LMFDB - Embedded newform 4056.2.a.ba.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.801938 q^{5} -1.69202 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.801938 q^{5} -1.69202 q^{7} +1.00000 q^{9} +1.24698 q^{11} +0.801938 q^{15} -0.445042 q^{17} +2.04892 q^{19} +1.69202 q^{21} +0.801938 q^{23} -4.35690 q^{25} -1.00000 q^{27} -2.46681 q^{29} -1.57673 q^{31} -1.24698 q^{33} +1.35690 q^{35} +9.54288 q^{37} +7.56465 q^{41} -0.286208 q^{43} -0.801938 q^{45} +0.542877 q^{47} -4.13706 q^{49} +0.445042 q^{51} -2.45473 q^{53} -1.00000 q^{55} -2.04892 q^{57} -14.3720 q^{59} +7.92692 q^{61} -1.69202 q^{63} -2.81402 q^{67} -0.801938 q^{69} +6.96077 q^{71} +3.69202 q^{73} +4.35690 q^{75} -2.10992 q^{77} -10.9661 q^{79} +1.00000 q^{81} -4.55496 q^{83} +0.356896 q^{85} +2.46681 q^{87} +8.02177 q^{89} +1.57673 q^{93} -1.64310 q^{95} +9.39612 q^{97} +1.24698 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^5 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9} - q^{11} - 2 q^{15} - q^{17} - 3 q^{19} - 2 q^{23} - 9 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} + q^{33} + 10 q^{37} + q^{41} - 9 q^{43} + 2 q^{45} - 17 q^{47} - 7 q^{49} + q^{51} + 15 q^{53} - 3 q^{55} + 3 q^{57} - 14 q^{59} - 5 q^{61} - 23 q^{67} + 2 q^{69} + 8 q^{71} + 6 q^{73} + 9 q^{75} - 7 q^{77} - 17 q^{79} + 3 q^{81} - 14 q^{83} - 3 q^{85} + 4 q^{87} + 21 q^{89} + 2 q^{93} - 9 q^{95} + 37 q^{97} - q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^5 + 3 * q^9 - q^11 - 2 * q^15 - q^17 - 3 * q^19 - 2 * q^23 - 9 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 + q^33 + 10 * q^37 + q^41 - 9 * q^43 + 2 * q^45 - 17 * q^47 - 7 * q^49 + q^51 + 15 * q^53 - 3 * q^55 + 3 * q^57 - 14 * q^59 - 5 * q^61 - 23 * q^67 + 2 * q^69 + 8 * q^71 + 6 * q^73 + 9 * q^75 - 7 * q^77 - 17 * q^79 + 3 * q^81 - 14 * q^83 - 3 * q^85 + 4 * q^87 + 21 * q^89 + 2 * q^93 - 9 * q^95 + 37 * q^97 - 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
$5$	−0.801938	−0.358637	−0.179319	−	0.983791i	$-0.557389\pi$
$5$	−0.801938	−0.358637	−0.179319	+	0.983791i	$0.557389\pi$
$6$	0	0
$7$	−1.69202	−0.639524	−0.319762	−	0.947498i	$-0.603603\pi$
$7$	−1.69202	−0.639524	−0.319762	+	0.947498i	$0.603603\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.24698	0.375978	0.187989	−	0.982171i	$-0.439803\pi$
$11$	1.24698	0.375978	0.187989	+	0.982171i	$0.439803\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.801938	0.207059
$16$	0	0
$17$	−0.445042	−0.107939	−0.0539693	−	0.998543i	$-0.517187\pi$
$17$	−0.445042	−0.107939	−0.0539693	+	0.998543i	$0.517187\pi$
$18$	0	0
$19$	2.04892	0.470054	0.235027	−	0.971989i	$-0.424482\pi$
$19$	2.04892	0.470054	0.235027	+	0.971989i	$0.424482\pi$
$20$	0	0
$21$	1.69202	0.369229
$22$	0	0
$23$	0.801938	0.167216	0.0836078	−	0.996499i	$-0.473356\pi$
$23$	0.801938	0.167216	0.0836078	+	0.996499i	$0.473356\pi$
$24$	0	0
$25$	−4.35690	−0.871379
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.46681	−0.458076	−0.229038	−	0.973418i	$-0.573558\pi$
$29$	−2.46681	−0.458076	−0.229038	+	0.973418i	$0.573558\pi$
$30$	0	0
$31$	−1.57673	−0.283189	−0.141594	−	0.989925i	$-0.545223\pi$
$31$	−1.57673	−0.283189	−0.141594	+	0.989925i	$0.545223\pi$
$32$	0	0
$33$	−1.24698	−0.217071
$34$	0	0
$35$	1.35690	0.229357
$36$	0	0
$37$	9.54288	1.56884	0.784420	−	0.620230i	$-0.212961\pi$
$37$	9.54288	1.56884	0.784420	+	0.620230i	$0.212961\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.56465	1.18140	0.590700	−	0.806892i	$-0.298852\pi$
$41$	7.56465	1.18140	0.590700	+	0.806892i	$0.298852\pi$
$42$	0	0
$43$	−0.286208	−0.0436464	−0.0218232	−	0.999762i	$-0.506947\pi$
$43$	−0.286208	−0.0436464	−0.0218232	+	0.999762i	$0.506947\pi$
$44$	0	0
$45$	−0.801938	−0.119546
$46$	0	0
$47$	0.542877	0.0791867	0.0395933	−	0.999216i	$-0.487394\pi$
$47$	0.542877	0.0791867	0.0395933	+	0.999216i	$0.487394\pi$
$48$	0	0
$49$	−4.13706	−0.591009
$50$	0	0
$51$	0.445042	0.0623183
$52$	0	0
$53$	−2.45473	−0.337183	−0.168592	−	0.985686i	$-0.553922\pi$
$53$	−2.45473	−0.337183	−0.168592	+	0.985686i	$0.553922\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−2.04892	−0.271386
$58$	0	0
$59$	−14.3720	−1.87107	−0.935535	−	0.353234i	$-0.885082\pi$
$59$	−14.3720	−1.87107	−0.935535	+	0.353234i	$0.885082\pi$
$60$	0	0
$61$	7.92692	1.01494	0.507469	−	0.861670i	$-0.330581\pi$
$61$	7.92692	1.01494	0.507469	+	0.861670i	$0.330581\pi$
$62$	0	0
$63$	−1.69202	−0.213175
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.81402	−0.343787	−0.171894	−	0.985116i	$-0.554989\pi$
$67$	−2.81402	−0.343787	−0.171894	+	0.985116i	$0.554989\pi$
$68$	0	0
$69$	−0.801938	−0.0965420
$70$	0	0
$71$	6.96077	0.826092	0.413046	−	0.910710i	$-0.364465\pi$
