LMFDB - Embedded newform 3072.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.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:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 48)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.74912$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.47363 q^{5} +2.55765 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.47363 q^{5} +2.55765 q^{7} +1.00000 q^{9} +0.669808 q^{11} +4.08402 q^{13} -2.47363 q^{15} +6.44549 q^{17} +6.44549 q^{19} -2.55765 q^{21} -2.82843 q^{23} +1.11882 q^{25} -1.00000 q^{27} +4.35480 q^{29} -6.55765 q^{31} -0.669808 q^{33} +6.32666 q^{35} -3.85970 q^{37} -4.08402 q^{39} +0.788632 q^{41} -0.550984 q^{43} +2.47363 q^{45} +2.82843 q^{47} -0.458440 q^{49} -6.44549 q^{51} +3.64520 q^{53} +1.65685 q^{55} -6.44549 q^{57} -5.65685 q^{59} +6.20285 q^{61} +2.55765 q^{63} +10.1023 q^{65} -2.99647 q^{67} +2.82843 q^{69} -5.11529 q^{71} -14.7721 q^{73} -1.11882 q^{75} +1.71313 q^{77} -6.32000 q^{79} +1.00000 q^{81} -0.907457 q^{83} +15.9437 q^{85} -4.35480 q^{87} -6.31724 q^{89} +10.4455 q^{91} +6.55765 q^{93} +15.9437 q^{95} +12.6533 q^{97} +0.669808 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 8 q^{13} - 4 q^{15} + 4 q^{21} + 4 q^{25} - 4 q^{27} + 12 q^{29} - 12 q^{31} + 16 q^{37} - 8 q^{39} + 4 q^{45} + 4 q^{49} + 20 q^{53} - 16 q^{55} + 16 q^{61} - 4 q^{63} - 8 q^{65} + 16 q^{67} + 8 q^{71} - 8 q^{73} - 4 q^{75} + 24 q^{77} - 12 q^{79} + 4 q^{81} + 24 q^{85} - 12 q^{87} - 8 q^{89} + 16 q^{91} + 12 q^{93} + 24 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 8 * q^13 - 4 * q^15 + 4 * q^21 + 4 * q^25 - 4 * q^27 + 12 * q^29 - 12 * q^31 + 16 * q^37 - 8 * q^39 + 4 * q^45 + 4 * q^49 + 20 * q^53 - 16 * q^55 + 16 * q^61 - 4 * q^63 - 8 * q^65 + 16 * q^67 + 8 * q^71 - 8 * q^73 - 4 * q^75 + 24 * q^77 - 12 * q^79 + 4 * q^81 + 24 * q^85 - 12 * q^87 - 8 * q^89 + 16 * q^91 + 12 * q^93 + 24 * q^95

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
$5$	2.47363	1.10624	0.553120	−	0.833102i	$-0.313437\pi$
$5$	2.47363	1.10624	0.553120	+	0.833102i	$0.313437\pi$
$6$	0	0
$7$	2.55765	0.966700	0.483350	−	0.875427i	$-0.339420\pi$
$7$	2.55765	0.966700	0.483350	+	0.875427i	$0.339420\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.669808	0.201955	0.100977	−	0.994889i	$-0.467803\pi$
$11$	0.669808	0.201955	0.100977	+	0.994889i	$0.467803\pi$
$12$	0	0
$13$	4.08402	1.13270	0.566352	−	0.824164i	$-0.308354\pi$
$13$	4.08402	1.13270	0.566352	+	0.824164i	$0.308354\pi$
$14$	0	0
$15$	−2.47363	−0.638687
$16$	0	0
$17$	6.44549	1.56326	0.781630	−	0.623742i	$-0.214389\pi$
$17$	6.44549	1.56326	0.781630	+	0.623742i	$0.214389\pi$
$18$	0	0
$19$	6.44549	1.47870	0.739348	−	0.673323i	$-0.235134\pi$
$19$	6.44549	1.47870	0.739348	+	0.673323i	$0.235134\pi$
$20$	0	0
$21$	−2.55765	−0.558124
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	1.11882	0.223765
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.35480	0.808666	0.404333	−	0.914612i	$-0.367504\pi$
$29$	4.35480	0.808666	0.404333	+	0.914612i	$0.367504\pi$
$30$	0	0
$31$	−6.55765	−1.17779	−0.588894	−	0.808210i	$-0.700437\pi$
$31$	−6.55765	−1.17779	−0.588894	+	0.808210i	$0.700437\pi$
$32$	0	0
$33$	−0.669808	−0.116599
$34$	0	0
$35$	6.32666	1.06940
$36$	0	0
$37$	−3.85970	−0.634531	−0.317265	−	0.948337i	$-0.602765\pi$
$37$	−3.85970	−0.634531	−0.317265	+	0.948337i	$0.602765\pi$
$38$	0	0
$39$	−4.08402	−0.653967
$40$	0	0
$41$	0.788632	0.123164	0.0615818	−	0.998102i	$-0.480385\pi$
$41$	0.788632	0.123164	0.0615818	+	0.998102i	$0.480385\pi$
$42$	0	0
