0.199930	−	0.979810i	$-0.435929\pi$
$151$	4.91356	0.399860	0.199930	+	0.979810i	$0.435929\pi$
$152$	0	0
$153$	5.92820	0.479266
$154$	0	0
$155$	−2.35846	−0.189436
$156$	0	0
$157$	21.6463	1.72756	0.863782	−	0.503866i	$-0.168090\pi$
$157$	21.6463	1.72756	0.863782	+	0.503866i	$0.168090\pi$
$158$	0	0
$159$	1.69629	0.134525
$160$	0	0
$161$	−4.58641	−0.361460
$162$	0	0
$163$	7.05422	0.552530	0.276265	−	0.961082i	$-0.410903\pi$
$163$	7.05422	0.552530	0.276265	+	0.961082i	$0.410903\pi$
$164$	0	0
$165$	1.71756	0.133712
$166$	0	0
$167$	−16.6885	−1.29139	−0.645696	−	0.763594i	$-0.723433\pi$
$167$	−16.6885	−1.29139	−0.645696	+	0.763594i	$0.723433\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−0.457112	−0.0349563
$172$	0	0
$173$	6.14111	0.466900	0.233450	−	0.972369i	$-0.424998\pi$
$173$	6.14111	0.466900	0.233450	+	0.972369i	$0.424998\pi$
$174$	0	0
$175$	8.30927	0.628122
$176$	0	0
$177$	−6.69391	−0.503145
$178$	0	0
$179$	12.3840	0.925626	0.462813	−	0.886456i	$-0.346840\pi$
$179$	12.3840	0.925626	0.462813	+	0.886456i	$0.346840\pi$
$180$	0	0
$181$	−14.5990	−1.08513	−0.542567	−	0.840013i	$-0.682547\pi$
$181$	−14.5990	−1.08513	−0.542567	+	0.840013i	$0.682547\pi$
$182$	0	0
$183$	−9.84116	−0.727479
$184$	0	0
$185$	2.57652	0.189429
$186$	0	0
$187$	24.9441	1.82409
$188$	0	0
$189$	1.71914	0.125049
$190$	0	0
$191$	11.1870	0.809464	0.404732	−	0.914435i	$-0.367365\pi$
$191$	11.1870	0.809464	0.404732	+	0.914435i	$0.367365\pi$
$192$	0	0
$193$	−3.76577	−0.271066	−0.135533	−	0.990773i	$-0.543275\pi$
$193$	−3.76577	−0.271066	−0.135533	+	0.990773i	$0.543275\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.0241	0.785434	0.392717	−	0.919659i	$-0.371535\pi$
$197$	11.0241	0.785434	0.392717	+	0.919659i	$0.371535\pi$
$198$	0	0
$199$	24.1418	1.71136	0.855682	−	0.517502i	$-0.173138\pi$
$199$	24.1418	1.71136	0.855682	+	0.517502i	$0.173138\pi$
$200$	0	0
$201$	10.2270	0.721358
$202$	0	0
$203$	0.676900	0.0475091
$204$	0	0
$205$	2.20145	0.153756
$206$	0	0
$207$	2.66784	0.185428
$208$	0	0
$209$	−1.92339	−0.133044
$210$	0	0
$211$	2.90884	0.200253	0.100126	−	0.994975i	$-0.468075\pi$
$211$	2.90884	0.200253	0.100126	+	0.994975i	$0.468075\pi$
$212$	0	0
$213$	−10.2118	−0.699700
$214$	0	0
$215$	−1.51694	−0.103455
$216$	0	0
$217$	−9.93284	−0.674285
$218$	0	0
$219$	11.8793	0.802727
$220$	0	0
$221$	0	0
$222$	0	0
$223$	28.7808	1.92731	0.963653	−	0.267158i	$-0.0860847\pi$
$223$	28.7808	1.92731	0.963653	+	0.267158i	$0.0860847\pi$
$224$	0	0
$225$	−4.83338	−0.322225
$226$	0	0
$227$	6.20051	0.411543	0.205771	−	0.978600i	$-0.434030\pi$
$227$	6.20051	0.411543	0.205771	+	0.978600i	$0.434030\pi$
$228$	0	0
$229$	−24.2058	−1.59956	−0.799781	−	0.600292i	$-0.795051\pi$
$229$	−24.2058	−1.59956	−0.799781	+	0.600292i	$0.795051\pi$
$230$	0	0
$231$	7.23365	0.475939
$232$	0	0
$233$	−24.3350	−1.59424	−0.797120	−	0.603822i	$-0.793644\pi$
$233$	−24.3350	−1.59424	−0.797120	+	0.603822i	$0.793644\pi$
$234$	0	0
$235$	2.47843	0.161675
$236$	0	0
$237$	−4.58878	−0.298073
$238$	0	0
$239$	12.8629	0.832030	0.416015	−	0.909358i	$-0.363426\pi$
$239$	12.8629	0.832030	0.416015	+	0.909358i	$0.363426\pi$
$240$	0	0
$241$	−22.8119	−1.46944	−0.734721	−	0.678369i	$-0.762687\pi$
$241$	−22.8119	−1.46944	−0.734721	+	0.678369i	$0.762687\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.65096	0.105476
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.6340	−0.673903
$250$	0	0
$251$	−23.5772	−1.48818	−0.744091	−	0.668079i	$-0.767117\pi$
$251$	−23.5772	−1.48818	−0.744091	+	0.668079i	$0.767117\pi$
$252$	0	0
$253$	11.2255	0.705742
$254$	0	0
$255$	2.41986	0.151538
$256$	0	0
$257$	22.0642	1.37632	0.688162	−	0.725557i	$-0.258418\pi$
$257$	22.0642	1.37632	0.688162	+	0.725557i	$0.258418\pi$
$258$	0	0
$259$	10.8512	0.674260