$71$	6.96077	0.826092	0.413046	+	0.910710i	$0.364465\pi$
$72$	0	0
$73$	3.69202	0.432118	0.216059	−	0.976380i	$-0.430680\pi$
$73$	3.69202	0.432118	0.216059	+	0.976380i	$0.430680\pi$
$74$	0	0
$75$	4.35690	0.503091
$76$	0	0
$77$	−2.10992	−0.240447
$78$	0	0
$79$	−10.9661	−1.23379	−0.616894	−	0.787046i	$-0.711609\pi$
$79$	−10.9661	−1.23379	−0.616894	+	0.787046i	$0.711609\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.55496	−0.499972	−0.249986	−	0.968250i	$-0.580426\pi$
$83$	−4.55496	−0.499972	−0.249986	+	0.968250i	$0.580426\pi$
$84$	0	0
$85$	0.356896	0.0387108
$86$	0	0
$87$	2.46681	0.264470
$88$	0	0
$89$	8.02177	0.850306	0.425153	−	0.905122i	$-0.360220\pi$
$89$	8.02177	0.850306	0.425153	+	0.905122i	$0.360220\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.57673	0.163499
$94$	0	0
$95$	−1.64310	−0.168579
$96$	0	0
$97$	9.39612	0.954032	0.477016	−	0.878895i	$-0.341718\pi$
$97$	9.39612	0.954032	0.477016	+	0.878895i	$0.341718\pi$
$98$	0	0
$99$	1.24698	0.125326
$100$	0	0
$101$	0.747644	0.0743933	0.0371967	−	0.999308i	$-0.488157\pi$
$101$	0.747644	0.0743933	0.0371967	+	0.999308i	$0.488157\pi$
$102$	0	0
$103$	−14.6353	−1.44206	−0.721031	−	0.692903i	$-0.756332\pi$
$103$	−14.6353	−1.44206	−0.721031	+	0.692903i	$0.756332\pi$
$104$	0	0
$105$	−1.35690	−0.132419
$106$	0	0
$107$	13.6189	1.31659	0.658296	−	0.752759i	$-0.271277\pi$
$107$	13.6189	1.31659	0.658296	+	0.752759i	$0.271277\pi$
$108$	0	0
$109$	−8.27844	−0.792931	−0.396465	−	0.918050i	$-0.629763\pi$
$109$	−8.27844	−0.792931	−0.396465	+	0.918050i	$0.629763\pi$
$110$	0	0
$111$	−9.54288	−0.905770
$112$	0	0
$113$	6.35450	0.597781	0.298891	−	0.954287i	$-0.403383\pi$
$113$	6.35450	0.597781	0.298891	+	0.954287i	$0.403383\pi$
$114$	0	0
$115$	−0.643104	−0.0599698
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.753020	0.0690293
$120$	0	0
$121$	−9.44504	−0.858640
$122$	0	0
$123$	−7.56465	−0.682081
$124$	0	0
$125$	7.50365	0.671147
$126$	0	0
$127$	−13.1957	−1.17093	−0.585463	−	0.810699i	$-0.699087\pi$
$127$	−13.1957	−1.17093	−0.585463	+	0.810699i	$0.699087\pi$
$128$	0	0
$129$	0.286208	0.0251992
$130$	0	0
$131$	−21.0586	−1.83990	−0.919949	−	0.392037i	$-0.871771\pi$
$131$	−21.0586	−1.83990	−0.919949	+	0.392037i	$0.871771\pi$
$132$	0	0
$133$	−3.46681	−0.300611
$134$	0	0
$135$	0.801938	0.0690198
$136$	0	0
$137$	−14.1293	−1.20715	−0.603574	−	0.797307i	$-0.706257\pi$
$137$	−14.1293	−1.20715	−0.603574	+	0.797307i	$0.706257\pi$
$138$	0	0
$139$	4.33273	0.367498	0.183749	−	0.982973i	$-0.441177\pi$
$139$	4.33273	0.367498	0.183749	+	0.982973i	$0.441177\pi$
$140$	0	0
$141$	−0.542877	−0.0457185
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.97823	0.164283
$146$	0	0
$147$	4.13706	0.341219
$148$	0	0
$149$	−15.8756	−1.30058	−0.650290	−	0.759686i	$-0.725353\pi$
$149$	−15.8756	−1.30058	−0.650290	+	0.759686i	$0.725353\pi$
$150$	0	0
$151$	−18.6799	−1.52015	−0.760076	−	0.649834i	$-0.774838\pi$
$151$	−18.6799	−1.52015	−0.760076	+	0.649834i	$0.774838\pi$
$152$	0	0
$153$	−0.445042	−0.0359795
$154$	0	0
$155$	1.26444	0.101562
$156$	0	0
$157$	0.481878	0.0384580	0.0192290	−	0.999815i	$-0.493879\pi$
$157$	0.481878	0.0384580	0.0192290	+	0.999815i	$0.493879\pi$
$158$	0	0
$159$	2.45473	0.194673
$160$	0	0
$161$	−1.35690	−0.106938
$162$	0	0
$163$	−14.8442	−1.16268	−0.581342	−	0.813659i	$-0.697472\pi$
$163$	−14.8442	−1.16268	−0.581342	+	0.813659i	$0.697472\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−4.97584	−0.385042	−0.192521	−	0.981293i	$-0.561666\pi$
$167$	−4.97584	−0.385042	−0.192521	+	0.981293i	$0.561666\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	2.04892	0.156685
$172$	0	0
$173$	−22.0465	−1.67617	−0.838083	−	0.545543i	$-0.816324\pi$
$173$	−22.0465	−1.67617	−0.838083	+	0.545543i	$0.816324\pi$
$174$	0	0
$175$	7.37196	0.557268
$176$	0	0
$177$	14.3720	1.08026
$178$	0	0
$179$	−7.59419	−0.567616	−0.283808	−	0.958881i	$-0.591598\pi$
$179$	−7.59419	−0.567616	−0.283808	+	0.958881i	$0.591598\pi$
$180$	0	0
$181$	−20.0761	−1.49224	−0.746121	−	0.665810i	$-0.768086\pi$
$181$	−20.0761	−1.49224	−0.746121	+	0.665810i	$0.768086\pi$
$182$	0	0
$183$	−7.92692	−0.585975
$184$	0	0
$185$	−7.65279	−0.562645
$186$	0	0
$187$	−0.554958	−0.0405826
$188$	0	0
$189$	1.69202	0.123076
$190$	0	0
$191$	8.55927	0.619327	0.309664	−	0.950846i	$-0.399784\pi$
$191$	8.55927	0.619327	0.309664	+	0.950846i	$0.399784\pi$
$192$	0	0
$193$	15.6353	1.12546	0.562728	−	0.826642i	$-0.309752\pi$
$193$	15.6353	1.12546	0.562728	+	0.826642i	$0.309752\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.7385	1.12132	0.560662	−	0.828044i	$-0.310547\pi$
$197$	15.7385	1.12132	0.560662	+	0.828044i	$0.310547\pi$
$198$	0	0
$199$	−15.8562	−1.12402	−0.562009	−	0.827131i	$-0.689971\pi$
