LMFDB - Embedded newform 4056.2.a.z.1.2

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:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.52892$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.133492 q^{5} -3.92434 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.133492 q^{5} -3.92434 q^{7} +1.00000 q^{9} -5.05784 q^{11} -0.133492 q^{15} -4.92434 q^{17} +6.79085 q^{19} +3.92434 q^{21} -5.05784 q^{23} -4.98218 q^{25} -1.00000 q^{27} -3.86651 q^{29} +1.13349 q^{31} +5.05784 q^{33} -0.523868 q^{35} +6.92434 q^{37} +5.19133 q^{41} -11.9243 q^{43} +0.133492 q^{45} -9.05784 q^{47} +8.40048 q^{49} +4.92434 q^{51} -1.59952 q^{53} -0.675180 q^{55} -6.79085 q^{57} +11.8487 q^{59} +5.79085 q^{61} -3.92434 q^{63} +4.07566 q^{67} +5.05784 q^{69} -1.05784 q^{71} -3.26698 q^{73} +4.98218 q^{75} +19.8487 q^{77} -7.24916 q^{79} +1.00000 q^{81} -11.3248 q^{83} -0.657360 q^{85} +3.86651 q^{87} +7.73302 q^{89} -1.13349 q^{93} +0.906524 q^{95} +7.13349 q^{97} -5.05784 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 6 q^{19} - 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} + 12 q^{35} + 6 q^{37} - 21 q^{43} - 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} - 6 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 27 q^{67} + 12 q^{71} - 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} - 18 q^{83} + 12 q^{85} + 12 q^{87} + 24 q^{89} - 3 q^{93} - 42 q^{95} + 21 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 6 * q^19 - 3 * q^21 + 15 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 + 12 * q^35 + 6 * q^37 - 21 * q^43 - 12 * q^47 + 24 * q^49 - 6 * q^53 - 18 * q^55 - 6 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 27 * q^67 + 12 * q^71 - 9 * q^73 - 15 * q^75 + 30 * q^77 + 9 * q^79 + 3 * q^81 - 18 * q^83 + 12 * q^85 + 12 * q^87 + 24 * q^89 - 3 * q^93 - 42 * q^95 + 21 * q^97

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
$5$	0.133492	0.0596994	0.0298497	−	0.999554i	$-0.490497\pi$
$5$	0.133492	0.0596994	0.0298497	+	0.999554i	$0.490497\pi$
$6$	0	0
$7$	−3.92434	−1.48326	−0.741631	−	0.670808i	$-0.765948\pi$
$7$	−3.92434	−1.48326	−0.741631	+	0.670808i	$0.765948\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.05784	−1.52499	−0.762497	−	0.646991i	$-0.776027\pi$
$11$	−5.05784	−1.52499	−0.762497	+	0.646991i	$0.776027\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.133492	−0.0344675
$16$	0	0
$17$	−4.92434	−1.19433	−0.597164	−	0.802119i	$-0.703706\pi$
$17$	−4.92434	−1.19433	−0.597164	+	0.802119i	$0.703706\pi$
$18$	0	0
$19$	6.79085	1.55793	0.778964	−	0.627068i	$-0.215745\pi$
$19$	6.79085	1.55793	0.778964	+	0.627068i	$0.215745\pi$
$20$	0	0
$21$	3.92434	0.856362
$22$	0	0
$23$	−5.05784	−1.05463	−0.527316	−	0.849669i	$-0.676802\pi$
$23$	−5.05784	−1.05463	−0.527316	+	0.849669i	$0.676802\pi$
$24$	0	0
$25$	−4.98218	−0.996436
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.86651	−0.717993	−0.358996	−	0.933339i	$-0.616881\pi$
$29$	−3.86651	−0.717993	−0.358996	+	0.933339i	$0.616881\pi$
$30$	0	0
$31$	1.13349	0.203581	0.101791	−	0.994806i	$-0.467543\pi$
$31$	1.13349	0.203581	0.101791	+	0.994806i	$0.467543\pi$
$32$	0	0
$33$	5.05784	0.880456
$34$	0	0
$35$	−0.523868	−0.0885499
$36$	0	0
$37$	6.92434	1.13836	0.569178	−	0.822215i	$-0.307262\pi$
$37$	6.92434	1.13836	0.569178	+	0.822215i	$0.307262\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.19133	0.810749	0.405375	−	0.914151i	$-0.367141\pi$
$41$	5.19133	0.810749	0.405375	+	0.914151i	$0.367141\pi$
$42$	0	0
$43$	−11.9243	−1.81845	−0.909223	−	0.416310i	$-0.863323\pi$
$43$	−11.9243	−1.81845	−0.909223	+	0.416310i	$0.863323\pi$
$44$	0	0
$45$	0.133492	0.0198998
$46$	0	0
$47$	−9.05784	−1.32122	−0.660611	−	0.750729i	$-0.729703\pi$
$47$	−9.05784	−1.32122	−0.660611	+	0.750729i	$0.729703\pi$
$48$	0	0
$49$	8.40048	1.20007
$50$	0	0
$51$	4.92434	0.689546
$52$	0	0
$53$	−1.59952	−0.219712	−0.109856	−	0.993948i	$-0.535039\pi$
$53$	−1.59952	−0.219712	−0.109856	+	0.993948i	$0.535039\pi$
$54$	0	0
$55$	−0.675180	−0.0910413
$56$	0	0
$57$	−6.79085	−0.899470
$58$	0	0
$59$	11.8487	1.54257	0.771284	−	0.636491i	$-0.219615\pi$
$59$	11.8487	1.54257	0.771284	+	0.636491i	$0.219615\pi$
$60$	0	0
$61$	5.79085	0.741443	0.370721	−	0.928744i	$-0.379111\pi$
$61$	5.79085	0.741443	0.370721	+	0.928744i	$0.379111\pi$
$62$	0	0
$63$	−3.92434	−0.494421
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.07566	0.497921	0.248960	−	0.968514i	$-0.419911\pi$
$67$	4.07566	0.497921	0.248960	+	0.968514i	$0.419911\pi$
$68$	0	0
$69$	5.05784	0.608892
$70$	0	0
$71$	−1.05784	−0.125542	−0.0627710	−	0.998028i	$-0.519994\pi$
$71$	−1.05784	−0.125542	−0.0627710	+	0.998028i	$0.519994\pi$
$72$	0	0
$73$	−3.26698	−0.382372	−0.191186	−	0.981554i	$-0.561233\pi$