$43$	−0.550984	−0.0840242	−0.0420121	−	0.999117i	$-0.513377\pi$
$43$	−0.550984	−0.0840242	−0.0420121	+	0.999117i	$0.513377\pi$
$44$	0	0
$45$	2.47363	0.368746
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−0.458440	−0.0654915
$50$	0	0
$51$	−6.44549	−0.902549
$52$	0	0
$53$	3.64520	0.500707	0.250353	−	0.968155i	$-0.419453\pi$
$53$	3.64520	0.500707	0.250353	+	0.968155i	$0.419453\pi$
$54$	0	0
$55$	1.65685	0.223410
$56$	0	0
$57$	−6.44549	−0.853726
$58$	0	0
$59$	−5.65685	−0.736460	−0.368230	−	0.929735i	$-0.620036\pi$
$59$	−5.65685	−0.736460	−0.368230	+	0.929735i	$0.620036\pi$
$60$	0	0
$61$	6.20285	0.794193	0.397097	−	0.917777i	$-0.370018\pi$
$61$	6.20285	0.794193	0.397097	+	0.917777i	$0.370018\pi$
$62$	0	0
$63$	2.55765	0.322233
$64$	0	0
$65$	10.1023	1.25304
$66$	0	0
$67$	−2.99647	−0.366077	−0.183039	−	0.983106i	$-0.558593\pi$
$67$	−2.99647	−0.366077	−0.183039	+	0.983106i	$0.558593\pi$
$68$	0	0
$69$	2.82843	0.340503
$70$	0	0
$71$	−5.11529	−0.607074	−0.303537	−	0.952820i	$-0.598168\pi$
$71$	−5.11529	−0.607074	−0.303537	+	0.952820i	$0.598168\pi$
$72$	0	0
$73$	−14.7721	−1.72895	−0.864475	−	0.502676i	$-0.832349\pi$
$73$	−14.7721	−1.72895	−0.864475	+	0.502676i	$0.832349\pi$
$74$	0	0
$75$	−1.11882	−0.129191
$76$	0	0
$77$	1.71313	0.195230
$78$	0	0
$79$	−6.32000	−0.711055	−0.355528	−	0.934666i	$-0.615699\pi$
$79$	−6.32000	−0.711055	−0.355528	+	0.934666i	$0.615699\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.907457	−0.0996063	−0.0498032	−	0.998759i	$-0.515859\pi$
$83$	−0.907457	−0.0996063	−0.0498032	+	0.998759i	$0.515859\pi$
$84$	0	0
$85$	15.9437	1.72934
$86$	0	0
$87$	−4.35480	−0.466884
$88$	0	0
$89$	−6.31724	−0.669626	−0.334813	−	0.942285i	$-0.608673\pi$
$89$	−6.31724	−0.669626	−0.334813	+	0.942285i	$0.608673\pi$
$90$	0	0
$91$	10.4455	1.09498
$92$	0	0
$93$	6.55765	0.679996
$94$	0	0
$95$	15.9437	1.63579
$96$	0	0
$97$	12.6533	1.28475	0.642375	−	0.766390i	$-0.277949\pi$
$97$	12.6533	1.28475	0.642375	+	0.766390i	$0.277949\pi$
$98$	0	0
$99$	0.669808	0.0673182
$100$	0	0
$101$	10.6417	1.05889	0.529443	−	0.848346i	$-0.322401\pi$
$101$	10.6417	1.05889	0.529443	+	0.848346i	$0.322401\pi$
$102$	0	0
$103$	−3.33686	−0.328790	−0.164395	−	0.986395i	$-0.552567\pi$
$103$	−3.33686	−0.328790	−0.164395	+	0.986395i	$0.552567\pi$
$104$	0	0
$105$	−6.32666	−0.617419
$106$	0	0
$107$	−19.8874	−1.92259	−0.961296	−	0.275518i	$-0.911151\pi$
$107$	−19.8874	−1.92259	−0.961296	+	0.275518i	$0.911151\pi$
$108$	0	0
$109$	3.91598	0.375083	0.187541	−	0.982257i	$-0.439948\pi$
$109$	3.91598	0.375083	0.187541	+	0.982257i	$0.439948\pi$
$110$	0	0
$111$	3.85970	0.366347
$112$	0	0
$113$	−2.23765	−0.210500	−0.105250	−	0.994446i	$-0.533564\pi$
$113$	−2.23765	−0.210500	−0.105250	+	0.994446i	$0.533564\pi$
$114$	0	0
$115$	−6.99647	−0.652424
$116$	0	0
$117$	4.08402	0.377568
$118$	0	0
$119$	16.4853	1.51120
$120$	0	0
$121$	−10.5514	−0.959214
$122$	0	0
$123$	−0.788632	−0.0711086
$124$	0	0
$125$	−9.60058	−0.858702
$126$	0	0
$127$	−12.2145	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$127$	−12.2145	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$128$	0	0
$129$	0.550984	0.0485114
$130$	0	0
$131$	−5.33962	−0.466524	−0.233262	−	0.972414i	$-0.574940\pi$
$131$	−5.33962	−0.466524	−0.233262	+	0.972414i	$0.574940\pi$
$132$	0	0
$133$	16.4853	1.42946
$134$	0	0
$135$	−2.47363	−0.212896
$136$	0	0
$137$	−5.10587	−0.436224	−0.218112	−	0.975924i	$-0.569990\pi$
$137$	−5.10587	−0.436224	−0.218112	+	0.975924i	$0.569990\pi$
$138$	0	0
$139$	16.6533	1.41252	0.706258	−	0.707954i	$-0.250382\pi$
$139$	16.6533	1.41252	0.706258	+	0.707954i	$0.250382\pi$