$199$	−15.8562	−1.12402	−0.562009	+	0.827131i	$0.689971\pi$
$200$	0	0
$201$	2.81402	0.198486
$202$	0	0
$203$	4.17390	0.292950
$204$	0	0
$205$	−6.06638	−0.423694
$206$	0	0
$207$	0.801938	0.0557385
$208$	0	0
$209$	2.55496	0.176730
$210$	0	0
$211$	−0.588810	−0.0405354	−0.0202677	−	0.999795i	$-0.506452\pi$
$211$	−0.588810	−0.0405354	−0.0202677	+	0.999795i	$0.506452\pi$
$212$	0	0
$213$	−6.96077	−0.476944
$214$	0	0
$215$	0.229521	0.0156532
$216$	0	0
$217$	2.66786	0.181106
$218$	0	0
$219$	−3.69202	−0.249484
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.42327	0.0953093	0.0476547	−	0.998864i	$-0.484825\pi$
$223$	1.42327	0.0953093	0.0476547	+	0.998864i	$0.484825\pi$
$224$	0	0
$225$	−4.35690	−0.290460
$226$	0	0
$227$	8.61463	0.571773	0.285887	−	0.958263i	$-0.407712\pi$
$227$	8.61463	0.571773	0.285887	+	0.958263i	$0.407712\pi$
$228$	0	0
$229$	−2.91723	−0.192776	−0.0963880	−	0.995344i	$-0.530729\pi$
$229$	−2.91723	−0.192776	−0.0963880	+	0.995344i	$0.530729\pi$
$230$	0	0
$231$	2.10992	0.138822
$232$	0	0
$233$	−0.225209	−0.0147540	−0.00737698	−	0.999973i	$-0.502348\pi$
$233$	−0.225209	−0.0147540	−0.00737698	+	0.999973i	$0.502348\pi$
$234$	0	0
$235$	−0.435353	−0.0283993
$236$	0	0
$237$	10.9661	0.712328
$238$	0	0
$239$	−0.457123	−0.0295689	−0.0147844	−	0.999891i	$-0.504706\pi$
$239$	−0.457123	−0.0295689	−0.0147844	+	0.999891i	$0.504706\pi$
$240$	0	0
$241$	1.33214	0.0858108	0.0429054	−	0.999079i	$-0.486339\pi$
$241$	1.33214	0.0858108	0.0429054	+	0.999079i	$0.486339\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.31767	0.211958
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.55496	0.288659
$250$	0	0
$251$	5.92692	0.374104	0.187052	−	0.982350i	$-0.440107\pi$
$251$	5.92692	0.374104	0.187052	+	0.982350i	$0.440107\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	−0.356896	−0.0223497
$256$	0	0
$257$	−0.252356	−0.0157416	−0.00787078	−	0.999969i	$-0.502505\pi$
$257$	−0.252356	−0.0157416	−0.00787078	+	0.999969i	$0.502505\pi$
$258$	0	0
$259$	−16.1468	−1.00331
$260$	0	0
$261$	−2.46681	−0.152692
$262$	0	0
$263$	20.1497	1.24249	0.621243	−	0.783618i	$-0.286628\pi$
$263$	20.1497	1.24249	0.621243	+	0.783618i	$0.286628\pi$
$264$	0	0
$265$	1.96854	0.120927
$266$	0	0
$267$	−8.02177	−0.490924
$268$	0	0
$269$	−13.2567	−0.808273	−0.404137	−	0.914699i	$-0.632428\pi$
$269$	−13.2567	−0.808273	−0.404137	+	0.914699i	$0.632428\pi$
$270$	0	0
$271$	−11.0489	−0.671174	−0.335587	−	0.942009i	$-0.608935\pi$
$271$	−11.0489	−0.671174	−0.335587	+	0.942009i	$0.608935\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.43296	−0.327620
$276$	0	0
$277$	−31.1987	−1.87455	−0.937273	−	0.348597i	$-0.886658\pi$
$277$	−31.1987	−1.87455	−0.937273	+	0.348597i	$0.886658\pi$
$278$	0	0
$279$	−1.57673	−0.0943963
$280$	0	0
$281$	30.9976	1.84916	0.924581	−	0.380985i	$-0.124415\pi$
$281$	30.9976	1.84916	0.924581	+	0.380985i	$0.124415\pi$
$282$	0	0
$283$	−27.3980	−1.62864	−0.814322	−	0.580413i	$-0.802891\pi$
$283$	−27.3980	−1.62864	−0.814322	+	0.580413i	$0.802891\pi$
$284$	0	0
$285$	1.64310	0.0973291
$286$	0	0
$287$	−12.7995	−0.755533
$288$	0	0
$289$	−16.8019	−0.988349
$290$	0	0
$291$	−9.39612	−0.550811
$292$	0	0
$293$	−7.41789	−0.433358	−0.216679	−	0.976243i	$-0.569523\pi$
$293$	−7.41789	−0.433358	−0.216679	+	0.976243i	$0.569523\pi$
$294$	0	0
$295$	11.5254	0.671036
$296$	0	0
$297$	−1.24698	−0.0723571
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0.484271	0.0279129
$302$	0	0
$303$	−0.747644	−0.0429510
$304$	0	0
$305$	−6.35690	−0.363995
$306$	0	0
$307$	−16.5211	−0.942909	−0.471455	−	0.881890i	$-0.656271\pi$
$307$	−16.5211	−0.942909	−0.471455	+	0.881890i	$0.656271\pi$
$308$	0	0
$309$	14.6353	0.832575
$310$	0	0
$311$	−20.9312	−1.18690	−0.593451	−	0.804870i	$-0.702235\pi$
$311$	−20.9312	−1.18690	−0.593451	+	0.804870i	$0.702235\pi$
$312$	0	0
$313$	28.3183	1.60064	0.800321	−	0.599571i	$-0.204662\pi$
$313$	28.3183	1.60064	0.800321	+	0.599571i	$0.204662\pi$
$314$	0	0
$315$	1.35690	0.0764524
$316$	0	0
$317$	−13.2731	−0.745489	−0.372745	−	0.927934i	$-0.621583\pi$
$317$	−13.2731	−0.745489	−0.372745	+	0.927934i	$0.621583\pi$
$318$	0	0
$319$	−3.07606	−0.172227
$320$	0	0
$321$	−13.6189	−0.760135
$322$	0	0
$323$	−0.911854	−0.0507369
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.27844	0.457799
$328$	0	0
$329$	−0.918559	−0.0506418
$330$	0	0
$331$	−12.5767	−0.691280	−0.345640	−	0.938367i	$-0.612338\pi$
$331$	−12.5767	−0.691280	−0.345640	+	0.938367i	$0.612338\pi$
$332$	0	0
$333$	9.54288	0.522946
$334$	0	0
$335$	2.25667	0.123295
$336$	0	0
$337$	−14.6974	−0.800618	−0.400309	−	0.916380i	$-0.631097\pi$
$337$	−14.6974	−0.800618	−0.400309	+	0.916380i	$0.631097\pi$
$338$	0	0
$339$	−6.35450	−0.345129
$340$	0	0