$73$	−3.26698	−0.382372	−0.191186	+	0.981554i	$0.561233\pi$
$74$	0	0
$75$	4.98218	0.575293
$76$	0	0
$77$	19.8487	2.26197
$78$	0	0
$79$	−7.24916	−0.815595	−0.407797	−	0.913072i	$-0.633703\pi$
$79$	−7.24916	−0.815595	−0.407797	+	0.913072i	$0.633703\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.3248	−1.24306	−0.621530	−	0.783390i	$-0.713489\pi$
$83$	−11.3248	−1.24306	−0.621530	+	0.783390i	$0.713489\pi$
$84$	0	0
$85$	−0.657360	−0.0713007
$86$	0	0
$87$	3.86651	0.414533
$88$	0	0
$89$	7.73302	0.819698	0.409849	−	0.912153i	$-0.365581\pi$
$89$	7.73302	0.819698	0.409849	+	0.912153i	$0.365581\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.13349	−0.117538
$94$	0	0
$95$	0.906524	0.0930074
$96$	0	0
$97$	7.13349	0.724296	0.362148	−	0.932121i	$-0.382043\pi$
$97$	7.13349	0.724296	0.362148	+	0.932121i	$0.382043\pi$
$98$	0	0
$99$	−5.05784	−0.508332
$100$	0	0
$101$	5.71520	0.568683	0.284342	−	0.958723i	$-0.408225\pi$
$101$	5.71520	0.568683	0.284342	+	0.958723i	$0.408225\pi$
$102$	0	0
$103$	6.19133	0.610050	0.305025	−	0.952344i	$-0.401335\pi$
$103$	6.19133	0.610050	0.305025	+	0.952344i	$0.401335\pi$
$104$	0	0
$105$	0.523868	0.0511243
$106$	0	0
$107$	−9.05784	−0.875654	−0.437827	−	0.899059i	$-0.644252\pi$
$107$	−9.05784	−0.875654	−0.437827	+	0.899059i	$0.644252\pi$
$108$	0	0
$109$	12.7152	1.21789	0.608947	−	0.793211i	$-0.291592\pi$
$109$	12.7152	1.21789	0.608947	+	0.793211i	$0.291592\pi$
$110$	0	0
$111$	−6.92434	−0.657230
$112$	0	0
$113$	18.9243	1.78025	0.890126	−	0.455714i	$-0.150616\pi$
$113$	18.9243	1.78025	0.890126	+	0.455714i	$0.150616\pi$
$114$	0	0
$115$	−0.675180	−0.0629609
$116$	0	0
$117$	0	0
$118$	0	0
$119$	19.3248	1.77150
$120$	0	0
$121$	14.5817	1.32561
$122$	0	0
$123$	−5.19133	−0.468086
$124$	0	0
$125$	−1.33254	−0.119186
$126$	0	0
$127$	−7.24916	−0.643259	−0.321630	−	0.946866i	$-0.604231\pi$
$127$	−7.24916	−0.643259	−0.321630	+	0.946866i	$0.604231\pi$
$128$	0	0
$129$	11.9243	1.04988
$130$	0	0
$131$	9.96436	0.870590	0.435295	−	0.900288i	$-0.356644\pi$
$131$	9.96436	0.870590	0.435295	+	0.900288i	$0.356644\pi$
$132$	0	0
$133$	−26.6496	−2.31082
$134$	0	0
$135$	−0.133492	−0.0114892
$136$	0	0
$137$	−9.45831	−0.808078	−0.404039	−	0.914742i	$-0.632394\pi$
$137$	−9.45831	−0.808078	−0.404039	+	0.914742i	$0.632394\pi$
$138$	0	0
$139$	−3.40048	−0.288425	−0.144212	−	0.989547i	$-0.546065\pi$
$139$	−3.40048	−0.288425	−0.144212	+	0.989547i	$0.546065\pi$
$140$	0	0
$141$	9.05784	0.762807
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.516148	−0.0428637
$146$	0	0
$147$	−8.40048	−0.692860
$148$	0	0
$149$	−3.71520	−0.304361	−0.152180	−	0.988353i	$-0.548629\pi$
$149$	−3.71520	−0.304361	−0.152180	+	0.988353i	$0.548629\pi$
$150$	0	0
$151$	3.32482	0.270570	0.135285	−	0.990807i	$-0.456805\pi$
$151$	3.32482	0.270570	0.135285	+	0.990807i	$0.456805\pi$
$152$	0	0
$153$	−4.92434	−0.398110
$154$	0	0
$155$	0.151312	0.0121537
$156$	0	0
$157$	2.32482	0.185541	0.0927704	−	0.995688i	$-0.470428\pi$
$157$	2.32482	0.185541	0.0927704	+	0.995688i	$0.470428\pi$
$158$	0	0
$159$	1.59952	0.126851
$160$	0	0
$161$	19.8487	1.56430
$162$	0	0
$163$	6.86651	0.537826	0.268913	−	0.963164i	$-0.413336\pi$
$163$	6.86651	0.537826	0.268913	+	0.963164i	$0.413336\pi$
$164$	0	0
$165$	0.675180	0.0525627
$166$	0	0
$167$	21.8130	1.68794	0.843972	−	0.536387i	$-0.180211\pi$
$167$	21.8130	1.68794	0.843972	+	0.536387i	$0.180211\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	6.79085	0.519309
$172$	0	0
$173$	−3.58170	−0.272312	−0.136156	−	0.990687i	$-0.543475\pi$
$173$	−3.58170	−0.272312	−0.136156	+	0.990687i	$0.543475\pi$
$174$	0	0
$175$	19.5518	1.47798
$176$	0	0
$177$	−11.8487	−0.890602
$178$	0	0
$179$	−1.20915	−0.0903760	−0.0451880	−	0.998979i	$-0.514389\pi$
$179$	−1.20915	−0.0903760	−0.0451880	+	0.998979i	$0.514389\pi$
$180$	0	0
$181$	18.9243	1.40664	0.703318	−	0.710876i	$-0.251701\pi$
$181$	18.9243	1.40664	0.703318	+	0.710876i	$0.251701\pi$
$182$	0	0
$183$	−5.79085	−0.428072
$184$	0	0
$185$	0.924344	0.0679591
$186$	0	0
$187$	24.9065	1.82135
$188$	0	0
$189$	3.92434	0.285454
$190$	0	0
$191$	7.84869	0.567911	0.283956	−	0.958837i	$-0.408353\pi$
$191$	7.84869	0.567911	0.283956	+	0.958837i	$0.408353\pi$
$192$	0	0
$193$	−7.11567	−0.512197	−0.256099	−	0.966651i	$-0.582437\pi$
$193$	−7.11567	−0.512197	−0.256099	+	0.966651i	$0.582437\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.4304	−1.66935	−0.834673	−	0.550746i	$-0.814343\pi$
$197$	−23.4304	−1.66935	−0.834673	+	0.550746i	$0.814343\pi$
$198$	0	0
$199$	−13.6574	−0.968145	−0.484072	−	0.875028i	$-0.660843\pi$
$199$	−13.6574	−0.968145	−0.484072	+	0.875028i	$0.660843\pi$
$200$	0	0
$201$	−4.07566	−0.287475
$202$	0	0
$203$	15.1735	1.06497