$140$	0	0
$141$	−2.82843	−0.238197
$142$	0	0
$143$	2.73551	0.228755
$144$	0	0
$145$	10.7721	0.894578
$146$	0	0
$147$	0.458440	0.0378115
$148$	0	0
$149$	11.1832	0.916166	0.458083	−	0.888909i	$-0.348536\pi$
$149$	11.1832	0.916166	0.458083	+	0.888909i	$0.348536\pi$
$150$	0	0
$151$	−14.6506	−1.19225	−0.596123	−	0.802893i	$-0.703293\pi$
$151$	−14.6506	−1.19225	−0.596123	+	0.802893i	$0.703293\pi$
$152$	0	0
$153$	6.44549	0.521087
$154$	0	0
$155$	−16.2212	−1.30292
$156$	0	0
$157$	4.45754	0.355750	0.177875	−	0.984053i	$-0.443078\pi$
$157$	4.45754	0.355750	0.177875	+	0.984053i	$0.443078\pi$
$158$	0	0
$159$	−3.64520	−0.289083
$160$	0	0
$161$	−7.23412	−0.570128
$162$	0	0
$163$	7.78510	0.609776	0.304888	−	0.952388i	$-0.401381\pi$
$163$	7.78510	0.609776	0.304888	+	0.952388i	$0.401381\pi$
$164$	0	0
$165$	−1.65685	−0.128986
$166$	0	0
$167$	20.1814	1.56168	0.780841	−	0.624730i	$-0.214791\pi$
$167$	20.1814	1.56168	0.780841	+	0.624730i	$0.214791\pi$
$168$	0	0
$169$	3.67923	0.283018
$170$	0	0
$171$	6.44549	0.492899
$172$	0	0
$173$	6.15639	0.468061	0.234031	−	0.972229i	$-0.424808\pi$
$173$	6.15639	0.468061	0.234031	+	0.972229i	$0.424808\pi$
$174$	0	0
$175$	2.86156	0.216313
$176$	0	0
$177$	5.65685	0.425195
$178$	0	0
$179$	18.7855	1.40409	0.702046	−	0.712131i	$-0.252270\pi$
$179$	18.7855	1.40409	0.702046	+	0.712131i	$0.252270\pi$
$180$	0	0
$181$	−8.97499	−0.667106	−0.333553	−	0.942731i	$-0.608248\pi$
$181$	−8.97499	−0.667106	−0.333553	+	0.942731i	$0.608248\pi$
$182$	0	0
$183$	−6.20285	−0.458528
$184$	0	0
$185$	−9.54745	−0.701943
$186$	0	0
$187$	4.31724	0.315708
$188$	0	0
$189$	−2.55765	−0.186041
$190$	0	0
$191$	−5.60058	−0.405243	−0.202622	−	0.979257i	$-0.564946\pi$
$191$	−5.60058	−0.405243	−0.202622	+	0.979257i	$0.564946\pi$
$192$	0	0
$193$	−19.4514	−1.40014	−0.700071	−	0.714074i	$-0.746848\pi$
$193$	−19.4514	−1.40014	−0.700071	+	0.714074i	$0.746848\pi$
$194$	0	0
$195$	−10.1023	−0.723444
$196$	0	0
$197$	1.75070	0.124732	0.0623659	−	0.998053i	$-0.480135\pi$
$197$	1.75070	0.124732	0.0623659	+	0.998053i	$0.480135\pi$
$198$	0	0
$199$	0.993710	0.0704422	0.0352211	−	0.999380i	$-0.488786\pi$
$199$	0.993710	0.0704422	0.0352211	+	0.999380i	$0.488786\pi$
$200$	0	0
$201$	2.99647	0.211355
$202$	0	0
$203$	11.1380	0.781738
$204$	0	0
$205$	1.95078	0.136248
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	4.31724	0.298630
$210$	0	0
$211$	−5.97409	−0.411273	−0.205637	−	0.978628i	$-0.565927\pi$
$211$	−5.97409	−0.411273	−0.205637	+	0.978628i	$0.565927\pi$
$212$	0	0
$213$	5.11529	0.350494
$214$	0	0
$215$	−1.36293	−0.0929509
$216$	0	0
$217$	−16.7721	−1.13857
$218$	0	0
$219$	14.7721	0.998209
$220$	0	0
$221$	26.3235	1.77071
$222$	0	0
$223$	23.7659	1.59148	0.795740	−	0.605639i	$-0.207082\pi$
$223$	23.7659	1.59148	0.795740	+	0.605639i	$0.207082\pi$
$224$	0	0
$225$	1.11882	0.0745883
$226$	0	0
$227$	−0.907457	−0.0602300	−0.0301150	−	0.999546i	$-0.509587\pi$
$227$	−0.907457	−0.0602300	−0.0301150	+	0.999546i	$0.509587\pi$
$228$	0	0
$229$	7.55579	0.499301	0.249650	−	0.968336i	$-0.419684\pi$
$229$	7.55579	0.499301	0.249650	+	0.968336i	$0.419684\pi$
$230$	0	0
$231$	−1.71313	−0.112716
$232$	0	0
$233$	−23.2271	−1.52166	−0.760828	−	0.648954i	$-0.775207\pi$
$233$	−23.2271	−1.52166	−0.760828	+	0.648954i	$0.775207\pi$
$234$	0	0
$235$	6.99647	0.456399
$236$	0	0
$237$	6.32000	0.410528
$238$	0	0
$239$	26.9213	1.74140	0.870698	−	0.491817i	$-0.163667\pi$
$239$	26.9213	1.74140	0.870698	+	0.491817i	$0.163667\pi$
$240$	0	0
$241$	10.3494	0.666664	0.333332	−	0.942809i	$-0.391827\pi$