$341$	−1.96615	−0.106473
$342$	0	0
$343$	18.8442	1.01749
$344$	0	0
$345$	0.643104	0.0346236
$346$	0	0
$347$	1.12737	0.0605206	0.0302603	−	0.999542i	$-0.490366\pi$
$347$	1.12737	0.0605206	0.0302603	+	0.999542i	$0.490366\pi$
$348$	0	0
$349$	21.4034	1.14570	0.572849	−	0.819661i	$-0.305838\pi$
$349$	21.4034	1.14570	0.572849	+	0.819661i	$0.305838\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.1933	1.50058	0.750288	−	0.661111i	$-0.229915\pi$
$353$	28.1933	1.50058	0.750288	+	0.661111i	$0.229915\pi$
$354$	0	0
$355$	−5.58211	−0.296267
$356$	0	0
$357$	−0.753020	−0.0398541
$358$	0	0
$359$	9.85623	0.520192	0.260096	−	0.965583i	$-0.416246\pi$
$359$	9.85623	0.520192	0.260096	+	0.965583i	$0.416246\pi$
$360$	0	0
$361$	−14.8019	−0.779049
$362$	0	0
$363$	9.44504	0.495736
$364$	0	0
$365$	−2.96077	−0.154974
$366$	0	0
$367$	−20.9119	−1.09159	−0.545795	−	0.837919i	$-0.683772\pi$
$367$	−20.9119	−1.09159	−0.545795	+	0.837919i	$0.683772\pi$
$368$	0	0
$369$	7.56465	0.393800
$370$	0	0
$371$	4.15346	0.215637
$372$	0	0
$373$	−9.38835	−0.486111	−0.243055	−	0.970012i	$-0.578150\pi$
$373$	−9.38835	−0.486111	−0.243055	+	0.970012i	$0.578150\pi$
$374$	0	0
$375$	−7.50365	−0.387487
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−11.1274	−0.571575	−0.285787	−	0.958293i	$-0.592255\pi$
$379$	−11.1274	−0.571575	−0.285787	+	0.958293i	$0.592255\pi$
$380$	0	0
$381$	13.1957	0.676035
$382$	0	0
$383$	5.48427	0.280233	0.140117	−	0.990135i	$-0.455252\pi$
$383$	5.48427	0.280233	0.140117	+	0.990135i	$0.455252\pi$
$384$	0	0
$385$	1.69202	0.0862334
$386$	0	0
$387$	−0.286208	−0.0145488
$388$	0	0
$389$	0.799545	0.0405385	0.0202693	−	0.999795i	$-0.493548\pi$
$389$	0.799545	0.0405385	0.0202693	+	0.999795i	$0.493548\pi$
$390$	0	0
$391$	−0.356896	−0.0180490
$392$	0	0
$393$	21.0586	1.06227
$394$	0	0
$395$	8.79417	0.442483
$396$	0	0
$397$	12.3860	0.621634	0.310817	−	0.950470i	$-0.399397\pi$
$397$	12.3860	0.621634	0.310817	+	0.950470i	$0.399397\pi$
$398$	0	0
$399$	3.46681	0.173558
$400$	0	0
$401$	−10.6732	−0.532996	−0.266498	−	0.963835i	$-0.585867\pi$
$401$	−10.6732	−0.532996	−0.266498	+	0.963835i	$0.585867\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−0.801938	−0.0398486
$406$	0	0
$407$	11.8998	0.589850
$408$	0	0
$409$	20.5472	1.01599	0.507997	−	0.861359i	$-0.330386\pi$
$409$	20.5472	1.01599	0.507997	+	0.861359i	$0.330386\pi$
$410$	0	0
$411$	14.1293	0.696947
$412$	0	0
$413$	24.3177	1.19659
$414$	0	0
$415$	3.65279	0.179309
$416$	0	0
$417$	−4.33273	−0.212175
$418$	0	0
$419$	28.3250	1.38376	0.691882	−	0.722010i	$-0.256782\pi$
$419$	28.3250	1.38376	0.691882	+	0.722010i	$0.256782\pi$
$420$	0	0
$421$	−20.3183	−0.990251	−0.495126	−	0.868821i	$-0.664878\pi$
$421$	−20.3183	−0.990251	−0.495126	+	0.868821i	$0.664878\pi$
$422$	0	0
$423$	0.542877	0.0263956
$424$	0	0
$425$	1.93900	0.0940554
$426$	0	0
$427$	−13.4125	−0.649077
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.2597	−1.02404	−0.512021	−	0.858973i	$-0.671103\pi$
$431$	−21.2597	−1.02404	−0.512021	+	0.858973i	$0.671103\pi$
$432$	0	0
$433$	17.0261	0.818221	0.409111	−	0.912485i	$-0.365839\pi$
$433$	17.0261	0.818221	0.409111	+	0.912485i	$0.365839\pi$
$434$	0	0
$435$	−1.97823	−0.0948489
$436$	0	0
$437$	1.64310	0.0786003
$438$	0	0
$439$	−16.4319	−0.784252	−0.392126	−	0.919912i	$-0.628260\pi$
$439$	−16.4319	−0.784252	−0.392126	+	0.919912i	$0.628260\pi$
$440$	0	0
$441$	−4.13706	−0.197003
$442$	0	0
$443$	−12.7385	−0.605227	−0.302613	−	0.953113i	$-0.597859\pi$
$443$	−12.7385	−0.605227	−0.302613	+	0.953113i	$0.597859\pi$
$444$	0	0
$445$	−6.43296	−0.304952
$446$	0	0
$447$	15.8756	0.750891
$448$	0	0
$449$	−10.3558	−0.488722	−0.244361	−	0.969684i	$-0.578578\pi$
$449$	−10.3558	−0.488722	−0.244361	+	0.969684i	$0.578578\pi$
$450$	0	0
$451$	9.43296	0.444181
$452$	0	0
$453$	18.6799	0.877660
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.60494	0.355744	0.177872	−	0.984054i	$-0.443079\pi$
$457$	7.60494	0.355744	0.177872	+	0.984054i	$0.443079\pi$
$458$	0	0
$459$	0.445042	0.0207728
$460$	0	0
$461$	−22.5797	−1.05164	−0.525821	−	0.850595i	$-0.676242\pi$
$461$	−22.5797	−1.05164	−0.525821	+	0.850595i	$0.676242\pi$
$462$	0	0
$463$	−32.3043	−1.50131	−0.750653	−	0.660697i	$-0.770261\pi$
$463$	−32.3043	−1.50131	−0.750653	+	0.660697i	$0.770261\pi$
$464$	0	0
$465$	−1.26444	−0.0586369
$466$	0	0
$467$	28.6039	1.32363	0.661815	−	0.749667i	$-0.269787\pi$
$467$	28.6039	1.32363	0.661815	+	0.749667i	$0.269787\pi$
$468$	0	0
$469$	4.76138	0.219860
$470$	0	0
$471$	−0.481878	−0.0222037
$472$	0	0
$473$	−0.356896	−0.0164101
$474$	0	0
$475$	−8.92692	−0.409595
$476$	0	0
$477$	−2.45473	−0.112394
$478$	0	0
$479$	−1.41013	−0.0644303	−0.0322151	−	0.999481i	$-0.510256\pi$