$260$	0	0
$261$	−0.393743	−0.0243721
$262$	0	0
$263$	6.67154	0.411385	0.205692	−	0.978617i	$-0.434055\pi$
$263$	6.67154	0.411385	0.205692	+	0.978617i	$0.434055\pi$
$264$	0	0
$265$	0.692419	0.0425349
$266$	0	0
$267$	−3.97450	−0.243236
$268$	0	0
$269$	13.9379	0.849806	0.424903	−	0.905239i	$-0.360308\pi$
$269$	13.9379	0.849806	0.424903	+	0.905239i	$0.360308\pi$
$270$	0	0
$271$	−14.2528	−0.865799	−0.432899	−	0.901442i	$-0.642509\pi$
$271$	−14.2528	−0.865799	−0.432899	+	0.901442i	$0.642509\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.3374	−1.22639
$276$	0	0
$277$	31.4833	1.89165	0.945825	−	0.324678i	$-0.105256\pi$
$277$	31.4833	1.89165	0.945825	+	0.324678i	$0.105256\pi$
$278$	0	0
$279$	5.77779	0.345907
$280$	0	0
$281$	−5.27630	−0.314758	−0.157379	−	0.987538i	$-0.550304\pi$
$281$	−5.27630	−0.314758	−0.157379	+	0.987538i	$0.550304\pi$
$282$	0	0
$283$	−2.06684	−0.122861	−0.0614304	−	0.998111i	$-0.519566\pi$
$283$	−2.06684	−0.122861	−0.0614304	+	0.998111i	$0.519566\pi$
$284$	0	0
$285$	−0.186591	−0.0110527
$286$	0	0
$287$	9.27159	0.547285
$288$	0	0
$289$	18.1435	1.06727
$290$	0	0
$291$	0.852514	0.0499753
$292$	0	0
$293$	−1.33522	−0.0780045	−0.0390022	−	0.999239i	$-0.512418\pi$
$293$	−1.33522	−0.0780045	−0.0390022	+	0.999239i	$0.512418\pi$
$294$	0	0
$295$	−2.73242	−0.159088
$296$	0	0
$297$	−4.20771	−0.244156
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−6.38871	−0.368239
$302$	0	0
$303$	−6.66833	−0.383085
$304$	0	0
$305$	−4.01711	−0.230019
$306$	0	0
$307$	−11.2398	−0.641491	−0.320746	−	0.947165i	$-0.603933\pi$
$307$	−11.2398	−0.641491	−0.320746	+	0.947165i	$0.603933\pi$
$308$	0	0
$309$	−14.3815	−0.818137
$310$	0	0
$311$	15.7439	0.892752	0.446376	−	0.894845i	$-0.352714\pi$
$311$	15.7439	0.892752	0.446376	+	0.894845i	$0.352714\pi$
$312$	0	0
$313$	6.62282	0.374344	0.187172	−	0.982327i	$-0.440068\pi$
$313$	6.62282	0.374344	0.187172	+	0.982327i	$0.440068\pi$
$314$	0	0
$315$	0.701746	0.0395389
$316$	0	0
$317$	−33.2261	−1.86616	−0.933080	−	0.359668i	$-0.882890\pi$
$317$	−33.2261	−1.86616	−0.933080	+	0.359668i	$0.882890\pi$
$318$	0	0
$319$	−1.65675	−0.0927604
$320$	0	0
$321$	11.1593	0.622853
$322$	0	0
$323$	−2.70985	−0.150780
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.0163	−0.719803
$328$	0	0
$329$	10.4381	0.575472
$330$	0	0
$331$	6.39552	0.351529	0.175765	−	0.984432i	$-0.443760\pi$
$331$	6.39552	0.351529	0.175765	+	0.984432i	$0.443760\pi$
$332$	0	0
$333$	−6.31197	−0.345894
$334$	0	0
$335$	4.17462	0.228084
$336$	0	0
$337$	33.6180	1.83129	0.915645	−	0.401987i	$-0.131680\pi$
$337$	33.6180	1.83129	0.915645	+	0.401987i	$0.131680\pi$
$338$	0	0
$339$	−14.0137	−0.761119
$340$	0	0
$341$	24.3112	1.31653
$342$	0	0
$343$	18.9872	1.02521
$344$	0	0
$345$	1.08900	0.0586298
$346$	0	0
$347$	16.3733	0.878964	0.439482	−	0.898251i	$-0.355162\pi$
$347$	16.3733	0.878964	0.439482	+	0.898251i	$0.355162\pi$
$348$	0	0
$349$	5.34034	0.285862	0.142931	−	0.989733i	$-0.454347\pi$
$349$	5.34034	0.285862	0.142931	+	0.989733i	$0.454347\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.6739	−0.940686	−0.470343	−	0.882484i	$-0.655870\pi$
$353$	−17.6739	−0.940686	−0.470343	+	0.882484i	$0.655870\pi$
$354$	0	0
$355$	−4.16840	−0.221236
$356$	0	0
$357$	10.1914	0.539387
$358$	0	0
$359$	−3.57017	−0.188426	−0.0942131	−	0.995552i	$-0.530033\pi$
$359$	−3.57017	−0.188426	−0.0942131	+	0.995552i	$0.530033\pi$
$360$	0	0
$361$	−18.7910	−0.989003
$362$	0	0
$363$	−6.70479	−0.351910
$364$	0	0
$365$	4.84906	0.253812
$366$	0	0
$367$	−28.4195	−1.48349	−0.741743	−	0.670684i	$-0.766001\pi$
$367$	−28.4195	−1.48349	−0.741743	+	0.670684i	$0.766001\pi$
$368$	0	0
$369$	−5.39314	−0.280756
$370$	0	0
$371$	2.91617	0.151400
$372$	0	0
$373$	34.5583	1.78936	0.894681	−	0.446705i	$-0.147403\pi$