$204$	0	0
$205$	0.693000	0.0484012
$206$	0	0
$207$	−5.05784	−0.351544
$208$	0	0
$209$	−34.3470	−2.37583
$210$	0	0
$211$	−19.2492	−1.32517	−0.662584	−	0.748988i	$-0.730540\pi$
$211$	−19.2492	−1.32517	−0.662584	+	0.748988i	$0.730540\pi$
$212$	0	0
$213$	1.05784	0.0724817
$214$	0	0
$215$	−1.59180	−0.108560
$216$	0	0
$217$	−4.44821	−0.301964
$218$	0	0
$219$	3.26698	0.220762
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−4.98218	−0.332145
$226$	0	0
$227$	23.0222	1.52804	0.764018	−	0.645194i	$-0.223224\pi$
$227$	23.0222	1.52804	0.764018	+	0.645194i	$0.223224\pi$
$228$	0	0
$229$	−8.11567	−0.536299	−0.268149	−	0.963377i	$-0.586412\pi$
$229$	−8.11567	−0.536299	−0.268149	+	0.963377i	$0.586412\pi$
$230$	0	0
$231$	−19.8487	−1.30595
$232$	0	0
$233$	6.53397	0.428054	0.214027	−	0.976828i	$-0.431342\pi$
$233$	6.53397	0.428054	0.214027	+	0.976828i	$0.431342\pi$
$234$	0	0
$235$	−1.20915	−0.0788761
$236$	0	0
$237$	7.24916	0.470884
$238$	0	0
$239$	−11.3248	−0.732542	−0.366271	−	0.930508i	$-0.619366\pi$
$239$	−11.3248	−0.732542	−0.366271	+	0.930508i	$0.619366\pi$
$240$	0	0
$241$	11.5995	0.747191	0.373596	−	0.927592i	$-0.378125\pi$
$241$	11.5995	0.747191	0.373596	+	0.927592i	$0.378125\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.12140	0.0716433
$246$	0	0
$247$	0	0
$248$	0	0
$249$	11.3248	0.717681
$250$	0	0
$251$	−10.1157	−0.638496	−0.319248	−	0.947671i	$-0.603430\pi$
$251$	−10.1157	−0.638496	−0.319248	+	0.947671i	$0.603430\pi$
$252$	0	0
$253$	25.5817	1.60831
$254$	0	0
$255$	0.657360	0.0411655
$256$	0	0
$257$	1.19133	0.0743130	0.0371565	−	0.999309i	$-0.488170\pi$
$257$	1.19133	0.0743130	0.0371565	+	0.999309i	$0.488170\pi$
$258$	0	0
$259$	−27.1735	−1.68848
$260$	0	0
$261$	−3.86651	−0.239331
$262$	0	0
$263$	20.9065	1.28915	0.644576	−	0.764540i	$-0.277034\pi$
$263$	20.9065	1.28915	0.644576	+	0.764540i	$0.277034\pi$
$264$	0	0
$265$	−0.213524	−0.0131166
$266$	0	0
$267$	−7.73302	−0.473253
$268$	0	0
$269$	−5.84869	−0.356601	−0.178300	−	0.983976i	$-0.557060\pi$
$269$	−5.84869	−0.356601	−0.178300	+	0.983976i	$0.557060\pi$
$270$	0	0
$271$	−23.0979	−1.40309	−0.701547	−	0.712623i	$-0.747507\pi$
$271$	−23.0979	−1.40309	−0.701547	+	0.712623i	$0.747507\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.1990	1.51956
$276$	0	0
$277$	4.80867	0.288925	0.144463	−	0.989510i	$-0.453855\pi$
$277$	4.80867	0.288925	0.144463	+	0.989510i	$0.453855\pi$
$278$	0	0
$279$	1.13349	0.0678604
$280$	0	0
$281$	−5.60962	−0.334642	−0.167321	−	0.985902i	$-0.553512\pi$
$281$	−5.60962	−0.334642	−0.167321	+	0.985902i	$0.553512\pi$
$282$	0	0
$283$	25.8887	1.53892	0.769462	−	0.638693i	$-0.220525\pi$
$283$	25.8887	1.53892	0.769462	+	0.638693i	$0.220525\pi$
$284$	0	0
$285$	−0.906524	−0.0536978
$286$	0	0
$287$	−20.3726	−1.20255
$288$	0	0
$289$	7.24916	0.426421
$290$	0	0
$291$	−7.13349	−0.418173
$292$	0	0
$293$	−25.4482	−1.48670	−0.743350	−	0.668902i	$-0.766764\pi$
$293$	−25.4482	−1.48670	−0.743350	+	0.668902i	$0.766764\pi$
$294$	0	0
$295$	1.58170	0.0920904
$296$	0	0
$297$	5.05784	0.293485
$298$	0	0
$299$	0	0
$300$	0	0
$301$	46.7952	2.69723
$302$	0	0
$303$	−5.71520	−0.328329
$304$	0	0
$305$	0.773032	0.0442637
$306$	0	0
$307$	−0.458312	−0.0261572	−0.0130786	−	0.999914i	$-0.504163\pi$
$307$	−0.458312	−0.0261572	−0.0130786	+	0.999914i	$0.504163\pi$
$308$	0	0
$309$	−6.19133	−0.352212
$310$	0	0
$311$	22.7909	1.29235	0.646175	−	0.763189i	$-0.276367\pi$
$311$	22.7909	1.29235	0.646175	+	0.763189i	$0.276367\pi$
$312$	0	0
$313$	28.5639	1.61453	0.807263	−	0.590192i	$-0.200948\pi$
$313$	28.5639	1.61453	0.807263	+	0.590192i	$0.200948\pi$
$314$	0	0
$315$	−0.523868	−0.0295166
$316$	0	0
$317$	−23.9465	−1.34497	−0.672486	−	0.740110i	$-0.734773\pi$
$317$	−23.9465	−1.34497	−0.672486	+	0.740110i	$0.734773\pi$
$318$	0	0
$319$	19.5562	1.09493
$320$	0	0
$321$	9.05784	0.505559
$322$	0	0
$323$	−33.4405	−1.86068
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.7152	−0.703152
$328$	0	0
$329$	35.5461	1.95972
$330$	0	0
$331$	25.1335	1.38146	0.690731	−	0.723112i	$-0.257289\pi$
$331$	25.1335	1.38146	0.690731	+	0.723112i	$0.257289\pi$
$332$	0	0
$333$	6.92434	0.379452
$334$	0	0
$335$	0.544067	0.0297256
$336$	0	0
$337$	28.9644	1.57779	0.788895	−	0.614529i	$-0.210654\pi$
$337$	28.9644	1.57779	0.788895	+	0.614529i	$0.210654\pi$
$338$	0	0
$339$	−18.9243	−1.02783
$340$	0	0
$341$	−5.73302	−0.310460
$342$	0	0
$343$	−5.49595	−0.296753
$344$	0	0
$345$	0.675180	0.0363505
$346$	0	0
$347$	4.67518	0.250977	0.125488	−	0.992095i	$-0.459950\pi$
$347$	4.67518	0.250977	0.125488	+	0.992095i	$0.459950\pi$
$348$	0	0
$349$	27.5161	1.47291	0.736453	−	0.676489i	$-0.236499\pi$
$349$	27.5161	1.47291	0.736453	+	0.676489i	$0.236499\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.5060	0.665630	0.332815	−	0.942992i	$-0.392002\pi$