$241$	10.3494	0.666664	0.333332	+	0.942809i	$0.391827\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.13401	−0.0724492
$246$	0	0
$247$	26.3235	1.67492
$248$	0	0
$249$	0.907457	0.0575077
$250$	0	0
$251$	13.7984	0.870949	0.435475	−	0.900201i	$-0.356581\pi$
$251$	13.7984	0.870949	0.435475	+	0.900201i	$0.356581\pi$
$252$	0	0
$253$	−1.89450	−0.119106
$254$	0	0
$255$	−15.9437	−0.998435
$256$	0	0
$257$	16.9965	1.06021	0.530105	−	0.847932i	$-0.322152\pi$
$257$	16.9965	1.06021	0.530105	+	0.847932i	$0.322152\pi$
$258$	0	0
$259$	−9.87175	−0.613401
$260$	0	0
$261$	4.35480	0.269555
$262$	0	0
$263$	29.9929	1.84944	0.924722	−	0.380643i	$-0.124297\pi$
$263$	29.9929	1.84944	0.924722	+	0.380643i	$0.124297\pi$
$264$	0	0
$265$	9.01686	0.553901
$266$	0	0
$267$	6.31724	0.386609
$268$	0	0
$269$	29.1332	1.77628	0.888142	−	0.459569i	$-0.151996\pi$
$269$	29.1332	1.77628	0.888142	+	0.459569i	$0.151996\pi$
$270$	0	0
$271$	−26.6506	−1.61891	−0.809453	−	0.587184i	$-0.800236\pi$
$271$	−26.6506	−1.61891	−0.809453	+	0.587184i	$0.800236\pi$
$272$	0	0
$273$	−10.4455	−0.632190
$274$	0	0
$275$	0.749397	0.0451904
$276$	0	0
$277$	−17.1430	−1.03003	−0.515013	−	0.857183i	$-0.672213\pi$
$277$	−17.1430	−1.03003	−0.515013	+	0.857183i	$0.672213\pi$
$278$	0	0
$279$	−6.55765	−0.392596
$280$	0	0
$281$	2.76588	0.164999	0.0824993	−	0.996591i	$-0.473710\pi$
$281$	2.76588	0.164999	0.0824993	+	0.996591i	$0.473710\pi$
$282$	0	0
$283$	−6.34315	−0.377061	−0.188530	−	0.982067i	$-0.560372\pi$
$283$	−6.34315	−0.377061	−0.188530	+	0.982067i	$0.560372\pi$
$284$	0	0
$285$	−15.9437	−0.944425
$286$	0	0
$287$	2.01704	0.119062
$288$	0	0
$289$	24.5443	1.44378
$290$	0	0
$291$	−12.6533	−0.741751
$292$	0	0
$293$	−11.6078	−0.678133	−0.339067	−	0.940762i	$-0.610111\pi$
$293$	−11.6078	−0.678133	−0.339067	+	0.940762i	$0.610111\pi$
$294$	0	0
$295$	−13.9929	−0.814700
$296$	0	0
$297$	−0.669808	−0.0388662
$298$	0	0
$299$	−11.5514	−0.668032
$300$	0	0
$301$	−1.40922	−0.0812262
$302$	0	0
$303$	−10.6417	−0.611348
$304$	0	0
$305$	15.3435	0.878567
$306$	0	0
$307$	−14.7855	−0.843852	−0.421926	−	0.906630i	$-0.638646\pi$
$307$	−14.7855	−0.843852	−0.421926	+	0.906630i	$0.638646\pi$
$308$	0	0
$309$	3.33686	0.189827
$310$	0	0
$311$	15.0761	0.854885	0.427442	−	0.904043i	$-0.359415\pi$
$311$	15.0761	0.854885	0.427442	+	0.904043i	$0.359415\pi$
$312$	0	0
$313$	23.0027	1.30019	0.650096	−	0.759852i	$-0.274729\pi$
$313$	23.0027	1.30019	0.650096	+	0.759852i	$0.274729\pi$
$314$	0	0
$315$	6.32666	0.356467
$316$	0	0
$317$	9.55855	0.536862	0.268431	−	0.963299i	$-0.413495\pi$
$317$	9.55855	0.536862	0.268431	+	0.963299i	$0.413495\pi$
$318$	0	0
$319$	2.91688	0.163314
$320$	0	0
$321$	19.8874	1.11001
$322$	0	0
$323$	41.5443	2.31159
$324$	0	0
$325$	4.56930	0.253459
$326$	0	0
$327$	−3.91598	−0.216554
$328$	0	0
$329$	7.23412	0.398830
$330$	0	0
$331$	−27.8079	−1.52846	−0.764229	−	0.644945i	$-0.776880\pi$
$331$	−27.8079	−1.52846	−0.764229	+	0.644945i	$0.776880\pi$
$332$	0	0
$333$	−3.85970	−0.211510
$334$	0	0
$335$	−7.41215	−0.404969
$336$	0	0
$337$	−3.00980	−0.163954	−0.0819771	−	0.996634i	$-0.526123\pi$
$337$	−3.00980	−0.163954	−0.0819771	+	0.996634i	$0.526123\pi$
$338$	0	0
$339$	2.23765	0.121532
$340$	0	0
$341$	−4.39236	−0.237860
$342$	0	0
$343$	−19.0761	−1.03001
$344$	0	0
$345$	6.99647	0.376677
$346$	0	0
$347$	−8.87449	−0.476408	−0.238204	−	0.971215i	$-0.576559\pi$
$347$	−8.87449	−0.476408	−0.238204	+	0.971215i	$0.576559\pi$
$348$	0	0
$349$	6.70698	0.359016	0.179508	−	0.983757i	$-0.442549\pi$
$349$	6.70698	0.359016	0.179508	+	0.983757i	$0.442549\pi$
$350$	0	0
$351$	−4.08402	−0.217989
$352$	0	0