$373$	34.5583	1.78936	0.894681	+	0.446705i	$0.147403\pi$
$374$	0	0
$375$	−4.01394	−0.207279
$376$	0	0
$377$	0	0
$378$	0	0
$379$	32.9822	1.69418	0.847090	−	0.531450i	$-0.178353\pi$
$379$	32.9822	1.69418	0.847090	+	0.531450i	$0.178353\pi$
$380$	0	0
$381$	−14.4832	−0.741996
$382$	0	0
$383$	8.20782	0.419400	0.209700	−	0.977766i	$-0.432751\pi$
$383$	8.20782	0.419400	0.209700	+	0.977766i	$0.432751\pi$
$384$	0	0
$385$	2.95274	0.150486
$386$	0	0
$387$	3.71622	0.188906
$388$	0	0
$389$	−23.7218	−1.20274	−0.601372	−	0.798969i	$-0.705379\pi$
$389$	−23.7218	−1.20274	−0.601372	+	0.798969i	$0.705379\pi$
$390$	0	0
$391$	15.8155	0.799825
$392$	0	0
$393$	14.1042	0.711463
$394$	0	0
$395$	−1.87312	−0.0942467
$396$	0	0
$397$	−13.7104	−0.688103	−0.344052	−	0.938951i	$-0.611800\pi$
$397$	−13.7104	−0.688103	−0.344052	+	0.938951i	$0.611800\pi$
$398$	0	0
$399$	−0.785842	−0.0393413
$400$	0	0
$401$	19.5018	0.973874	0.486937	−	0.873437i	$-0.338114\pi$
$401$	19.5018	0.973874	0.486937	+	0.873437i	$0.338114\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−0.408195	−0.0202834
$406$	0	0
$407$	−26.5589	−1.31648
$408$	0	0
$409$	−29.1514	−1.44144	−0.720721	−	0.693225i	$-0.756189\pi$
$409$	−29.1514	−1.44144	−0.720721	+	0.693225i	$0.756189\pi$
$410$	0	0
$411$	19.8683	0.980031
$412$	0	0
$413$	−11.5078	−0.566262
$414$	0	0
$415$	−4.34075	−0.213079
$416$	0	0
$417$	−16.2590	−0.796205
$418$	0	0
$419$	8.29539	0.405256	0.202628	−	0.979256i	$-0.435052\pi$
$419$	8.29539	0.405256	0.202628	+	0.979256i	$0.435052\pi$
$420$	0	0
$421$	−7.69502	−0.375032	−0.187516	−	0.982262i	$-0.560044\pi$
$421$	−7.69502	−0.375032	−0.187516	+	0.982262i	$0.560044\pi$
$422$	0	0
$423$	−6.07169	−0.295216
$424$	0	0
$425$	−28.6532	−1.38988
$426$	0	0
$427$	−16.9184	−0.818737
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.8938	−1.34360	−0.671798	−	0.740735i	$-0.734478\pi$
$431$	−27.8938	−1.34360	−0.671798	+	0.740735i	$0.734478\pi$
$432$	0	0
$433$	13.8146	0.663885	0.331943	−	0.943300i	$-0.392296\pi$
$433$	13.8146	0.663885	0.331943	+	0.943300i	$0.392296\pi$
$434$	0	0
$435$	−0.160724	−0.00770612
$436$	0	0
$437$	−1.21950	−0.0583368
$438$	0	0
$439$	28.2824	1.34985	0.674924	−	0.737888i	$-0.264177\pi$
$439$	28.2824	1.34985	0.674924	+	0.737888i	$0.264177\pi$
$440$	0	0
$441$	−4.04454	−0.192597
$442$	0	0
$443$	−18.0920	−0.859579	−0.429790	−	0.902929i	$-0.641412\pi$
$443$	−18.0920	−0.859579	−0.429790	+	0.902929i	$0.641412\pi$
$444$	0	0
$445$	−1.62237	−0.0769078
$446$	0	0
$447$	−9.42331	−0.445707
$448$	0	0
$449$	1.75662	0.0829000	0.0414500	−	0.999141i	$-0.486802\pi$
$449$	1.75662	0.0829000	0.0414500	+	0.999141i	$0.486802\pi$
$450$	0	0
$451$	−22.6928	−1.06856
$452$	0	0
$453$	−4.91356	−0.230859
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.9219	0.651239	0.325619	−	0.945501i	$-0.394427\pi$
$457$	13.9219	0.651239	0.325619	+	0.945501i	$0.394427\pi$
$458$	0	0
$459$	−5.92820	−0.276705
$460$	0	0
$461$	29.6887	1.38274	0.691370	−	0.722501i	$-0.257008\pi$
$461$	29.6887	1.38274	0.691370	+	0.722501i	$0.257008\pi$
$462$	0	0
$463$	14.9638	0.695425	0.347712	−	0.937601i	$-0.386959\pi$
$463$	14.9638	0.695425	0.347712	+	0.937601i	$0.386959\pi$
$464$	0	0
$465$	2.35846	0.109371
$466$	0	0
$467$	24.9088	1.15264	0.576321	−	0.817224i	$-0.304488\pi$
$467$	24.9088	1.15264	0.576321	+	0.817224i	$0.304488\pi$
$468$	0	0
$469$	17.5817	0.811848
$470$	0	0
$471$	−21.6463	−0.997409
$472$	0	0
$473$	15.6368	0.718979
$474$	0	0
$475$	2.20940	0.101374
$476$	0	0
$477$	−1.69629	−0.0776679
$478$	0	0
$479$	2.18443	0.0998091	0.0499046	−	0.998754i	$-0.484108\pi$
$479$	2.18443	0.0998091	0.0499046	+	0.998754i	$0.484108\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	4.58641	0.208689
$484$	0	0
$485$	0.347992	0.0158015