$479$	−1.41013	−0.0644303	−0.0322151	+	0.999481i	$0.510256\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.35690	0.0617409
$484$	0	0
$485$	−7.53511	−0.342152
$486$	0	0
$487$	25.8310	1.17051	0.585257	−	0.810848i	$-0.300994\pi$
$487$	25.8310	1.17051	0.585257	+	0.810848i	$0.300994\pi$
$488$	0	0
$489$	14.8442	0.671276
$490$	0	0
$491$	−15.1987	−0.685906	−0.342953	−	0.939353i	$-0.611427\pi$
$491$	−15.1987	−0.685906	−0.342953	+	0.939353i	$0.611427\pi$
$492$	0	0
$493$	1.09783	0.0494440
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	−11.7778	−0.528305
$498$	0	0
$499$	18.7332	0.838612	0.419306	−	0.907845i	$-0.362273\pi$
$499$	18.7332	0.838612	0.419306	+	0.907845i	$0.362273\pi$
$500$	0	0
$501$	4.97584	0.222304
$502$	0	0
$503$	−0.161818	−0.00721509	−0.00360754	−	0.999993i	$-0.501148\pi$
$503$	−0.161818	−0.00721509	−0.00360754	+	0.999993i	$0.501148\pi$
$504$	0	0
$505$	−0.599564	−0.0266802
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.40688	−0.283980	−0.141990	−	0.989868i	$-0.545350\pi$
$509$	−6.40688	−0.283980	−0.141990	+	0.989868i	$0.545350\pi$
$510$	0	0
$511$	−6.24698	−0.276350
$512$	0	0
$513$	−2.04892	−0.0904619
$514$	0	0
$515$	11.7366	0.517178
$516$	0	0
$517$	0.676956	0.0297725
$518$	0	0
$519$	22.0465	0.967735
$520$	0	0
$521$	38.1051	1.66942	0.834708	−	0.550693i	$-0.185636\pi$
$521$	38.1051	1.66942	0.834708	+	0.550693i	$0.185636\pi$
$522$	0	0
$523$	−27.1933	−1.18908	−0.594539	−	0.804066i	$-0.702666\pi$
$523$	−27.1933	−1.18908	−0.594539	+	0.804066i	$0.702666\pi$
$524$	0	0
$525$	−7.37196	−0.321739
$526$	0	0
$527$	0.701710	0.0305670
$528$	0	0
$529$	−22.3569	−0.972039
$530$	0	0
$531$	−14.3720	−0.623690
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−10.9215	−0.472179
$536$	0	0
$537$	7.59419	0.327713
$538$	0	0
$539$	−5.15883	−0.222207
$540$	0	0
$541$	−36.5284	−1.57048	−0.785239	−	0.619192i	$-0.787460\pi$
$541$	−36.5284	−1.57048	−0.785239	+	0.619192i	$0.787460\pi$
$542$	0	0
$543$	20.0761	0.861546
$544$	0	0
$545$	6.63879	0.284375
$546$	0	0
$547$	6.89200	0.294681	0.147340	−	0.989086i	$-0.452929\pi$
$547$	6.89200	0.294681	0.147340	+	0.989086i	$0.452929\pi$
$548$	0	0
$549$	7.92692	0.338313
$550$	0	0
$551$	−5.05429	−0.215320
$552$	0	0
$553$	18.5550	0.789037
$554$	0	0
$555$	7.65279	0.324843
$556$	0	0
$557$	−29.4983	−1.24988	−0.624941	−	0.780672i	$-0.714877\pi$
$557$	−29.4983	−1.24988	−0.624941	+	0.780672i	$0.714877\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.554958	0.0234304
$562$	0	0
$563$	32.8135	1.38293	0.691463	−	0.722412i	$-0.256966\pi$
$563$	32.8135	1.38293	0.691463	+	0.722412i	$0.256966\pi$
$564$	0	0
$565$	−5.09592	−0.214387
$566$	0	0
$567$	−1.69202	−0.0710582
$568$	0	0
$569$	28.5392	1.19642	0.598212	−	0.801338i	$-0.295878\pi$
$569$	28.5392	1.19642	0.598212	+	0.801338i	$0.295878\pi$
$570$	0	0
$571$	−1.94677	−0.0814698	−0.0407349	−	0.999170i	$-0.512970\pi$
$571$	−1.94677	−0.0814698	−0.0407349	+	0.999170i	$0.512970\pi$
$572$	0	0
$573$	−8.55927	−0.357569
$574$	0	0
$575$	−3.49396	−0.145708
$576$	0	0
$577$	14.5442	0.605483	0.302742	−	0.953073i	$-0.402098\pi$
$577$	14.5442	0.605483	0.302742	+	0.953073i	$0.402098\pi$
$578$	0	0
$579$	−15.6353	−0.649782
$580$	0	0
$581$	7.70709	0.319744
$582$	0	0
$583$	−3.06100	−0.126774
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.2620	−0.547383	−0.273692	−	0.961818i	$-0.588245\pi$
$587$	−13.2620	−0.547383	−0.273692	+	0.961818i	$0.588245\pi$
$588$	0	0
$589$	−3.23059	−0.133114
$590$	0	0
$591$	−15.7385	−0.647397
$592$	0	0
$593$	−13.6775	−0.561670	−0.280835	−	0.959756i	$-0.590611\pi$
$593$	−13.6775	−0.561670	−0.280835	+	0.959756i	$0.590611\pi$
$594$	0	0
$595$	−0.603875	−0.0247565
$596$	0	0
$597$	15.8562	0.648952
$598$	0	0
$599$	−45.0834	−1.84206	−0.921028	−	0.389496i	$-0.872649\pi$
$599$	−45.0834	−1.84206	−0.921028	+	0.389496i	$0.872649\pi$
$600$	0	0
$601$	−9.09305	−0.370913	−0.185457	−	0.982652i	$-0.559376\pi$
$601$	−9.09305	−0.370913	−0.185457	+	0.982652i	$0.559376\pi$
$602$	0	0
$603$	−2.81402	−0.114596
$604$	0	0
$605$	7.57434	0.307941
$606$	0	0
$607$	22.4534	0.911355	0.455678	−	0.890145i	$-0.349397\pi$
$607$	22.4534	0.911355	0.455678	+	0.890145i	$0.349397\pi$
$608$	0	0
$609$	−4.17390	−0.169135
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−6.43296	−0.259825	−0.129912	−	0.991525i	$-0.541470\pi$
$613$	−6.43296	−0.259825	−0.129912	+	0.991525i	$0.541470\pi$
$614$	0	0
$615$	6.06638	0.244620
$616$	0	0
$617$	23.2319	0.935282	0.467641	−	0.883919i	$-0.345104\pi$
$617$	23.2319	0.935282	0.467641	+	0.883919i	$0.345104\pi$
$618$	0	0
$619$	−35.4088	−1.42320	−0.711600	−	0.702585i	$-0.752029\pi$
$619$	−35.4088	−1.42320	−0.711600	+	0.702585i	$0.752029\pi$
$620$	0	0
$621$	−0.801938	−0.0321807
$622$	0	0
$623$	−13.5730	−0.543791