$353$	12.5060	0.665630	0.332815	+	0.942992i	$0.392002\pi$
$354$	0	0
$355$	−0.141213	−0.00749478
$356$	0	0
$357$	−19.3248	−1.02278
$358$	0	0
$359$	10.7909	0.569519	0.284760	−	0.958599i	$-0.408086\pi$
$359$	10.7909	0.569519	0.284760	+	0.958599i	$0.408086\pi$
$360$	0	0
$361$	27.1157	1.42714
$362$	0	0
$363$	−14.5817	−0.765341
$364$	0	0
$365$	−0.436116	−0.0228274
$366$	0	0
$367$	7.92434	0.413647	0.206824	−	0.978378i	$-0.433687\pi$
$367$	7.92434	0.413647	0.206824	+	0.978378i	$0.433687\pi$
$368$	0	0
$369$	5.19133	0.270250
$370$	0	0
$371$	6.27708	0.325890
$372$	0	0
$373$	−33.3726	−1.72797	−0.863983	−	0.503521i	$-0.832037\pi$
$373$	−33.3726	−1.72797	−0.863983	+	0.503521i	$0.832037\pi$
$374$	0	0
$375$	1.33254	0.0688121
$376$	0	0
$377$	0	0
$378$	0	0
$379$	14.0501	0.721706	0.360853	−	0.932623i	$-0.382486\pi$
$379$	14.0501	0.721706	0.360853	+	0.932623i	$0.382486\pi$
$380$	0	0
$381$	7.24916	0.371386
$382$	0	0
$383$	−12.2313	−0.624992	−0.312496	−	0.949919i	$-0.601165\pi$
$383$	−12.2313	−0.624992	−0.312496	+	0.949919i	$0.601165\pi$
$384$	0	0
$385$	2.64964	0.135038
$386$	0	0
$387$	−11.9243	−0.606148
$388$	0	0
$389$	−28.4805	−1.44402	−0.722010	−	0.691883i	$-0.756781\pi$
$389$	−28.4805	−1.44402	−0.722010	+	0.691883i	$0.756781\pi$
$390$	0	0
$391$	24.9065	1.25958
$392$	0	0
$393$	−9.96436	−0.502635
$394$	0	0
$395$	−0.967705	−0.0486905
$396$	0	0
$397$	27.2135	1.36581	0.682904	−	0.730508i	$-0.260717\pi$
$397$	27.2135	1.36581	0.682904	+	0.730508i	$0.260717\pi$
$398$	0	0
$399$	26.6496	1.33415
$400$	0	0
$401$	18.2391	0.910815	0.455408	−	0.890283i	$-0.349494\pi$
$401$	18.2391	0.910815	0.455408	+	0.890283i	$0.349494\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.133492	0.00663327
$406$	0	0
$407$	−35.0222	−1.73599
$408$	0	0
$409$	−28.0323	−1.38611	−0.693054	−	0.720886i	$-0.743735\pi$
$409$	−28.0323	−1.38611	−0.693054	+	0.720886i	$0.743735\pi$
$410$	0	0
$411$	9.45831	0.466544
$412$	0	0
$413$	−46.4983	−2.28803
$414$	0	0
$415$	−1.51177	−0.0742100
$416$	0	0
$417$	3.40048	0.166522
$418$	0	0
$419$	−15.6974	−0.766867	−0.383433	−	0.923568i	$-0.625258\pi$
$419$	−15.6974	−0.766867	−0.383433	+	0.923568i	$0.625258\pi$
$420$	0	0
$421$	11.5239	0.561639	0.280819	−	0.959761i	$-0.409394\pi$
$421$	11.5239	0.561639	0.280819	+	0.959761i	$0.409394\pi$
$422$	0	0
$423$	−9.05784	−0.440407
$424$	0	0
$425$	24.5340	1.19007
$426$	0	0
$427$	−22.7253	−1.09975
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−23.3248	−1.12352	−0.561759	−	0.827301i	$-0.689875\pi$
$431$	−23.3248	−1.12352	−0.561759	+	0.827301i	$0.689875\pi$
$432$	0	0
$433$	−5.23134	−0.251402	−0.125701	−	0.992068i	$-0.540118\pi$
$433$	−5.23134	−0.251402	−0.125701	+	0.992068i	$0.540118\pi$
$434$	0	0
$435$	0.516148	0.0247474
$436$	0	0
$437$	−34.3470	−1.64304
$438$	0	0
$439$	23.7730	1.13462	0.567312	−	0.823503i	$-0.307983\pi$
$439$	23.7730	1.13462	0.567312	+	0.823503i	$0.307983\pi$
$440$	0	0
$441$	8.40048	0.400023
$442$	0	0
$443$	−10.1157	−0.480610	−0.240305	−	0.970697i	$-0.577247\pi$
$443$	−10.1157	−0.480610	−0.240305	+	0.970697i	$0.577247\pi$
$444$	0	0
$445$	1.03230	0.0489355
$446$	0	0
$447$	3.71520	0.175723
$448$	0	0
$449$	6.53397	0.308357	0.154179	−	0.988043i	$-0.450727\pi$
$449$	6.53397	0.308357	0.154179	+	0.988043i	$0.450727\pi$
$450$	0	0
$451$	−26.2569	−1.23639
$452$	0	0
$453$	−3.32482	−0.156214
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.0800	−1.54742	−0.773709	−	0.633541i	$-0.781601\pi$
$457$	−33.0800	−1.54742	−0.773709	+	0.633541i	$0.781601\pi$
$458$	0	0
$459$	4.92434	0.229849
$460$	0	0
$461$	−0.551788	−0.0256993	−0.0128497	−	0.999917i	$-0.504090\pi$
$461$	−0.551788	−0.0256993	−0.0128497	+	0.999917i	$0.504090\pi$
$462$	0	0
$463$	2.34264	0.108872	0.0544359	−	0.998517i	$-0.482664\pi$
$463$	2.34264	0.108872	0.0544359	+	0.998517i	$0.482664\pi$
$464$	0	0
$465$	−0.151312	−0.00701693
$466$	0	0
$467$	15.1735	0.702146	0.351073	−	0.936348i	$-0.385817\pi$
$467$	15.1735	0.702146	0.351073	+	0.936348i	$0.385817\pi$
$468$	0	0
$469$	−15.9943	−0.738547
$470$	0	0
$471$	−2.32482	−0.107122
$472$	0	0
$473$	60.3114	2.77312
$474$	0	0
$475$	−33.8332	−1.55238
$476$	0	0
$477$	−1.59952	−0.0732372
$478$	0	0
$479$	−1.58170	−0.0722699	−0.0361350	−	0.999347i	$-0.511505\pi$
$479$	−1.58170	−0.0722699	−0.0361350	+	0.999347i	$0.511505\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−19.8487	−0.903147
$484$	0	0
$485$	0.952264	0.0432401
$486$	0	0
$487$	17.2091	0.779821	0.389910	−	0.920853i	$-0.372506\pi$
$487$	17.2091	0.779821	0.389910	+	0.920853i	$0.372506\pi$
$488$	0	0
$489$	−6.86651	−0.310514
$490$	0	0
$491$	−23.7075	−1.06990	−0.534952	−	0.844883i	$-0.679670\pi$