$486$	0	0
$487$	−31.6451	−1.43398	−0.716988	−	0.697085i	$-0.754480\pi$
$487$	−31.6451	−1.43398	−0.716988	+	0.697085i	$0.754480\pi$
$488$	0	0
$489$	−7.05422	−0.319003
$490$	0	0
$491$	14.8650	0.670847	0.335424	−	0.942067i	$-0.391121\pi$
$491$	14.8650	0.670847	0.335424	+	0.942067i	$0.391121\pi$
$492$	0	0
$493$	−2.33418	−0.105126
$494$	0	0
$495$	−1.71756	−0.0771988
$496$	0	0
$497$	−17.5555	−0.787473
$498$	0	0
$499$	26.1171	1.16916	0.584581	−	0.811335i	$-0.301259\pi$
$499$	26.1171	1.16916	0.584581	+	0.811335i	$0.301259\pi$
$500$	0	0
$501$	16.6885	0.745586
$502$	0	0
$503$	−29.8164	−1.32945	−0.664723	−	0.747089i	$-0.731451\pi$
$503$	−29.8164	−1.32945	−0.664723	+	0.747089i	$0.731451\pi$
$504$	0	0
$505$	−2.72198	−0.121126
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.7787	1.27559	0.637796	−	0.770205i	$-0.279846\pi$
$509$	28.7787	1.27559	0.637796	+	0.770205i	$0.279846\pi$
$510$	0	0
$511$	20.4222	0.903424
$512$	0	0
$513$	0.457112	0.0201820
$514$	0	0
$515$	−5.87047	−0.258684
$516$	0	0
$517$	−25.5479	−1.12360
$518$	0	0
$519$	−6.14111	−0.269565
$520$	0	0
$521$	−35.2690	−1.54516	−0.772581	−	0.634917i	$-0.781034\pi$
$521$	−35.2690	−1.54516	−0.772581	+	0.634917i	$0.781034\pi$
$522$	0	0
$523$	13.7508	0.601280	0.300640	−	0.953738i	$-0.402800\pi$
$523$	13.7508	0.601280	0.300640	+	0.953738i	$0.402800\pi$
$524$	0	0
$525$	−8.30927	−0.362646
$526$	0	0
$527$	34.2518	1.49203
$528$	0	0
$529$	−15.8826	−0.690548
$530$	0	0
$531$	6.69391	0.290491
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.55518	0.196938
$536$	0	0
$537$	−12.3840	−0.534410
$538$	0	0
$539$	−17.0183	−0.733028
$540$	0	0
$541$	1.62953	0.0700590	0.0350295	−	0.999386i	$-0.488847\pi$
$541$	1.62953	0.0700590	0.0350295	+	0.999386i	$0.488847\pi$
$542$	0	0
$543$	14.5990	0.626502
$544$	0	0
$545$	−5.31319	−0.227592
$546$	0	0
$547$	17.0684	0.729794	0.364897	−	0.931048i	$-0.381104\pi$
$547$	17.0684	0.729794	0.364897	+	0.931048i	$0.381104\pi$
$548$	0	0
$549$	9.84116	0.420010
$550$	0	0
$551$	0.179985	0.00766760
$552$	0	0
$553$	−7.88877	−0.335465
$554$	0	0
$555$	−2.57652	−0.109367
$556$	0	0
$557$	26.7943	1.13531	0.567656	−	0.823266i	$-0.307850\pi$
$557$	26.7943	1.13531	0.567656	+	0.823266i	$0.307850\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−24.9441	−1.05314
$562$	0	0
$563$	−8.92758	−0.376253	−0.188126	−	0.982145i	$-0.560241\pi$
$563$	−8.92758	−0.376253	−0.188126	+	0.982145i	$0.560241\pi$
$564$	0	0
$565$	−5.72032	−0.240656
$566$	0	0
$567$	−1.71914	−0.0721973
$568$	0	0
$569$	−13.8842	−0.582058	−0.291029	−	0.956714i	$-0.593998\pi$
$569$	−13.8842	−0.582058	−0.291029	+	0.956714i	$0.593998\pi$
$570$	0	0
$571$	1.76966	0.0740580	0.0370290	−	0.999314i	$-0.488211\pi$
$571$	1.76966	0.0740580	0.0370290	+	0.999314i	$0.488211\pi$
$572$	0	0
$573$	−11.1870	−0.467344
$574$	0	0
$575$	−12.8947	−0.537746
$576$	0	0
$577$	8.77335	0.365239	0.182620	−	0.983184i	$-0.441542\pi$
$577$	8.77335	0.365239	0.182620	+	0.983184i	$0.441542\pi$
$578$	0	0
$579$	3.76577	0.156500
$580$	0	0
$581$	−18.2814	−0.758440
$582$	0	0
$583$	−7.13750	−0.295605
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.6953	0.482716	0.241358	−	0.970436i	$-0.422407\pi$
$587$	11.6953	0.482716	0.241358	+	0.970436i	$0.422407\pi$
$588$	0	0
$589$	−2.64110	−0.108825
$590$	0	0
$591$	−11.0241	−0.453470
$592$	0	0
$593$	23.8102	0.977770	0.488885	−	0.872348i	$-0.337404\pi$
$593$	23.8102	0.977770	0.488885	+	0.872348i	$0.337404\pi$
$594$	0	0
$595$	4.16009	0.170547
$596$	0	0
$597$	−24.1418	−0.988056
$598$	0	0
$599$	−11.0976	−0.453436	−0.226718	−	0.973960i	$-0.572800\pi$
$599$	−11.0976	−0.453436	−0.226718	+	0.973960i	$0.572800\pi$
$600$	0	0
$601$	1.10004	0.0448716	0.0224358	−	0.999748i	$-0.492858\pi$
$601$	1.10004	0.0448716	0.0224358	+	0.999748i	$0.492858\pi$