$624$	0	0
$625$	15.7670	0.630681
$626$	0	0
$627$	−2.55496	−0.102035
$628$	0	0
$629$	−4.24698	−0.169338
$630$	0	0
$631$	25.0925	0.998915	0.499457	−	0.866338i	$-0.333533\pi$
$631$	25.0925	0.998915	0.499457	+	0.866338i	$0.333533\pi$
$632$	0	0
$633$	0.588810	0.0234031
$634$	0	0
$635$	10.5821	0.419938
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.96077	0.275364
$640$	0	0
$641$	37.8743	1.49594	0.747972	−	0.663730i	$-0.231028\pi$
$641$	37.8743	1.49594	0.747972	+	0.663730i	$0.231028\pi$
$642$	0	0
$643$	−7.60925	−0.300080	−0.150040	−	0.988680i	$-0.547940\pi$
$643$	−7.60925	−0.300080	−0.150040	+	0.988680i	$0.547940\pi$
$644$	0	0
$645$	−0.229521	−0.00903739
$646$	0	0
$647$	29.2771	1.15100	0.575501	−	0.817801i	$-0.304807\pi$
$647$	29.2771	1.15100	0.575501	+	0.817801i	$0.304807\pi$
$648$	0	0
$649$	−17.9215	−0.703482
$650$	0	0
$651$	−2.66786	−0.104562
$652$	0	0
$653$	9.12093	0.356930	0.178465	−	0.983946i	$-0.442887\pi$
$653$	9.12093	0.356930	0.178465	+	0.983946i	$0.442887\pi$
$654$	0	0
$655$	16.8877	0.659857
$656$	0	0
$657$	3.69202	0.144039
$658$	0	0
$659$	−36.9197	−1.43819	−0.719094	−	0.694912i	$-0.755443\pi$
$659$	−36.9197	−1.43819	−0.719094	+	0.694912i	$0.755443\pi$
$660$	0	0
$661$	30.5217	1.18716	0.593578	−	0.804776i	$-0.297715\pi$
$661$	30.5217	1.18716	0.593578	+	0.804776i	$0.297715\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.78017	0.107810
$666$	0	0
$667$	−1.97823	−0.0765974
$668$	0	0
$669$	−1.42327	−0.0550269
$670$	0	0
$671$	9.88471	0.381595
$672$	0	0
$673$	−15.8280	−0.610125	−0.305063	−	0.952332i	$-0.598677\pi$
$673$	−15.8280	−0.610125	−0.305063	+	0.952332i	$0.598677\pi$
$674$	0	0
$675$	4.35690	0.167697
$676$	0	0
$677$	36.3002	1.39513	0.697565	−	0.716521i	$-0.254267\pi$
$677$	36.3002	1.39513	0.697565	+	0.716521i	$0.254267\pi$
$678$	0	0
$679$	−15.8984	−0.610126
$680$	0	0
$681$	−8.61463	−0.330113
$682$	0	0
$683$	35.7463	1.36779	0.683897	−	0.729578i	$-0.260284\pi$
$683$	35.7463	1.36779	0.683897	+	0.729578i	$0.260284\pi$
$684$	0	0
$685$	11.3308	0.432928
$686$	0	0
$687$	2.91723	0.111299
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−42.5370	−1.61818	−0.809092	−	0.587682i	$-0.800041\pi$
$691$	−42.5370	−1.61818	−0.809092	+	0.587682i	$0.800041\pi$
$692$	0	0
$693$	−2.10992	−0.0801491
$694$	0	0
$695$	−3.47458	−0.131798
$696$	0	0
$697$	−3.36658	−0.127518
$698$	0	0
$699$	0.225209	0.00851820
$700$	0	0
$701$	49.1396	1.85598	0.927988	−	0.372610i	$-0.121537\pi$
$701$	49.1396	1.85598	0.927988	+	0.372610i	$0.121537\pi$
$702$	0	0
$703$	19.5526	0.737439
$704$	0	0
$705$	0.435353	0.0163963
$706$	0	0
$707$	−1.26503	−0.0475763
$708$	0	0
$709$	43.4413	1.63147	0.815737	−	0.578424i	$-0.196332\pi$
$709$	43.4413	1.63147	0.815737	+	0.578424i	$0.196332\pi$
$710$	0	0
$711$	−10.9661	−0.411263
$712$	0	0
$713$	−1.26444	−0.0473536
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.457123	0.0170716
$718$	0	0
$719$	31.6474	1.18025	0.590125	−	0.807312i	$-0.299079\pi$
$719$	31.6474	1.18025	0.590125	+	0.807312i	$0.299079\pi$
$720$	0	0
$721$	24.7633	0.922233
$722$	0	0
$723$	−1.33214	−0.0495429
$724$	0	0
$725$	10.7476	0.399157
$726$	0	0
$727$	34.7506	1.28883	0.644415	−	0.764676i	$-0.277101\pi$
$727$	34.7506	1.28883	0.644415	+	0.764676i	$0.277101\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.127375	0.00471112
$732$	0	0
$733$	−9.35152	−0.345406	−0.172703	−	0.984974i	$-0.555250\pi$
$733$	−9.35152	−0.345406	−0.172703	+	0.984974i	$0.555250\pi$
$734$	0	0
$735$	−3.31767	−0.122374
$736$	0	0
$737$	−3.50902	−0.129257
$738$	0	0
$739$	24.5609	0.903488	0.451744	−	0.892148i	$-0.350802\pi$
$739$	24.5609	0.903488	0.451744	+	0.892148i	$0.350802\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.6179	−0.572964	−0.286482	−	0.958086i	$-0.592486\pi$
$743$	−15.6179	−0.572964	−0.286482	+	0.958086i	$0.592486\pi$
$744$	0	0
$745$	12.7313	0.466437
$746$	0	0
$747$	−4.55496	−0.166657
$748$	0	0
$749$	−23.0435	−0.841993
$750$	0	0
$751$	17.8006	0.649553	0.324777	−	0.945791i	$-0.394711\pi$
$751$	17.8006	0.649553	0.324777	+	0.945791i	$0.394711\pi$
$752$	0	0
$753$	−5.92692	−0.215989
$754$	0	0
$755$	14.9801	0.545183
$756$	0	0
$757$	13.5211	0.491433	0.245716	−	0.969342i	$-0.420977\pi$
$757$	13.5211	0.491433	0.245716	+	0.969342i	$0.420977\pi$
$758$	0	0
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	−34.0605	−1.23469	−0.617347	−	0.786691i	$-0.711792\pi$
$761$	−34.0605	−1.23469	−0.617347	+	0.786691i	$0.711792\pi$
$762$	0	0
$763$	14.0073	0.507098
$764$	0	0
$765$	0.356896	0.0129036
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−23.3927	−0.843561	−0.421781	−	0.906698i	$-0.638595\pi$
$769$	−23.3927	−0.843561	−0.421781	+	0.906698i	$0.638595\pi$
$770$	0	0
$771$	0.252356	0.00908839
$772$	0	0