$491$	−23.7075	−1.06990	−0.534952	+	0.844883i	$0.679670\pi$
$492$	0	0
$493$	19.0400	0.857519
$494$	0	0
$495$	−0.675180	−0.0303471
$496$	0	0
$497$	4.15131	0.186212
$498$	0	0
$499$	−6.11567	−0.273775	−0.136888	−	0.990587i	$-0.543710\pi$
$499$	−6.11567	−0.273775	−0.136888	+	0.990587i	$0.543710\pi$
$500$	0	0
$501$	−21.8130	−0.974535
$502$	0	0
$503$	−35.7075	−1.59212	−0.796059	−	0.605219i	$-0.793085\pi$
$503$	−35.7075	−1.59212	−0.796059	+	0.605219i	$0.793085\pi$
$504$	0	0
$505$	0.762933	0.0339501
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.2135	−1.51649	−0.758244	−	0.651971i	$-0.773942\pi$
$509$	−34.2135	−1.51649	−0.758244	+	0.651971i	$0.773942\pi$
$510$	0	0
$511$	12.8208	0.567157
$512$	0	0
$513$	−6.79085	−0.299823
$514$	0	0
$515$	0.826492	0.0364196
$516$	0	0
$517$	45.8130	2.01486
$518$	0	0
$519$	3.58170	0.157219
$520$	0	0
$521$	38.4704	1.68542	0.842710	−	0.538368i	$-0.180959\pi$
$521$	38.4704	1.68542	0.842710	+	0.538368i	$0.180959\pi$
$522$	0	0
$523$	5.44049	0.237896	0.118948	−	0.992900i	$-0.462048\pi$
$523$	5.44049	0.237896	0.118948	+	0.992900i	$0.462048\pi$
$524$	0	0
$525$	−19.5518	−0.853310
$526$	0	0
$527$	−5.58170	−0.243143
$528$	0	0
$529$	2.58170	0.112248
$530$	0	0
$531$	11.8487	0.514189
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1.20915	−0.0522760
$536$	0	0
$537$	1.20915	0.0521786
$538$	0	0
$539$	−42.4882	−1.83010
$540$	0	0
$541$	37.3369	1.60524	0.802620	−	0.596491i	$-0.203439\pi$
$541$	37.3369	1.60524	0.802620	+	0.596491i	$0.203439\pi$
$542$	0	0
$543$	−18.9243	−0.812121
$544$	0	0
$545$	1.69738	0.0727076
$546$	0	0
$547$	−19.2391	−0.822603	−0.411301	−	0.911499i	$-0.634926\pi$
$547$	−19.2391	−0.822603	−0.411301	+	0.911499i	$0.634926\pi$
$548$	0	0
$549$	5.79085	0.247148
$550$	0	0
$551$	−26.2569	−1.11858
$552$	0	0
$553$	28.4482	1.20974
$554$	0	0
$555$	−0.924344	−0.0392362
$556$	0	0
$557$	41.7952	1.77092	0.885460	−	0.464715i	$-0.153843\pi$
$557$	41.7952	1.77092	0.885460	+	0.464715i	$0.153843\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−24.9065	−1.05155
$562$	0	0
$563$	31.0121	1.30700	0.653502	−	0.756925i	$-0.273299\pi$
$563$	31.0121	1.30700	0.653502	+	0.756925i	$0.273299\pi$
$564$	0	0
$565$	2.52625	0.106280
$566$	0	0
$567$	−3.92434	−0.164807
$568$	0	0
$569$	−35.8130	−1.50136	−0.750681	−	0.660665i	$-0.770274\pi$
$569$	−35.8130	−1.50136	−0.750681	+	0.660665i	$0.770274\pi$
$570$	0	0
$571$	−25.2091	−1.05497	−0.527485	−	0.849564i	$-0.676865\pi$
$571$	−25.2091	−1.05497	−0.527485	+	0.849564i	$0.676865\pi$
$572$	0	0
$573$	−7.84869	−0.327884
$574$	0	0
$575$	25.1990	1.05087
$576$	0	0
$577$	36.8988	1.53612	0.768059	−	0.640380i	$-0.221223\pi$
$577$	36.8988	1.53612	0.768059	+	0.640380i	$0.221223\pi$
$578$	0	0
$579$	7.11567	0.295717
$580$	0	0
$581$	44.4425	1.84379
$582$	0	0
$583$	8.09013	0.335059
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.26698	0.0935684	0.0467842	−	0.998905i	$-0.485103\pi$
$587$	2.26698	0.0935684	0.0467842	+	0.998905i	$0.485103\pi$
$588$	0	0
$589$	7.69738	0.317165
$590$	0	0
$591$	23.4304	0.963798
$592$	0	0
$593$	−3.49395	−0.143479	−0.0717397	−	0.997423i	$-0.522855\pi$
$593$	−3.49395	−0.143479	−0.0717397	+	0.997423i	$0.522855\pi$
$594$	0	0
$595$	2.57971	0.105758
$596$	0	0
$597$	13.6574	0.558959
$598$	0	0
$599$	17.1990	0.702734	0.351367	−	0.936238i	$-0.385717\pi$
$599$	17.1990	0.702734	0.351367	+	0.936238i	$0.385717\pi$
$600$	0	0
$601$	−2.51615	−0.102636	−0.0513179	−	0.998682i	$-0.516342\pi$
$601$	−2.51615	−0.102636	−0.0513179	+	0.998682i	$0.516342\pi$
$602$	0	0
$603$	4.07566	0.165974
$604$	0	0
$605$	1.94654	0.0791381
$606$	0	0
$607$	−5.04774	−0.204881	−0.102441	−	0.994739i	$-0.532665\pi$
$607$	−5.04774	−0.204881	−0.102441	+	0.994739i	$0.532665\pi$
$608$	0	0
$609$	−15.1735	−0.614862
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−3.25688	−0.131544	−0.0657722	−	0.997835i	$-0.520951\pi$
$613$	−3.25688	−0.131544	−0.0657722	+	0.997835i	$0.520951\pi$
$614$	0	0
$615$	−0.693000	−0.0279445
$616$	0	0
$617$	−24.2391	−0.975828	−0.487914	−	0.872892i	$-0.662242\pi$
$617$	−24.2391	−0.975828	−0.487914	+	0.872892i	$0.662242\pi$
$618$	0	0
$619$	8.90215	0.357808	0.178904	−	0.983867i	$-0.442745\pi$
$619$	8.90215	0.357808	0.178904	+	0.983867i	$0.442745\pi$
$620$	0	0
$621$	5.05784	0.202964
$622$	0	0
$623$	−30.3470	−1.21583
$624$	0	0
$625$	24.7330	0.989321
$626$	0	0
$627$	34.3470	1.37169
$628$	0	0
$629$	−34.0979	−1.35957
$630$	0	0
$631$	−20.2969	−0.808007	−0.404003	−	0.914757i	$-0.632382\pi$
$631$	−20.2969	−0.808007	−0.404003	+	0.914757i	$0.632382\pi$
$632$	0	0
$633$	19.2492	0.765086
$634$	0	0
$635$	−0.967705	−0.0384022
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.05784	−0.0418473
$640$	0	0
$641$	−15.7253	−0.621112	−0.310556	−	0.950555i	$-0.600515\pi$