$602$	0	0
$603$	−10.2270	−0.416476
$604$	0	0
$605$	−2.73686	−0.111269
$606$	0	0
$607$	−21.9770	−0.892017	−0.446009	−	0.895029i	$-0.647155\pi$
$607$	−21.9770	−0.892017	−0.446009	+	0.895029i	$0.647155\pi$
$608$	0	0
$609$	−0.676900	−0.0274294
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.0498	1.05214	0.526071	−	0.850441i	$-0.323665\pi$
$613$	26.0498	1.05214	0.526071	+	0.850441i	$0.323665\pi$
$614$	0	0
$615$	−2.20145	−0.0887712
$616$	0	0
$617$	15.4623	0.622487	0.311244	−	0.950330i	$-0.399254\pi$
$617$	15.4623	0.622487	0.311244	+	0.950330i	$0.399254\pi$
$618$	0	0
$619$	25.5564	1.02720	0.513598	−	0.858031i	$-0.328312\pi$
$619$	25.5564	1.02720	0.513598	+	0.858031i	$0.328312\pi$
$620$	0	0
$621$	−2.66784	−0.107057
$622$	0	0
$623$	−6.83274	−0.273748
$624$	0	0
$625$	22.5284	0.901137
$626$	0	0
$627$	1.92339	0.0768130
$628$	0	0
$629$	−37.4186	−1.49198
$630$	0	0
$631$	9.07889	0.361425	0.180712	−	0.983536i	$-0.442160\pi$
$631$	9.07889	0.361425	0.180712	+	0.983536i	$0.442160\pi$
$632$	0	0
$633$	−2.90884	−0.115616
$634$	0	0
$635$	−5.91196	−0.234609
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.2118	0.403972
$640$	0	0
$641$	27.9190	1.10274	0.551368	−	0.834262i	$-0.314106\pi$
$641$	27.9190	1.10274	0.551368	+	0.834262i	$0.314106\pi$
$642$	0	0
$643$	−30.9790	−1.22169	−0.610846	−	0.791749i	$-0.709171\pi$
$643$	−30.9790	−1.22169	−0.610846	+	0.791749i	$0.709171\pi$
$644$	0	0
$645$	1.51694	0.0597295
$646$	0	0
$647$	−50.4618	−1.98386	−0.991929	−	0.126797i	$-0.959530\pi$
$647$	−50.4618	−1.98386	−0.991929	+	0.126797i	$0.959530\pi$
$648$	0	0
$649$	28.1660	1.10561
$650$	0	0
$651$	9.93284	0.389299
$652$	0	0
$653$	−50.8069	−1.98823	−0.994113	−	0.108346i	$-0.965445\pi$
$653$	−50.8069	−1.98823	−0.994113	+	0.108346i	$0.965445\pi$
$654$	0	0
$655$	5.75727	0.224955
$656$	0	0
$657$	−11.8793	−0.463455
$658$	0	0
$659$	41.9078	1.63250	0.816249	−	0.577700i	$-0.196050\pi$
$659$	41.9078	1.63250	0.816249	+	0.577700i	$0.196050\pi$
$660$	0	0
$661$	−12.9638	−0.504232	−0.252116	−	0.967697i	$-0.581127\pi$
$661$	−12.9638	−0.504232	−0.252116	+	0.967697i	$0.581127\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.320777	−0.0124392
$666$	0	0
$667$	−1.05044	−0.0406734
$668$	0	0
$669$	−28.7808	−1.11273
$670$	0	0
$671$	41.4087	1.59856
$672$	0	0
$673$	−34.3636	−1.32462	−0.662309	−	0.749231i	$-0.730423\pi$
$673$	−34.3636	−1.32462	−0.662309	+	0.749231i	$0.730423\pi$
$674$	0	0
$675$	4.83338	0.186037
$676$	0	0
$677$	7.48438	0.287648	0.143824	−	0.989603i	$-0.454060\pi$
$677$	7.48438	0.287648	0.143824	+	0.989603i	$0.454060\pi$
$678$	0	0
$679$	1.46559	0.0562444
$680$	0	0
$681$	−6.20051	−0.237604
$682$	0	0
$683$	21.7659	0.832849	0.416425	−	0.909170i	$-0.363283\pi$
$683$	21.7659	0.832849	0.416425	+	0.909170i	$0.363283\pi$
$684$	0	0
$685$	8.11014	0.309873
$686$	0	0
$687$	24.2058	0.923508
$688$	0	0
$689$	0	0
$690$	0	0
$691$	25.0986	0.954797	0.477399	−	0.878687i	$-0.341580\pi$
$691$	25.0986	0.954797	0.477399	+	0.878687i	$0.341580\pi$
$692$	0	0
$693$	−7.23365	−0.274784
$694$	0	0
$695$	−6.63683	−0.251749
$696$	0	0
$697$	−31.9716	−1.21101
$698$	0	0
$699$	24.3350	0.920434
$700$	0	0
$701$	15.9934	0.604063	0.302031	−	0.953298i	$-0.402335\pi$
$701$	15.9934	0.604063	0.302031	+	0.953298i	$0.402335\pi$
$702$	0	0
$703$	2.88528	0.108820
$704$	0	0
$705$	−2.47843	−0.0933433
$706$	0	0
$707$	−11.4638	−0.431141
$708$	0	0
$709$	47.1473	1.77066	0.885328	−	0.464967i	$-0.153934\pi$
$709$	47.1473	1.77066	0.885328	+	0.464967i	$0.153934\pi$
$710$	0	0
$711$	4.58878	0.172093
$712$	0	0
$713$	15.4142	0.577268
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12.8629	−0.480373
$718$	0	0
$719$	−29.3625	−1.09504	−0.547519	−	0.836793i	$-0.684427\pi$
$719$	−29.3625	−1.09504	−0.547519	+	0.836793i	$0.684427\pi$