$353$	−8.75882	−0.466185	−0.233093	−	0.972455i	$-0.574884\pi$
$353$	−8.75882	−0.466185	−0.233093	+	0.972455i	$0.574884\pi$
$354$	0	0
$355$	−12.6533	−0.671569
$356$	0	0
$357$	−16.4853	−0.872494
$358$	0	0
$359$	−32.7917	−1.73068	−0.865341	−	0.501184i	$-0.832898\pi$
$359$	−32.7917	−1.73068	−0.865341	+	0.501184i	$0.832898\pi$
$360$	0	0
$361$	22.5443	1.18654
$362$	0	0
$363$	10.5514	0.553803
$364$	0	0
$365$	−36.5408	−1.91263
$366$	0	0
$367$	20.6435	1.07758	0.538791	−	0.842439i	$-0.318881\pi$
$367$	20.6435	1.07758	0.538791	+	0.842439i	$0.318881\pi$
$368$	0	0
$369$	0.788632	0.0410546
$370$	0	0
$371$	9.32313	0.484033
$372$	0	0
$373$	23.4995	1.21676	0.608379	−	0.793646i	$-0.291820\pi$
$373$	23.4995	1.21676	0.608379	+	0.793646i	$0.291820\pi$
$374$	0	0
$375$	9.60058	0.495772
$376$	0	0
$377$	17.7851	0.915979
$378$	0	0
$379$	−11.0004	−0.565051	−0.282526	−	0.959260i	$-0.591172\pi$
$379$	−11.0004	−0.565051	−0.282526	+	0.959260i	$0.591172\pi$
$380$	0	0
$381$	12.2145	0.625768
$382$	0	0
$383$	17.2037	0.879070	0.439535	−	0.898225i	$-0.355143\pi$
$383$	17.2037	0.879070	0.439535	+	0.898225i	$0.355143\pi$
$384$	0	0
$385$	4.23765	0.215971
$386$	0	0
$387$	−0.550984	−0.0280081
$388$	0	0
$389$	−33.7311	−1.71023	−0.855116	−	0.518436i	$-0.826514\pi$
$389$	−33.7311	−1.71023	−0.855116	+	0.518436i	$0.826514\pi$
$390$	0	0
$391$	−18.2306	−0.921961
$392$	0	0
$393$	5.33962	0.269348
$394$	0	0
$395$	−15.6333	−0.786597
$396$	0	0
$397$	−14.5201	−0.728742	−0.364371	−	0.931254i	$-0.618716\pi$
$397$	−14.5201	−0.728742	−0.364371	+	0.931254i	$0.618716\pi$
$398$	0	0
$399$	−16.4853	−0.825296
$400$	0	0
$401$	−32.2274	−1.60936	−0.804681	−	0.593708i	$-0.797663\pi$
$401$	−32.2274	−1.60936	−0.804681	+	0.593708i	$0.797663\pi$
$402$	0	0
$403$	−26.7816	−1.33409
$404$	0	0
$405$	2.47363	0.122915
$406$	0	0
$407$	−2.58526	−0.128146
$408$	0	0
$409$	−11.5702	−0.572110	−0.286055	−	0.958213i	$-0.592344\pi$
$409$	−11.5702	−0.572110	−0.286055	+	0.958213i	$0.592344\pi$
$410$	0	0
$411$	5.10587	0.251854
$412$	0	0
$413$	−14.4682	−0.711935
$414$	0	0
$415$	−2.24471	−0.110188
$416$	0	0
$417$	−16.6533	−0.815517
$418$	0	0
$419$	9.54193	0.466154	0.233077	−	0.972458i	$-0.425121\pi$
$419$	9.54193	0.466154	0.233077	+	0.972458i	$0.425121\pi$
$420$	0	0
$421$	−24.3583	−1.18715	−0.593576	−	0.804778i	$-0.702284\pi$
$421$	−24.3583	−1.18715	−0.593576	+	0.804778i	$0.702284\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	7.21137	0.349803
$426$	0	0
$427$	15.8647	0.767746
$428$	0	0
$429$	−2.73551	−0.132072
$430$	0	0
$431$	40.7088	1.96087	0.980437	−	0.196832i	$-0.0630654\pi$
$431$	40.7088	1.96087	0.980437	+	0.196832i	$0.0630654\pi$
$432$	0	0
$433$	−7.31371	−0.351474	−0.175737	−	0.984437i	$-0.556231\pi$
$433$	−7.31371	−0.351474	−0.175737	+	0.984437i	$0.556231\pi$
$434$	0	0
$435$	−10.7721	−0.516485
$436$	0	0
$437$	−18.2306	−0.872087
$438$	0	0
$439$	−17.7122	−0.845356	−0.422678	−	0.906280i	$-0.638910\pi$
$439$	−17.7122	−0.845356	−0.422678	+	0.906280i	$0.638910\pi$
$440$	0	0
$441$	−0.458440	−0.0218305
$442$	0	0
$443$	−22.1953	−1.05453	−0.527264	−	0.849701i	$-0.676782\pi$
$443$	−22.1953	−1.05453	−0.527264	+	0.849701i	$0.676782\pi$
$444$	0	0
$445$	−15.6265	−0.740766
$446$	0	0
$447$	−11.1832	−0.528949
$448$	0	0
$449$	−28.3400	−1.33745	−0.668723	−	0.743511i	$-0.733159\pi$
$449$	−28.3400	−1.33745	−0.668723	+	0.743511i	$0.733159\pi$
$450$	0	0
$451$	0.528232	0.0248735
$452$	0	0
$453$	14.6506	0.688344
$454$	0	0
$455$	25.8382	1.21131
$456$	0	0
$457$	−17.3396	−0.811113	−0.405557	−	0.914070i	$-0.632922\pi$
$457$	−17.3396	−0.811113	−0.405557	+	0.914070i	$0.632922\pi$
$458$	0	0