$641$	−15.7253	−0.621112	−0.310556	+	0.950555i	$0.600515\pi$
$642$	0	0
$643$	−1.66746	−0.0657582	−0.0328791	−	0.999459i	$-0.510468\pi$
$643$	−1.66746	−0.0657582	−0.0328791	+	0.999459i	$0.510468\pi$
$644$	0	0
$645$	1.59180	0.0626772
$646$	0	0
$647$	37.4304	1.47154	0.735770	−	0.677231i	$-0.236820\pi$
$647$	37.4304	1.47154	0.735770	+	0.677231i	$0.236820\pi$
$648$	0	0
$649$	−59.9287	−2.35241
$650$	0	0
$651$	4.44821	0.174339
$652$	0	0
$653$	−46.6140	−1.82415	−0.912073	−	0.410027i	$-0.865519\pi$
$653$	−46.6140	−1.82415	−0.912073	+	0.410027i	$0.865519\pi$
$654$	0	0
$655$	1.33016	0.0519737
$656$	0	0
$657$	−3.26698	−0.127457
$658$	0	0
$659$	3.46603	0.135017	0.0675087	−	0.997719i	$-0.478495\pi$
$659$	3.46603	0.135017	0.0675087	+	0.997719i	$0.478495\pi$
$660$	0	0
$661$	−18.9543	−0.737235	−0.368618	−	0.929581i	$-0.620169\pi$
$661$	−18.9543	−0.737235	−0.368618	+	0.929581i	$0.620169\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.55751	−0.137954
$666$	0	0
$667$	19.5562	0.757218
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−29.2892	−1.13070
$672$	0	0
$673$	13.9321	0.537042	0.268521	−	0.963274i	$-0.413465\pi$
$673$	13.9321	0.537042	0.268521	+	0.963274i	$0.413465\pi$
$674$	0	0
$675$	4.98218	0.191764
$676$	0	0
$677$	−2.68528	−0.103204	−0.0516018	−	0.998668i	$-0.516433\pi$
$677$	−2.68528	−0.103204	−0.0516018	+	0.998668i	$0.516433\pi$
$678$	0	0
$679$	−27.9943	−1.07432
$680$	0	0
$681$	−23.0222	−0.882212
$682$	0	0
$683$	2.26698	0.0867437	0.0433719	−	0.999059i	$-0.486190\pi$
$683$	2.26698	0.0867437	0.0433719	+	0.999059i	$0.486190\pi$
$684$	0	0
$685$	−1.26261	−0.0482418
$686$	0	0
$687$	8.11567	0.309632
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−2.49395	−0.0948744	−0.0474372	−	0.998874i	$-0.515105\pi$
$691$	−2.49395	−0.0948744	−0.0474372	+	0.998874i	$0.515105\pi$
$692$	0	0
$693$	19.8487	0.753989
$694$	0	0
$695$	−0.453936	−0.0172188
$696$	0	0
$697$	−25.5639	−0.968301
$698$	0	0
$699$	−6.53397	−0.247137
$700$	0	0
$701$	−37.8487	−1.42953	−0.714763	−	0.699367i	$-0.753465\pi$
$701$	−37.8487	−1.42953	−0.714763	+	0.699367i	$0.753465\pi$
$702$	0	0
$703$	47.0222	1.77348
$704$	0	0
$705$	1.20915	0.0455391
$706$	0	0
$707$	−22.4284	−0.843507
$708$	0	0
$709$	−16.4049	−0.616097	−0.308049	−	0.951371i	$-0.599676\pi$
$709$	−16.4049	−0.616097	−0.308049	+	0.951371i	$0.599676\pi$
$710$	0	0
$711$	−7.24916	−0.271865
$712$	0	0
$713$	−5.73302	−0.214703
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.3248	0.422933
$718$	0	0
$719$	−16.0800	−0.599684	−0.299842	−	0.953989i	$-0.596934\pi$
$719$	−16.0800	−0.599684	−0.299842	+	0.953989i	$0.596934\pi$
$720$	0	0
$721$	−24.2969	−0.904864
$722$	0	0
$723$	−11.5995	−0.431391
$724$	0	0
$725$	19.2636	0.715434
$726$	0	0
$727$	−44.6695	−1.65670	−0.828349	−	0.560212i	$-0.810720\pi$
$727$	−44.6695	−1.65670	−0.828349	+	0.560212i	$0.810720\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	58.7196	2.17182
$732$	0	0
$733$	−27.9422	−1.03207	−0.516034	−	0.856568i	$-0.672592\pi$
$733$	−27.9422	−1.03207	−0.516034	+	0.856568i	$0.672592\pi$
$734$	0	0
$735$	−1.12140	−0.0413633
$736$	0	0
$737$	−20.6140	−0.759326
$738$	0	0
$739$	11.4660	0.421785	0.210892	−	0.977509i	$-0.432363\pi$
$739$	11.4660	0.421785	0.210892	+	0.977509i	$0.432363\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.3147	−0.708588	−0.354294	−	0.935134i	$-0.615279\pi$
$743$	−19.3147	−0.708588	−0.354294	+	0.935134i	$0.615279\pi$
$744$	0	0
$745$	−0.495949	−0.0181702
$746$	0	0
$747$	−11.3248	−0.414353
$748$	0	0
$749$	35.5461	1.29882
$750$	0	0
$751$	−9.97446	−0.363973	−0.181987	−	0.983301i	$-0.558253\pi$
$751$	−9.97446	−0.363973	−0.181987	+	0.983301i	$0.558253\pi$
$752$	0	0
$753$	10.1157	0.368636
$754$	0	0
$755$	0.443837	0.0161529
$756$	0	0
$757$	−22.2313	−0.808012	−0.404006	−	0.914756i	$-0.632383\pi$
$757$	−22.2313	−0.808012	−0.404006	+	0.914756i	$0.632383\pi$
$758$	0	0
$759$	−25.5817	−0.928557
$760$	0	0
$761$	−8.26698	−0.299678	−0.149839	−	0.988710i	$-0.547876\pi$
$761$	−8.26698	−0.299678	−0.149839	+	0.988710i	$0.547876\pi$
$762$	0	0
$763$	−49.8988	−1.80646
$764$	0	0
$765$	−0.657360	−0.0237669
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−14.5138	−0.523380	−0.261690	−	0.965152i	$-0.584280\pi$
$769$	−14.5138	−0.523380	−0.261690	+	0.965152i	$0.584280\pi$
$770$	0	0
$771$	−1.19133	−0.0429046
$772$	0	0
$773$	−24.1157	−0.867380	−0.433690	−	0.901062i	$-0.642789\pi$
$773$	−24.1157	−0.867380	−0.433690	+	0.901062i	$0.642789\pi$
$774$	0	0
$775$	−5.64726	−0.202856
$776$	0	0
$777$	27.1735	0.974844
$778$	0	0
$779$	35.2535	1.26309
$780$	0	0
$781$	5.35036	0.191451
$782$	0	0
$783$	3.86651	0.138178
$784$	0	0
$785$	0.310345	0.0110767
$786$	0	0
$787$	35.0979	1.25110	0.625552	−	0.780183i	$-0.284874\pi$