$720$	0	0
$721$	−24.7239	−0.920767
$722$	0	0
$723$	22.8119	0.848383
$724$	0	0
$725$	1.90311	0.0706796
$726$	0	0
$727$	11.5849	0.429661	0.214831	−	0.976651i	$-0.431080\pi$
$727$	11.5849	0.429661	0.214831	+	0.976651i	$0.431080\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	22.0305	0.814826
$732$	0	0
$733$	−0.102529	−0.00378699	−0.00189349	−	0.999998i	$-0.500603\pi$
$733$	−0.102529	−0.00378699	−0.00189349	+	0.999998i	$0.500603\pi$
$734$	0	0
$735$	−1.65096	−0.0608967
$736$	0	0
$737$	−43.0323	−1.58511
$738$	0	0
$739$	11.6846	0.429825	0.214912	−	0.976633i	$-0.431053\pi$
$739$	11.6846	0.429825	0.214912	+	0.976633i	$0.431053\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.1397	−1.54596	−0.772978	−	0.634433i	$-0.781234\pi$
$743$	−42.1397	−1.54596	−0.772978	+	0.634433i	$0.781234\pi$
$744$	0	0
$745$	−3.84655	−0.140927
$746$	0	0
$747$	10.6340	0.389078
$748$	0	0
$749$	19.1845	0.700986
$750$	0	0
$751$	−39.4654	−1.44011	−0.720056	−	0.693916i	$-0.755884\pi$
$751$	−39.4654	−1.44011	−0.720056	+	0.693916i	$0.755884\pi$
$752$	0	0
$753$	23.5772	0.859202
$754$	0	0
$755$	−2.00569	−0.0729945
$756$	0	0
$757$	−31.7797	−1.15505	−0.577526	−	0.816372i	$-0.695982\pi$
$757$	−31.7797	−1.15505	−0.577526	+	0.816372i	$0.695982\pi$
$758$	0	0
$759$	−11.2255	−0.407460
$760$	0	0
$761$	−49.8492	−1.80703	−0.903517	−	0.428552i	$-0.859024\pi$
$761$	−49.8492	−1.80703	−0.903517	+	0.428552i	$0.859024\pi$
$762$	0	0
$763$	−22.3769	−0.810098
$764$	0	0
$765$	−2.41986	−0.0874902
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−23.3249	−0.841116	−0.420558	−	0.907266i	$-0.638166\pi$
$769$	−23.3249	−0.841116	−0.420558	+	0.907266i	$0.638166\pi$
$770$	0	0
$771$	−22.0642	−0.794621
$772$	0	0
$773$	−22.8565	−0.822093	−0.411046	−	0.911614i	$-0.634837\pi$
$773$	−22.8565	−0.822093	−0.411046	+	0.911614i	$0.634837\pi$
$774$	0	0
$775$	−27.9262	−1.00314
$776$	0	0
$777$	−10.8512	−0.389284
$778$	0	0
$779$	2.46527	0.0883276
$780$	0	0
$781$	42.9682	1.53752
$782$	0	0
$783$	0.393743	0.0140712
$784$	0	0
$785$	−8.83592	−0.315367
$786$	0	0
$787$	−5.63621	−0.200909	−0.100455	−	0.994942i	$-0.532030\pi$
$787$	−5.63621	−0.200909	−0.100455	+	0.994942i	$0.532030\pi$
$788$	0	0
$789$	−6.67154	−0.237513
$790$	0	0
$791$	−24.0915	−0.856596
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−0.692419	−0.0245576
$796$	0	0
$797$	−37.8118	−1.33936	−0.669681	−	0.742649i	$-0.733569\pi$
$797$	−37.8118	−1.33936	−0.669681	+	0.742649i	$0.733569\pi$
$798$	0	0
$799$	−35.9942	−1.27338
$800$	0	0
$801$	3.97450	0.140432
$802$	0	0
$803$	−49.9845	−1.76392
$804$	0	0
$805$	1.87215	0.0659846
$806$	0	0
$807$	−13.9379	−0.490636
$808$	0	0
$809$	25.2309	0.887072	0.443536	−	0.896257i	$-0.353724\pi$
$809$	25.2309	0.887072	0.443536	+	0.896257i	$0.353724\pi$
$810$	0	0
$811$	0.715704	0.0251318	0.0125659	−	0.999921i	$-0.496000\pi$
$811$	0.715704	0.0251318	0.0125659	+	0.999921i	$0.496000\pi$
$812$	0	0
$813$	14.2528	0.499869
$814$	0	0
$815$	−2.87950	−0.100864
$816$	0	0
$817$	−1.69873	−0.0594310
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36.3316	1.26798	0.633991	−	0.773341i	$-0.281416\pi$
$821$	36.3316	1.26798	0.633991	+	0.773341i	$0.281416\pi$
$822$	0	0
$823$	−10.3172	−0.359634	−0.179817	−	0.983700i	$-0.557551\pi$
$823$	−10.3172	−0.359634	−0.179817	+	0.983700i	$0.557551\pi$
$824$	0	0
$825$	20.3374	0.708058
$826$	0	0
$827$	−25.8450	−0.898720	−0.449360	−	0.893351i	$-0.648348\pi$
$827$	−25.8450	−0.898720	−0.449360	+	0.893351i	$0.648348\pi$
$828$	0	0
$829$	−34.2183	−1.18845	−0.594226	−	0.804298i	$-0.702541\pi$
$829$	−34.2183	−1.18845	−0.594226	+	0.804298i	$0.702541\pi$
$830$	0	0
$831$	−31.4833	−1.09214
$832$	0	0
$833$	−23.9769	−0.830749
$834$	0	0
$835$	6.81215	0.235744
$836$	0	0
$837$	−5.77779	−0.199709