$459$	−6.44549	−0.300850
$460$	0	0
$461$	−2.39404	−0.111501	−0.0557507	−	0.998445i	$-0.517755\pi$
$461$	−2.39404	−0.111501	−0.0557507	+	0.998445i	$0.517755\pi$
$462$	0	0
$463$	2.70238	0.125590	0.0627951	−	0.998026i	$-0.479999\pi$
$463$	2.70238	0.125590	0.0627951	+	0.998026i	$0.479999\pi$
$464$	0	0
$465$	16.2212	0.752239
$466$	0	0
$467$	24.2023	1.11995	0.559975	−	0.828510i	$-0.310811\pi$
$467$	24.2023	1.11995	0.559975	+	0.828510i	$0.310811\pi$
$468$	0	0
$469$	−7.66391	−0.353887
$470$	0	0
$471$	−4.45754	−0.205393
$472$	0	0
$473$	−0.369053	−0.0169691
$474$	0	0
$475$	7.21137	0.330880
$476$	0	0
$477$	3.64520	0.166902
$478$	0	0
$479$	22.2251	1.01549	0.507745	−	0.861508i	$-0.330479\pi$
$479$	22.2251	1.01549	0.507745	+	0.861508i	$0.330479\pi$
$480$	0	0
$481$	−15.7631	−0.718735
$482$	0	0
$483$	7.23412	0.329164
$484$	0	0
$485$	31.2996	1.42124
$486$	0	0
$487$	−13.9839	−0.633672	−0.316836	−	0.948480i	$-0.602620\pi$
$487$	−13.9839	−0.633672	−0.316836	+	0.948480i	$0.602620\pi$
$488$	0	0
$489$	−7.78510	−0.352055
$490$	0	0
$491$	10.2306	0.461700	0.230850	−	0.972989i	$-0.425849\pi$
$491$	10.2306	0.461700	0.230850	+	0.972989i	$0.425849\pi$
$492$	0	0
$493$	28.0688	1.26416
$494$	0	0
$495$	1.65685	0.0744701
$496$	0	0
$497$	−13.0831	−0.586858
$498$	0	0
$499$	−3.66391	−0.164019	−0.0820097	−	0.996632i	$-0.526134\pi$
$499$	−3.66391	−0.164019	−0.0820097	+	0.996632i	$0.526134\pi$
$500$	0	0
$501$	−20.1814	−0.901637
$502$	0	0
$503$	39.6443	1.76765	0.883825	−	0.467817i	$-0.154959\pi$
$503$	39.6443	1.76765	0.883825	+	0.467817i	$0.154959\pi$
$504$	0	0
$505$	26.3235	1.17138
$506$	0	0
$507$	−3.67923	−0.163400
$508$	0	0
$509$	−28.6909	−1.27170	−0.635851	−	0.771812i	$-0.719351\pi$
$509$	−28.6909	−1.27170	−0.635851	+	0.771812i	$0.719351\pi$
$510$	0	0
$511$	−37.7819	−1.67137
$512$	0	0
$513$	−6.44549	−0.284575
$514$	0	0
$515$	−8.25413	−0.363721
$516$	0	0
$517$	1.89450	0.0833201
$518$	0	0
$519$	−6.15639	−0.270235
$520$	0	0
$521$	23.1784	1.01546	0.507732	−	0.861515i	$-0.330484\pi$
$521$	23.1784	1.01546	0.507732	+	0.861515i	$0.330484\pi$
$522$	0	0
$523$	−8.18193	−0.357771	−0.178885	−	0.983870i	$-0.557249\pi$
$523$	−8.18193	−0.357771	−0.178885	+	0.983870i	$0.557249\pi$
$524$	0	0
$525$	−2.86156	−0.124889
$526$	0	0
$527$	−42.2672	−1.84119
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	−5.65685	−0.245487
$532$	0	0
$533$	3.22079	0.139508
$534$	0	0
$535$	−49.1941	−2.12685
$536$	0	0
$537$	−18.7855	−0.810653
$538$	0	0
$539$	−0.307067	−0.0132263
$540$	0	0
$541$	−6.43715	−0.276755	−0.138377	−	0.990380i	$-0.544189\pi$
$541$	−6.43715	−0.276755	−0.138377	+	0.990380i	$0.544189\pi$
$542$	0	0
$543$	8.97499	0.385154
$544$	0	0
$545$	9.68667	0.414931
$546$	0	0
$547$	−39.2239	−1.67709	−0.838546	−	0.544830i	$-0.816594\pi$
$547$	−39.2239	−1.67709	−0.838546	+	0.544830i	$0.816594\pi$
$548$	0	0
$549$	6.20285	0.264731
$550$	0	0
$551$	28.0688	1.19577
$552$	0	0
$553$	−16.1643	−0.687377
$554$	0	0
$555$	9.54745	0.405267
$556$	0	0
$557$	−1.66224	−0.0704315	−0.0352157	−	0.999380i	$-0.511212\pi$
$557$	−1.66224	−0.0704315	−0.0352157	+	0.999380i	$0.511212\pi$
$558$	0	0
$559$	−2.25023	−0.0951745
$560$	0	0
$561$	−4.31724	−0.182274
$562$	0	0
$563$	40.6368	1.71264	0.856319	−	0.516447i	$-0.172746\pi$
$563$	40.6368	1.71264	0.856319	+	0.516447i	$0.172746\pi$
$564$	0	0
$565$	−5.53511	−0.232864
$566$	0	0
$567$	2.55765	0.107411
$568$	0	0
$569$	−27.0004	−1.13191	−0.565957	−	0.824435i	$-0.691493\pi$
$569$	−27.0004	−1.13191	−0.565957	+	0.824435i	$0.691493\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	0	0
$573$	5.60058	0.233967
$574$	0	0