$787$	35.0979	1.25110	0.625552	+	0.780183i	$0.284874\pi$
$788$	0	0
$789$	−20.9065	−0.744292
$790$	0	0
$791$	−74.2656	−2.64058
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.213524	0.00757290
$796$	0	0
$797$	−45.0323	−1.59513	−0.797563	−	0.603236i	$-0.793878\pi$
$797$	−45.0323	−1.59513	−0.797563	+	0.603236i	$0.793878\pi$
$798$	0	0
$799$	44.6039	1.57797
$800$	0	0
$801$	7.73302	0.273233
$802$	0	0
$803$	16.5239	0.583115
$804$	0	0
$805$	2.64964	0.0933875
$806$	0	0
$807$	5.84869	0.205884
$808$	0	0
$809$	47.8208	1.68129	0.840644	−	0.541588i	$-0.182177\pi$
$809$	47.8208	1.68129	0.840644	+	0.541588i	$0.182177\pi$
$810$	0	0
$811$	−6.56388	−0.230489	−0.115245	−	0.993337i	$-0.536765\pi$
$811$	−6.56388	−0.230489	−0.115245	+	0.993337i	$0.536765\pi$
$812$	0	0
$813$	23.0979	0.810077
$814$	0	0
$815$	0.916623	0.0321079
$816$	0	0
$817$	−80.9765	−2.83301
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.81305	0.133076	0.0665381	−	0.997784i	$-0.478805\pi$
$821$	3.81305	0.133076	0.0665381	+	0.997784i	$0.478805\pi$
$822$	0	0
$823$	−10.3470	−0.360674	−0.180337	−	0.983605i	$-0.557719\pi$
$823$	−10.3470	−0.360674	−0.180337	+	0.983605i	$0.557719\pi$
$824$	0	0
$825$	−25.1990	−0.877318
$826$	0	0
$827$	34.6496	1.20489	0.602443	−	0.798162i	$-0.294194\pi$
$827$	34.6496	1.20489	0.602443	+	0.798162i	$0.294194\pi$
$828$	0	0
$829$	−15.1056	−0.524638	−0.262319	−	0.964981i	$-0.584487\pi$
$829$	−15.1056	−0.524638	−0.262319	+	0.964981i	$0.584487\pi$
$830$	0	0
$831$	−4.80867	−0.166811
$832$	0	0
$833$	−41.3668	−1.43328
$834$	0	0
$835$	2.91187	0.100769
$836$	0	0
$837$	−1.13349	−0.0391792
$838$	0	0
$839$	44.8964	1.55000	0.774998	−	0.631963i	$-0.217751\pi$
$839$	44.8964	1.55000	0.774998	+	0.631963i	$0.217751\pi$
$840$	0	0
$841$	−14.0501	−0.484487
$842$	0	0
$843$	5.60962	0.193206
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−57.2236	−1.96623
$848$	0	0
$849$	−25.8887	−0.888498
$850$	0	0
$851$	−35.0222	−1.20055
$852$	0	0
$853$	−18.9543	−0.648982	−0.324491	−	0.945889i	$-0.605193\pi$
$853$	−18.9543	−0.648982	−0.324491	+	0.945889i	$0.605193\pi$
$854$	0	0
$855$	0.906524	0.0310025
$856$	0	0
$857$	−24.2391	−0.827991	−0.413995	−	0.910279i	$-0.635867\pi$
$857$	−24.2391	−0.827991	−0.413995	+	0.910279i	$0.635867\pi$
$858$	0	0
$859$	14.1913	0.484202	0.242101	−	0.970251i	$-0.422163\pi$
$859$	14.1913	0.484202	0.242101	+	0.970251i	$0.422163\pi$
$860$	0	0
$861$	20.3726	0.694295
$862$	0	0
$863$	−31.7075	−1.07934	−0.539668	−	0.841878i	$-0.681450\pi$
$863$	−31.7075	−1.07934	−0.539668	+	0.841878i	$0.681450\pi$
$864$	0	0
$865$	−0.478129	−0.0162569
$866$	0	0
$867$	−7.24916	−0.246195
$868$	0	0
$869$	36.6651	1.24378
$870$	0	0
$871$	0	0
$872$	0	0
$873$	7.13349	0.241432
$874$	0	0
$875$	5.22935	0.176784
$876$	0	0
$877$	40.7374	1.37560	0.687802	−	0.725898i	$-0.258576\pi$
$877$	40.7374	1.37560	0.687802	+	0.725898i	$0.258576\pi$
$878$	0	0
$879$	25.4482	0.858347
$880$	0	0
$881$	−20.3904	−0.686969	−0.343485	−	0.939158i	$-0.611607\pi$
$881$	−20.3904	−0.686969	−0.343485	+	0.939158i	$0.611607\pi$
$882$	0	0
$883$	−3.93444	−0.132405	−0.0662023	−	0.997806i	$-0.521088\pi$
$883$	−3.93444	−0.132405	−0.0662023	+	0.997806i	$0.521088\pi$
$884$	0	0
$885$	−1.58170	−0.0531684
$886$	0	0
$887$	−15.3147	−0.514218	−0.257109	−	0.966382i	$-0.582770\pi$
$887$	−15.3147	−0.514218	−0.257109	+	0.966382i	$0.582770\pi$
$888$	0	0
$889$	28.4482	0.954122
$890$	0	0
$891$	−5.05784	−0.169444
$892$	0	0
$893$	−61.5104	−2.05837
$894$	0	0
$895$	−0.161411	−0.00539539
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.38266	−0.146170
$900$	0	0
$901$	7.87661	0.262408
$902$	0	0
$903$	−46.7952	−1.55725
$904$	0	0
$905$	2.52625	0.0839753
$906$	0	0
$907$	−50.5784	−1.67943	−0.839713	−	0.543030i	$-0.817277\pi$
$907$	−50.5784	−1.67943	−0.839713	+	0.543030i	$0.817277\pi$
$908$	0	0
$909$	5.71520	0.189561
$910$	0	0
$911$	10.3470	0.342812	0.171406	−	0.985200i	$-0.445169\pi$
$911$	10.3470	0.342812	0.171406	+	0.985200i	$0.445169\pi$
$912$	0	0
$913$	57.2791	1.89566
$914$	0	0
$915$	−0.773032	−0.0255556
$916$	0	0
$917$	−39.1036	−1.29131
$918$	0	0
$919$	2.95226	0.0973862	0.0486931	−	0.998814i	$-0.484494\pi$
$919$	2.95226	0.0973862	0.0486931	+	0.998814i	$0.484494\pi$
$920$	0	0
$921$	0.458312	0.0151019
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−34.4983	−1.13430
$926$	0	0
$927$	6.19133	0.203350
$928$	0	0
$929$	8.35474	0.274110	0.137055	−	0.990563i	$-0.456236\pi$
$929$	8.35474	0.274110	0.137055	+	0.990563i	$0.456236\pi$
$930$	0	0
$931$	57.0464	1.86962
$932$	0	0
$933$	−22.7909	−0.746139
$934$	0	0
$935$	3.32482	0.108733
$936$	0	0
$937$	20.3648	0.665290	0.332645	−	0.943052i	$-0.392059\pi$
$937$	20.3648	0.665290	0.332645	+	0.943052i	$0.392059\pi$
$938$	0	0