$773$	−45.3696	−1.63183	−0.815915	−	0.578172i	$-0.803766\pi$
$773$	−45.3696	−1.63183	−0.815915	+	0.578172i	$0.803766\pi$
$774$	0	0
$775$	6.86964	0.246765
$776$	0	0
$777$	16.1468	0.579262
$778$	0	0
$779$	15.4993	0.555321
$780$	0	0
$781$	8.67994	0.310593
$782$	0	0
$783$	2.46681	0.0881567
$784$	0	0
$785$	−0.386436	−0.0137925
$786$	0	0
$787$	26.0103	0.927166	0.463583	−	0.886053i	$-0.346564\pi$
$787$	26.0103	0.927166	0.463583	+	0.886053i	$0.346564\pi$
$788$	0	0
$789$	−20.1497	−0.717350
$790$	0	0
$791$	−10.7520	−0.382296
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1.96854	−0.0698170
$796$	0	0
$797$	−40.3323	−1.42864	−0.714321	−	0.699818i	$-0.753264\pi$
$797$	−40.3323	−1.42864	−0.714321	+	0.699818i	$0.753264\pi$
$798$	0	0
$799$	−0.241603	−0.00854729
$800$	0	0
$801$	8.02177	0.283435
$802$	0	0
$803$	4.60388	0.162467
$804$	0	0
$805$	1.08815	0.0383521
$806$	0	0
$807$	13.2567	0.466657
$808$	0	0
$809$	−54.3309	−1.91017	−0.955087	−	0.296326i	$-0.904239\pi$
$809$	−54.3309	−1.91017	−0.955087	+	0.296326i	$0.904239\pi$
$810$	0	0
$811$	3.88338	0.136364	0.0681819	−	0.997673i	$-0.478280\pi$
$811$	3.88338	0.136364	0.0681819	+	0.997673i	$0.478280\pi$
$812$	0	0
$813$	11.0489	0.387502
$814$	0	0
$815$	11.9041	0.416982
$816$	0	0
$817$	−0.586417	−0.0205161
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.21206	0.321503	0.160752	−	0.986995i	$-0.448608\pi$
$821$	9.21206	0.321503	0.160752	+	0.986995i	$0.448608\pi$
$822$	0	0
$823$	25.2174	0.879025	0.439512	−	0.898237i	$-0.355151\pi$
$823$	25.2174	0.879025	0.439512	+	0.898237i	$0.355151\pi$
$824$	0	0
$825$	5.43296	0.189151
$826$	0	0
$827$	−7.47411	−0.259900	−0.129950	−	0.991521i	$-0.541482\pi$
$827$	−7.47411	−0.259900	−0.129950	+	0.991521i	$0.541482\pi$
$828$	0	0
$829$	3.30665	0.114845	0.0574224	−	0.998350i	$-0.481712\pi$
$829$	3.30665	0.114845	0.0574224	+	0.998350i	$0.481712\pi$
$830$	0	0
$831$	31.1987	1.08227
$832$	0	0
$833$	1.84117	0.0637926
$834$	0	0
$835$	3.99031	0.138090
$836$	0	0
$837$	1.57673	0.0544997
$838$	0	0
$839$	35.5483	1.22726	0.613631	−	0.789593i	$-0.289708\pi$
$839$	35.5483	1.22726	0.613631	+	0.789593i	$0.289708\pi$
$840$	0	0
$841$	−22.9148	−0.790167
$842$	0	0
$843$	−30.9976	−1.06761
$844$	0	0
$845$	0	0
$846$	0	0
$847$	15.9812	0.549121
$848$	0	0
$849$	27.3980	0.940298
$850$	0	0
$851$	7.65279	0.262334
$852$	0	0
$853$	−35.1885	−1.20483	−0.602415	−	0.798183i	$-0.705795\pi$
$853$	−35.1885	−1.20483	−0.602415	+	0.798183i	$0.705795\pi$
$854$	0	0
$855$	−1.64310	−0.0561930
$856$	0	0
$857$	33.7888	1.15420	0.577102	−	0.816672i	$-0.304184\pi$
$857$	33.7888	1.15420	0.577102	+	0.816672i	$0.304184\pi$
$858$	0	0
$859$	29.7006	1.01337	0.506686	−	0.862130i	$-0.330870\pi$
$859$	29.7006	1.01337	0.506686	+	0.862130i	$0.330870\pi$
$860$	0	0
$861$	12.7995	0.436207
$862$	0	0
$863$	4.84979	0.165089	0.0825444	−	0.996587i	$-0.473695\pi$
$863$	4.84979	0.165089	0.0825444	+	0.996587i	$0.473695\pi$
$864$	0	0
$865$	17.6799	0.601136
$866$	0	0
$867$	16.8019	0.570624
$868$	0	0
$869$	−13.6746	−0.463878
$870$	0	0
$871$	0	0
$872$	0	0
$873$	9.39612	0.318011
$874$	0	0
$875$	−12.6963	−0.429214
$876$	0	0
$877$	18.8974	0.638119	0.319060	−	0.947735i	$-0.396633\pi$
$877$	18.8974	0.638119	0.319060	+	0.947735i	$0.396633\pi$
$878$	0	0
$879$	7.41789	0.250199
$880$	0	0
$881$	15.1260	0.509609	0.254805	−	0.966993i	$-0.417989\pi$
$881$	15.1260	0.509609	0.254805	+	0.966993i	$0.417989\pi$
$882$	0	0
$883$	−4.93038	−0.165920	−0.0829602	−	0.996553i	$-0.526437\pi$
$883$	−4.93038	−0.165920	−0.0829602	+	0.996553i	$0.526437\pi$
$884$	0	0
$885$	−11.5254	−0.387423
$886$	0	0
$887$	47.7697	1.60395	0.801975	−	0.597357i	$-0.203782\pi$
$887$	47.7697	1.60395	0.801975	+	0.597357i	$0.203782\pi$
$888$	0	0
$889$	22.3274	0.748835
$890$	0	0
$891$	1.24698	0.0417754
$892$	0	0
$893$	1.11231	0.0372220
$894$	0	0
$895$	6.09006	0.203568
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.88949	0.129722
$900$	0	0
$901$	1.09246	0.0363950
$902$	0	0
$903$	−0.484271	−0.0161155
$904$	0	0
$905$	16.0998	0.535174
$906$	0	0
$907$	−1.73019	−0.0574499	−0.0287249	−	0.999587i	$-0.509145\pi$
$907$	−1.73019	−0.0574499	−0.0287249	+	0.999587i	$0.509145\pi$
$908$	0	0
$909$	0.747644	0.0247978
$910$	0	0
$911$	3.78794	0.125500	0.0627500	−	0.998029i	$-0.480013\pi$
$911$	3.78794	0.125500	0.0627500	+	0.998029i	$0.480013\pi$
$912$	0	0
$913$	−5.67994	−0.187979
$914$	0	0
$915$	6.35690	0.210152
$916$	0	0
$917$	35.6316	1.17666
$918$	0	0
$919$	−54.6349	−1.80224	−0.901119	−	0.433572i	$-0.857253\pi$
$919$	−54.6349	−1.80224	−0.901119	+	0.433572i	$0.857253\pi$
$920$	0	0
$921$	16.5211	0.544389
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−41.5773	−1.36705
$926$	0	0
$927$	−14.6353	−0.480687
$928$	0	0