$575$	−3.16451	−0.131969
$576$	0	0
$577$	−37.6372	−1.56686	−0.783429	−	0.621481i	$-0.786531\pi$
$577$	−37.6372	−1.56686	−0.783429	+	0.621481i	$0.786531\pi$
$578$	0	0
$579$	19.4514	0.808372
$580$	0	0
$581$	−2.32095	−0.0962894
$582$	0	0
$583$	2.44158	0.101120
$584$	0	0
$585$	10.1023	0.417680
$586$	0	0
$587$	−44.2047	−1.82452	−0.912261	−	0.409609i	$-0.865665\pi$
$587$	−44.2047	−1.82452	−0.912261	+	0.409609i	$0.865665\pi$
$588$	0	0
$589$	−42.2672	−1.74159
$590$	0	0
$591$	−1.75070	−0.0720140
$592$	0	0
$593$	3.59611	0.147675	0.0738373	−	0.997270i	$-0.476475\pi$
$593$	3.59611	0.147675	0.0738373	+	0.997270i	$0.476475\pi$
$594$	0	0
$595$	40.7784	1.67175
$596$	0	0
$597$	−0.993710	−0.0406698
$598$	0	0
$599$	−22.0296	−0.900104	−0.450052	−	0.893002i	$-0.648595\pi$
$599$	−22.0296	−0.900104	−0.450052	+	0.893002i	$0.648595\pi$
$600$	0	0
$601$	10.7721	0.439405	0.219703	−	0.975567i	$-0.429491\pi$
$601$	10.7721	0.439405	0.219703	+	0.975567i	$0.429491\pi$
$602$	0	0
$603$	−2.99647	−0.122026
$604$	0	0
$605$	−26.1001	−1.06112
$606$	0	0
$607$	−5.47453	−0.222204	−0.111102	−	0.993809i	$-0.535438\pi$
$607$	−5.47453	−0.222204	−0.111102	+	0.993809i	$0.535438\pi$
$608$	0	0
$609$	−11.1380	−0.451336
$610$	0	0
$611$	11.5514	0.467318
$612$	0	0
$613$	−14.8562	−0.600035	−0.300018	−	0.953934i	$-0.596993\pi$
$613$	−14.8562	−0.600035	−0.300018	+	0.953934i	$0.596993\pi$
$614$	0	0
$615$	−1.95078	−0.0786631
$616$	0	0
$617$	22.2235	0.894686	0.447343	−	0.894363i	$-0.352370\pi$
$617$	22.2235	0.894686	0.447343	+	0.894363i	$0.352370\pi$
$618$	0	0
$619$	16.4612	0.661631	0.330815	−	0.943696i	$-0.392676\pi$
$619$	16.4612	0.661631	0.330815	+	0.943696i	$0.392676\pi$
$620$	0	0
$621$	2.82843	0.113501
$622$	0	0
$623$	−16.1573	−0.647327
$624$	0	0
$625$	−29.3424	−1.17369
$626$	0	0
$627$	−4.31724	−0.172414
$628$	0	0
$629$	−24.8776	−0.991937
$630$	0	0
$631$	4.06977	0.162015	0.0810075	−	0.996713i	$-0.474186\pi$
$631$	4.06977	0.162015	0.0810075	+	0.996713i	$0.474186\pi$
$632$	0	0
$633$	5.97409	0.237449
$634$	0	0
$635$	−30.2141	−1.19901
$636$	0	0
$637$	−1.87228	−0.0741824
$638$	0	0
$639$	−5.11529	−0.202358
$640$	0	0
$641$	−8.41958	−0.332553	−0.166277	−	0.986079i	$-0.553174\pi$
$641$	−8.41958	−0.332553	−0.166277	+	0.986079i	$0.553174\pi$
$642$	0	0
$643$	−10.4266	−0.411186	−0.205593	−	0.978638i	$-0.565912\pi$
$643$	−10.4266	−0.411186	−0.205593	+	0.978638i	$0.565912\pi$
$644$	0	0
$645$	1.36293	0.0536652
$646$	0	0
$647$	−11.6132	−0.456560	−0.228280	−	0.973595i	$-0.573310\pi$
$647$	−11.6132	−0.456560	−0.228280	+	0.973595i	$0.573310\pi$
$648$	0	0
$649$	−3.78901	−0.148731
$650$	0	0
$651$	16.7721	0.657352
$652$	0	0
$653$	−2.73012	−0.106838	−0.0534190	−	0.998572i	$-0.517012\pi$
$653$	−2.73012	−0.106838	−0.0534190	+	0.998572i	$0.517012\pi$
$654$	0	0
$655$	−13.2082	−0.516088
$656$	0	0
$657$	−14.7721	−0.576316
$658$	0	0
$659$	−31.5514	−1.22907	−0.614533	−	0.788891i	$-0.710656\pi$
$659$	−31.5514	−1.22907	−0.614533	+	0.788891i	$0.710656\pi$
$660$	0	0
$661$	−15.1368	−0.588752	−0.294376	−	0.955690i	$-0.595112\pi$
$661$	−15.1368	−0.588752	−0.294376	+	0.955690i	$0.595112\pi$
$662$	0	0
$663$	−26.3235	−1.02232
$664$	0	0
$665$	40.7784	1.58132
$666$	0	0
$667$	−12.3172	−0.476925
$668$	0	0
$669$	−23.7659	−0.918841
$670$	0	0
$671$	4.15472	0.160391
$672$	0	0
$673$	−20.6345	−0.795401	−0.397700	−	0.917515i	$-0.630192\pi$
$673$	−20.6345	−0.795401	−0.397700	+	0.917515i	$0.630192\pi$
$674$	0	0
$675$	−1.11882	−0.0430636
$676$	0	0
$677$	37.9357	1.45799	0.728994	−	0.684520i	$-0.239988\pi$
$677$	37.9357	1.45799	0.728994	+	0.684520i	$0.239988\pi$
$678$	0	0