$939$	−28.5639	−0.932147
$940$	0	0
$941$	−10.5138	−0.342739	−0.171370	−	0.985207i	$-0.554819\pi$
$941$	−10.5138	−0.342739	−0.171370	+	0.985207i	$0.554819\pi$
$942$	0	0
$943$	−26.2569	−0.855042
$944$	0	0
$945$	0.523868	0.0170414
$946$	0	0
$947$	−33.5817	−1.09126	−0.545629	−	0.838027i	$-0.683709\pi$
$947$	−33.5817	−1.09126	−0.545629	+	0.838027i	$0.683709\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	23.9465	0.776520
$952$	0	0
$953$	58.9765	1.91043	0.955217	−	0.295905i	$-0.0956212\pi$
$953$	58.9765	1.91043	0.955217	+	0.295905i	$0.0956212\pi$
$954$	0	0
$955$	1.04774	0.0339040
$956$	0	0
$957$	−19.5562	−0.632161
$958$	0	0
$959$	37.1177	1.19859
$960$	0	0
$961$	−29.7152	−0.958555
$962$	0	0
$963$	−9.05784	−0.291885
$964$	0	0
$965$	−0.949885	−0.0305779
$966$	0	0
$967$	58.9509	1.89573	0.947867	−	0.318667i	$-0.103235\pi$
$967$	58.9509	1.89573	0.947867	+	0.318667i	$0.103235\pi$
$968$	0	0
$969$	33.4405	1.07426
$970$	0	0
$971$	52.1601	1.67390	0.836948	−	0.547282i	$-0.184338\pi$
$971$	52.1601	1.67390	0.836948	+	0.547282i	$0.184338\pi$
$972$	0	0
$973$	13.3446	0.427809
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.5538	−0.753552	−0.376776	−	0.926304i	$-0.622967\pi$
$977$	−23.5538	−0.753552	−0.376776	+	0.926304i	$0.622967\pi$
$978$	0	0
$979$	−39.1123	−1.25004
$980$	0	0
$981$	12.7152	0.405965
$982$	0	0
$983$	−0.453936	−0.0144783	−0.00723916	−	0.999974i	$-0.502304\pi$
$983$	−0.453936	−0.0144783	−0.00723916	+	0.999974i	$0.502304\pi$
$984$	0	0
$985$	−3.12777	−0.0996590
$986$	0	0
$987$	−35.5461	−1.13144
$988$	0	0
$989$	60.3114	1.91779
$990$	0	0
$991$	−1.72292	−0.0547303	−0.0273651	−	0.999626i	$-0.508712\pi$
$991$	−1.72292	−0.0547303	−0.0273651	+	0.999626i	$0.508712\pi$
$992$	0	0
$993$	−25.1335	−0.797587
$994$	0	0
$995$	−1.82315	−0.0577977
$996$	0	0
$997$	−12.3248	−0.390331	−0.195165	−	0.980770i	$-0.562524\pi$
$997$	−12.3248	−0.390331	−0.195165	+	0.980770i	$0.562524\pi$
$998$	0	0
$999$	−6.92434	−0.219077

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.z.1.2		3
4.3	odd	2		8112.2.a.ck.1.2		3
13.3	even	3		312.2.q.e.217.2	✓	6
13.5	odd	4		4056.2.c.m.337.3		6
13.8	odd	4		4056.2.c.m.337.4		6
13.9	even	3		312.2.q.e.289.2	yes	6
13.12	even	2		4056.2.a.y.1.2		3
39.29	odd	6		936.2.t.h.217.2		6
39.35	odd	6		936.2.t.h.289.2		6
52.3	odd	6		624.2.q.j.529.2		6
52.35	odd	6		624.2.q.j.289.2		6
52.51	odd	2		8112.2.a.cl.1.2		3
156.35	even	6		1872.2.t.u.289.2		6
156.107	even	6		1872.2.t.u.1153.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.e.217.2	✓	6	13.3	even	3
312.2.q.e.289.2	yes	6	13.9	even	3
624.2.q.j.289.2		6	52.35	odd	6
624.2.q.j.529.2		6	52.3	odd	6
936.2.t.h.217.2		6	39.29	odd	6
936.2.t.h.289.2		6	39.35	odd	6
1872.2.t.u.289.2		6	156.35	even	6
1872.2.t.u.1153.2		6	156.107	even	6
4056.2.a.y.1.2		3	13.12	even	2
4056.2.a.z.1.2		3	1.1	even	1	trivial
4056.2.c.m.337.3		6	13.5	odd	4
4056.2.c.m.337.4		6	13.8	odd	4
8112.2.a.ck.1.2		3	4.3	odd	2
8112.2.a.cl.1.2		3	52.51	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.133492 q^{5} -3.92434 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.133492 q^{5} -3.92434 q^{7} +1.00000 q^{9} -5.05784 q^{11} -0.133492 q^{15} -4.92434 q^{17} +6.79085 q^{19} +3.92434 q^{21} -5.05784 q^{23} -4.98218 q^{25} -1.00000 q^{27} -3.86651 q^{29} +1.13349 q^{31} +5.05784 q^{33} -0.523868 q^{35} +6.92434 q^{37} +5.19133 q^{41} -11.9243 q^{43} +0.133492 q^{45} -9.05784 q^{47} +8.40048 q^{49} +4.92434 q^{51} -1.59952 q^{53} -0.675180 q^{55} -6.79085 q^{57} +11.8487 q^{59} +5.79085 q^{61} -3.92434 q^{63} +4.07566 q^{67} +5.05784 q^{69} -1.05784 q^{71} -3.26698 q^{73} +4.98218 q^{75} +19.8487 q^{77} -7.24916 q^{79} +1.00000 q^{81} -11.3248 q^{83} -0.657360 q^{85} +3.86651 q^{87} +7.73302 q^{89} -1.13349 q^{93} +0.906524 q^{95} +7.13349 q^{97} -5.05784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 6 q^{19} - 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} + 12 q^{35} + 6 q^{37} - 21 q^{43} - 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} - 6 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 27 q^{67} + 12 q^{71} - 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} - 18 q^{83} + 12 q^{85} + 12 q^{87} + 24 q^{89} - 3 q^{93} - 42 q^{95} + 21 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 6 * q^19 - 3 * q^21 + 15 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 + 12 * q^35 + 6 * q^37 - 21 * q^43 - 12 * q^47 + 24 * q^49 - 6 * q^53 - 18 * q^55 - 6 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 27 * q^67 + 12 * q^71 - 9 * q^73 - 15 * q^75 + 30 * q^77 + 9 * q^79 + 3 * q^81 - 18 * q^83 + 12 * q^85 + 12 * q^87 + 24 * q^89 - 3 * q^93 - 42 * q^95 + 21 * q^97

Introduction

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