$838$	0	0
$839$	33.2937	1.14943	0.574714	−	0.818354i	$-0.305113\pi$
$839$	33.2937	1.14943	0.574714	+	0.818354i	$0.305113\pi$
$840$	0	0
$841$	−28.8450	−0.994654
$842$	0	0
$843$	5.27630	0.181726
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11.5265	−0.396055
$848$	0	0
$849$	2.06684	0.0709337
$850$	0	0
$851$	−16.8394	−0.577246
$852$	0	0
$853$	−27.1881	−0.930903	−0.465452	−	0.885073i	$-0.654108\pi$
$853$	−27.1881	−0.930903	−0.465452	+	0.885073i	$0.654108\pi$
$854$	0	0
$855$	0.186591	0.00638128
$856$	0	0
$857$	−27.8642	−0.951824	−0.475912	−	0.879493i	$-0.657882\pi$
$857$	−27.8642	−0.951824	−0.475912	+	0.879493i	$0.657882\pi$
$858$	0	0
$859$	0.611552	0.0208659	0.0104329	−	0.999946i	$-0.496679\pi$
$859$	0.611552	0.0208659	0.0104329	+	0.999946i	$0.496679\pi$
$860$	0	0
$861$	−9.27159	−0.315975
$862$	0	0
$863$	−38.1667	−1.29921	−0.649605	−	0.760272i	$-0.725066\pi$
$863$	−38.1667	−1.29921	−0.649605	+	0.760272i	$0.725066\pi$
$864$	0	0
$865$	−2.50677	−0.0852328
$866$	0	0
$867$	−18.1435	−0.616186
$868$	0	0
$869$	19.3082	0.654987
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−0.852514	−0.0288532
$874$	0	0
$875$	−6.90053	−0.233281
$876$	0	0
$877$	5.87470	0.198375	0.0991873	−	0.995069i	$-0.468376\pi$
$877$	5.87470	0.198375	0.0991873	+	0.995069i	$0.468376\pi$
$878$	0	0
$879$	1.33522	0.0450359
$880$	0	0
$881$	−7.95236	−0.267922	−0.133961	−	0.990987i	$-0.542770\pi$
$881$	−7.95236	−0.267922	−0.133961	+	0.990987i	$0.542770\pi$
$882$	0	0
$883$	16.6716	0.561044	0.280522	−	0.959848i	$-0.409493\pi$
$883$	16.6716	0.561044	0.280522	+	0.959848i	$0.409493\pi$
$884$	0	0
$885$	2.73242	0.0918494
$886$	0	0
$887$	7.30830	0.245389	0.122694	−	0.992445i	$-0.460847\pi$
$887$	7.30830	0.245389	0.122694	+	0.992445i	$0.460847\pi$
$888$	0	0
$889$	−24.8987	−0.835074
$890$	0	0
$891$	4.20771	0.140963
$892$	0	0
$893$	2.77545	0.0928767
$894$	0	0
$895$	−5.05510	−0.168973
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.27496	−0.0758742
$900$	0	0
$901$	−10.0560	−0.335013
$902$	0	0
$903$	6.38871	0.212603
$904$	0	0
$905$	5.95923	0.198092
$906$	0	0
$907$	−12.5309	−0.416081	−0.208041	−	0.978120i	$-0.566709\pi$
$907$	−12.5309	−0.416081	−0.208041	+	0.978120i	$0.566709\pi$
$908$	0	0
$909$	6.66833	0.221174
$910$	0	0
$911$	14.9087	0.493946	0.246973	−	0.969022i	$-0.420564\pi$
$911$	14.9087	0.493946	0.246973	+	0.969022i	$0.420564\pi$
$912$	0	0
$913$	44.7448	1.48084
$914$	0	0
$915$	4.01711	0.132802
$916$	0	0
$917$	24.2472	0.800712
$918$	0	0
$919$	7.54795	0.248984	0.124492	−	0.992221i	$-0.460270\pi$
$919$	7.54795	0.248984	0.124492	+	0.992221i	$0.460270\pi$
$920$	0	0
$921$	11.2398	0.370365
$922$	0	0
$923$	0	0
$924$	0	0
$925$	30.5081	1.00310
$926$	0	0
$927$	14.3815	0.472352
$928$	0	0
$929$	−41.5159	−1.36209	−0.681047	−	0.732240i	$-0.738475\pi$
$929$	−41.5159	−1.36209	−0.681047	+	0.732240i	$0.738475\pi$
$930$	0	0
$931$	1.84881	0.0605924
$932$	0	0
$933$	−15.7439	−0.515431
$934$	0	0
$935$	−10.1821	−0.332989
$936$	0	0
$937$	−22.3945	−0.731597	−0.365798	−	0.930694i	$-0.619204\pi$
$937$	−22.3945	−0.731597	−0.365798	+	0.930694i	$0.619204\pi$
$938$	0	0
$939$	−6.62282	−0.216128
$940$	0	0
$941$	18.9475	0.617672	0.308836	−	0.951115i	$-0.400061\pi$
$941$	18.9475	0.617672	0.308836	+	0.951115i	$0.400061\pi$
$942$	0	0
$943$	−14.3881	−0.468540
$944$	0	0
$945$	−0.701746	−0.0228278
$946$	0	0
$947$	−3.53653	−0.114922	−0.0574609	−	0.998348i	$-0.518300\pi$
$947$	−3.53653	−0.114922	−0.0574609	+	0.998348i	$0.518300\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	33.2261	1.07743
$952$	0	0
$953$	8.33340	0.269945	0.134973	−	0.990849i	$-0.456905\pi$
$953$	8.33340	0.269945	0.134973	+	0.990849i	$0.456905\pi$
$954$	0	0
$955$	−4.56648	−0.147768
$956$	0	0
$957$	1.65675	0.0535552
$958$	0	0