$929$	−10.4517	−0.342911	−0.171455	−	0.985192i	$-0.554847\pi$
$929$	−10.4517	−0.342911	−0.171455	+	0.985192i	$0.554847\pi$
$930$	0	0
$931$	−8.47650	−0.277806
$932$	0	0
$933$	20.9312	0.685258
$934$	0	0
$935$	0.445042	0.0145544
$936$	0	0
$937$	30.3830	0.992569	0.496284	−	0.868160i	$-0.334697\pi$
$937$	30.3830	0.992569	0.496284	+	0.868160i	$0.334697\pi$
$938$	0	0
$939$	−28.3183	−0.924131
$940$	0	0
$941$	−13.7457	−0.448098	−0.224049	−	0.974578i	$-0.571928\pi$
$941$	−13.7457	−0.448098	−0.224049	+	0.974578i	$0.571928\pi$
$942$	0	0
$943$	6.06638	0.197548
$944$	0	0
$945$	−1.35690	−0.0441398
$946$	0	0
$947$	42.6939	1.38737	0.693683	−	0.720280i	$-0.255987\pi$
$947$	42.6939	1.38737	0.693683	+	0.720280i	$0.255987\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	13.2731	0.430409
$952$	0	0
$953$	42.7329	1.38425	0.692127	−	0.721775i	$-0.256674\pi$
$953$	42.7329	1.38425	0.692127	+	0.721775i	$0.256674\pi$
$954$	0	0
$955$	−6.86400	−0.222114
$956$	0	0
$957$	3.07606	0.0994350
$958$	0	0
$959$	23.9071	0.771999
$960$	0	0
$961$	−28.5139	−0.919804
$962$	0	0
$963$	13.6189	0.438864
$964$	0	0
$965$	−12.5386	−0.403631
$966$	0	0
$967$	−34.7458	−1.11735	−0.558675	−	0.829386i	$-0.688690\pi$
$967$	−34.7458	−1.11735	−0.558675	+	0.829386i	$0.688690\pi$
$968$	0	0
$969$	0.911854	0.0292930
$970$	0	0
$971$	−28.7706	−0.923292	−0.461646	−	0.887064i	$-0.652741\pi$
$971$	−28.7706	−0.923292	−0.461646	+	0.887064i	$0.652741\pi$
$972$	0	0
$973$	−7.33108	−0.235024
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.798954	−0.0255608	−0.0127804	−	0.999918i	$-0.504068\pi$
$977$	−0.798954	−0.0255608	−0.0127804	+	0.999918i	$0.504068\pi$
$978$	0	0
$979$	10.0030	0.319697
$980$	0	0
$981$	−8.27844	−0.264310
$982$	0	0
$983$	24.2825	0.774491	0.387246	−	0.921977i	$-0.373427\pi$
$983$	24.2825	0.774491	0.387246	+	0.921977i	$0.373427\pi$
$984$	0	0
$985$	−12.6213	−0.402149
$986$	0	0
$987$	0.918559	0.0292380
$988$	0	0
$989$	−0.229521	−0.00729835
$990$	0	0
$991$	−22.4762	−0.713981	−0.356991	−	0.934108i	$-0.616197\pi$
$991$	−22.4762	−0.713981	−0.356991	+	0.934108i	$0.616197\pi$
$992$	0	0
$993$	12.5767	0.399110
$994$	0	0
$995$	12.7157	0.403115
$996$	0	0
$997$	−23.1812	−0.734156	−0.367078	−	0.930190i	$-0.619642\pi$
$997$	−23.1812	−0.734156	−0.367078	+	0.930190i	$0.619642\pi$
$998$	0	0
$999$	−9.54288	−0.301923

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.ba.1.1	yes	3
4.3	odd	2		8112.2.a.cn.1.1		3
13.5	odd	4		4056.2.c.n.337.5		6
13.8	odd	4		4056.2.c.n.337.2		6
13.12	even	2		4056.2.a.x.1.3	✓	3
52.51	odd	2		8112.2.a.ci.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.x.1.3	✓	3	13.12	even	2
4056.2.a.ba.1.1	yes	3	1.1	even	1	trivial
4056.2.c.n.337.2		6	13.8	odd	4
4056.2.c.n.337.5		6	13.5	odd	4
8112.2.a.ci.1.3		3	52.51	odd	2
8112.2.a.cn.1.1		3	4.3	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.801938 q^{5} -1.69202 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.801938 q^{5} -1.69202 q^{7} +1.00000 q^{9} +1.24698 q^{11} +0.801938 q^{15} -0.445042 q^{17} +2.04892 q^{19} +1.69202 q^{21} +0.801938 q^{23} -4.35690 q^{25} -1.00000 q^{27} -2.46681 q^{29} -1.57673 q^{31} -1.24698 q^{33} +1.35690 q^{35} +9.54288 q^{37} +7.56465 q^{41} -0.286208 q^{43} -0.801938 q^{45} +0.542877 q^{47} -4.13706 q^{49} +0.445042 q^{51} -2.45473 q^{53} -1.00000 q^{55} -2.04892 q^{57} -14.3720 q^{59} +7.92692 q^{61} -1.69202 q^{63} -2.81402 q^{67} -0.801938 q^{69} +6.96077 q^{71} +3.69202 q^{73} +4.35690 q^{75} -2.10992 q^{77} -10.9661 q^{79} +1.00000 q^{81} -4.55496 q^{83} +0.356896 q^{85} +2.46681 q^{87} +8.02177 q^{89} +1.57673 q^{93} -1.64310 q^{95} +9.39612 q^{97} +1.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^5 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9} - q^{11} - 2 q^{15} - q^{17} - 3 q^{19} - 2 q^{23} - 9 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} + q^{33} + 10 q^{37} + q^{41} - 9 q^{43} + 2 q^{45} - 17 q^{47} - 7 q^{49} + q^{51} + 15 q^{53} - 3 q^{55} + 3 q^{57} - 14 q^{59} - 5 q^{61} - 23 q^{67} + 2 q^{69} + 8 q^{71} + 6 q^{73} + 9 q^{75} - 7 q^{77} - 17 q^{79} + 3 q^{81} - 14 q^{83} - 3 q^{85} + 4 q^{87} + 21 q^{89} + 2 q^{93} - 9 q^{95} + 37 q^{97} - q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^5 + 3 * q^9 - q^11 - 2 * q^15 - q^17 - 3 * q^19 - 2 * q^23 - 9 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 + q^33 + 10 * q^37 + q^41 - 9 * q^43 + 2 * q^45 - 17 * q^47 - 7 * q^49 + q^51 + 15 * q^53 - 3 * q^55 + 3 * q^57 - 14 * q^59 - 5 * q^61 - 23 * q^67 + 2 * q^69 + 8 * q^71 + 6 * q^73 + 9 * q^75 - 7 * q^77 - 17 * q^79 + 3 * q^81 - 14 * q^83 - 3 * q^85 + 4 * q^87 + 21 * q^89 + 2 * q^93 - 9 * q^95 + 37 * q^97 - q^99

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