$959$	34.1565	1.10297
$960$	0	0
$961$	2.38281	0.0768647
$962$	0	0
$963$	−11.1593	−0.359604
$964$	0	0
$965$	1.53717	0.0494832
$966$	0	0
$967$	−30.6874	−0.986839	−0.493420	−	0.869791i	$-0.664253\pi$
$967$	−30.6874	−0.986839	−0.493420	+	0.869791i	$0.664253\pi$
$968$	0	0
$969$	2.70985	0.0870530
$970$	0	0
$971$	−44.7886	−1.43734	−0.718668	−	0.695353i	$-0.755248\pi$
$971$	−44.7886	−1.43734	−0.718668	+	0.695353i	$0.755248\pi$
$972$	0	0
$973$	−27.9515	−0.896084
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.5920	0.338867	0.169433	−	0.985542i	$-0.445806\pi$
$977$	10.5920	0.338867	0.169433	+	0.985542i	$0.445806\pi$
$978$	0	0
$979$	16.7235	0.534486
$980$	0	0
$981$	13.0163	0.415579
$982$	0	0
$983$	11.1134	0.354463	0.177232	−	0.984169i	$-0.443286\pi$
$983$	11.1134	0.354463	0.177232	+	0.984169i	$0.443286\pi$
$984$	0	0
$985$	−4.49998	−0.143381
$986$	0	0
$987$	−10.4381	−0.332249
$988$	0	0
$989$	9.91429	0.315256
$990$	0	0
$991$	−26.6921	−0.847903	−0.423951	−	0.905685i	$-0.639357\pi$
$991$	−26.6921	−0.847903	−0.423951	+	0.905685i	$0.639357\pi$
$992$	0	0
$993$	−6.39552	−0.202956
$994$	0	0
$995$	−9.85455	−0.312410
$996$	0	0
$997$	1.66767	0.0528156	0.0264078	−	0.999651i	$-0.491593\pi$
$997$	1.66767	0.0528156	0.0264078	+	0.999651i	$0.491593\pi$
$998$	0	0
$999$	6.31197	0.199702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.bg.1.3	yes	6
4.3	odd	2		8112.2.a.cw.1.3		6
13.5	odd	4		4056.2.c.q.337.7		12
13.8	odd	4		4056.2.c.q.337.6		12
13.12	even	2		4056.2.a.bf.1.4	✓	6
52.51	odd	2		8112.2.a.cv.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bf.1.4	✓	6	13.12	even	2
4056.2.a.bg.1.3	yes	6	1.1	even	1	trivial
4056.2.c.q.337.6		12	13.8	odd	4
4056.2.c.q.337.7		12	13.5	odd	4
8112.2.a.cv.1.4		6	52.51	odd	2
8112.2.a.cw.1.3		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.408195 q^{5} -1.71914 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.408195 q^{5} -1.71914 q^{7} +1.00000 q^{9} +4.20771 q^{11} +0.408195 q^{15} +5.92820 q^{17} -0.457112 q^{19} +1.71914 q^{21} +2.66784 q^{23} -4.83338 q^{25} -1.00000 q^{27} -0.393743 q^{29} +5.77779 q^{31} -4.20771 q^{33} +0.701746 q^{35} -6.31197 q^{37} -5.39314 q^{41} +3.71622 q^{43} -0.408195 q^{45} -6.07169 q^{47} -4.04454 q^{49} -5.92820 q^{51} -1.69629 q^{53} -1.71756 q^{55} +0.457112 q^{57} +6.69391 q^{59} +9.84116 q^{61} -1.71914 q^{63} -10.2270 q^{67} -2.66784 q^{69} +10.2118 q^{71} -11.8793 q^{73} +4.83338 q^{75} -7.23365 q^{77} +4.58878 q^{79} +1.00000 q^{81} +10.6340 q^{83} -2.41986 q^{85} +0.393743 q^{87} +3.97450 q^{89} -5.77779 q^{93} +0.186591 q^{95} -0.852514 q^{97} +4.20771 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9	\( 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9} + 8 q^{11} - q^{15} - 5 q^{17} + 19 q^{19} - 7 q^{21} - 6 q^{23} + 17 q^{25} - 6 q^{27} + 3 q^{29} + 9 q^{31} - 8 q^{33} + 4 q^{35} + 6 q^{37} - 15 q^{41} + 11 q^{43} + q^{45} + 5 q^{47} + 5 q^{49} + 5 q^{51} + 12 q^{53} + 7 q^{55} - 19 q^{57} - 5 q^{59} + 17 q^{61} + 7 q^{63} + 13 q^{67} + 6 q^{69} + 26 q^{71} - 49 q^{73} - 17 q^{75} - 25 q^{77} + 14 q^{79} + 6 q^{81} - 3 q^{83} - 19 q^{85} - 3 q^{87} + 13 q^{89} - 9 q^{93} + 43 q^{95} - 14 q^{97} + 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9 + 8 * q^11 - q^15 - 5 * q^17 + 19 * q^19 - 7 * q^21 - 6 * q^23 + 17 * q^25 - 6 * q^27 + 3 * q^29 + 9 * q^31 - 8 * q^33 + 4 * q^35 + 6 * q^37 - 15 * q^41 + 11 * q^43 + q^45 + 5 * q^47 + 5 * q^49 + 5 * q^51 + 12 * q^53 + 7 * q^55 - 19 * q^57 - 5 * q^59 + 17 * q^61 + 7 * q^63 + 13 * q^67 + 6 * q^69 + 26 * q^71 - 49 * q^73 - 17 * q^75 - 25 * q^77 + 14 * q^79 + 6 * q^81 - 3 * q^83 - 19 * q^85 - 3 * q^87 + 13 * q^89 - 9 * q^93 + 43 * q^95 - 14 * q^97 + 8 * q^99

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