$679$	32.3627	1.24197
$680$	0	0
$681$	0.907457	0.0347738
$682$	0	0
$683$	18.2471	0.698205	0.349102	−	0.937085i	$-0.386487\pi$
$683$	18.2471	0.698205	0.349102	+	0.937085i	$0.386487\pi$
$684$	0	0
$685$	−12.6300	−0.482568
$686$	0	0
$687$	−7.55579	−0.288271
$688$	0	0
$689$	14.8871	0.567152
$690$	0	0
$691$	30.2533	1.15089	0.575446	−	0.817840i	$-0.304829\pi$
$691$	30.2533	1.15089	0.575446	+	0.817840i	$0.304829\pi$
$692$	0	0
$693$	1.71313	0.0650765
$694$	0	0
$695$	41.1941	1.56258
$696$	0	0
$697$	5.08312	0.192537
$698$	0	0
$699$	23.2271	0.878528
$700$	0	0
$701$	−20.0875	−0.758696	−0.379348	−	0.925254i	$-0.623852\pi$
$701$	−20.0875	−0.758696	−0.379348	+	0.925254i	$0.623852\pi$
$702$	0	0
$703$	−24.8776	−0.938278
$704$	0	0
$705$	−6.99647	−0.263502
$706$	0	0
$707$	27.2176	1.02362
$708$	0	0
$709$	−41.7864	−1.56932	−0.784660	−	0.619926i	$-0.787163\pi$
$709$	−41.7864	−1.56932	−0.784660	+	0.619926i	$0.787163\pi$
$710$	0	0
$711$	−6.32000	−0.237018
$712$	0	0
$713$	18.5478	0.694622
$714$	0	0
$715$	6.76663	0.253058
$716$	0	0
$717$	−26.9213	−1.00540
$718$	0	0
$719$	28.3683	1.05796	0.528979	−	0.848635i	$-0.322575\pi$
$719$	28.3683	1.05796	0.528979	+	0.848635i	$0.322575\pi$
$720$	0	0
$721$	−8.53450	−0.317841
$722$	0	0
$723$	−10.3494	−0.384899
$724$	0	0
$725$	4.87226	0.180951
$726$	0	0
$727$	20.4843	0.759722	0.379861	−	0.925044i	$-0.375972\pi$
$727$	20.4843	0.759722	0.379861	+	0.925044i	$0.375972\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.55136	−0.131352
$732$	0	0
$733$	−48.0777	−1.77579	−0.887895	−	0.460045i	$-0.847833\pi$
$733$	−48.0777	−1.77579	−0.887895	+	0.460045i	$0.847833\pi$
$734$	0	0
$735$	1.13401	0.0418286
$736$	0	0
$737$	−2.00706	−0.0739310
$738$	0	0
$739$	21.4459	0.788899	0.394449	−	0.918918i	$-0.370935\pi$
$739$	21.4459	0.788899	0.394449	+	0.918918i	$0.370935\pi$
$740$	0	0
$741$	−26.3235	−0.967018
$742$	0	0
$743$	−2.17431	−0.0797677	−0.0398839	−	0.999204i	$-0.512699\pi$
$743$	−2.17431	−0.0797677	−0.0398839	+	0.999204i	$0.512699\pi$
$744$	0	0
$745$	27.6631	1.01350
$746$	0	0
$747$	−0.907457	−0.0332021
$748$	0	0
$749$	−50.8651	−1.85857
$750$	0	0
$751$	29.8980	1.09099	0.545497	−	0.838113i	$-0.316341\pi$
$751$	29.8980	1.09099	0.545497	+	0.838113i	$0.316341\pi$
$752$	0	0
$753$	−13.7984	−0.502843
$754$	0	0
$755$	−36.2400	−1.31891
$756$	0	0
$757$	21.6791	0.787939	0.393970	−	0.919123i	$-0.371101\pi$
$757$	21.6791	0.787939	0.393970	+	0.919123i	$0.371101\pi$
$758$	0	0
$759$	1.89450	0.0687661
$760$	0	0
$761$	4.29449	0.155675	0.0778375	−	0.996966i	$-0.475198\pi$
$761$	4.29449	0.155675	0.0778375	+	0.996966i	$0.475198\pi$
$762$	0	0
$763$	10.0157	0.362592
$764$	0	0
$765$	15.9437	0.576446
$766$	0	0
$767$	−23.1027	−0.834191
$768$	0	0
$769$	33.8819	1.22181	0.610907	−	0.791703i	$-0.290805\pi$
$769$	33.8819	1.22181	0.610907	+	0.791703i	$0.290805\pi$
$770$	0	0
$771$	−16.9965	−0.612113
$772$	0	0
$773$	−49.5300	−1.78147	−0.890736	−	0.454521i	$-0.849810\pi$
$773$	−49.5300	−1.78147	−0.890736	+	0.454521i	$0.849810\pi$
$774$	0	0
$775$	−7.33686	−0.263548
$776$	0	0
$777$	9.87175	0.354147
$778$	0	0
$779$	5.08312	0.182122
$780$	0	0
$781$	−3.42627	−0.122601
$782$	0	0
$783$	−4.35480	−0.155628
$784$	0	0
$785$	11.0263	0.393545
$786$	0	0
$787$	−34.0953	−1.21537	−0.607683	−	0.794180i	$-0.707901\pi$
$787$	−34.0953	−1.21537	−0.607683	+	0.794180i	$0.707901\pi$
$788$	0	0
$789$	−29.9929	−1.06778
$790$	0	0
$791$	−5.72312	−0.203491
$792$	0	0
$793$	25.3326	0.899585
$794$	0	0
$795$	−9.01686	−0.319795
$796$	0	0
$797$	40.6901	1.44132	0.720659	−	0.693290i	$-0.243840\pi$
$797$	40.6901	1.44132	0.720659	+	0.693290i	$0.243840\pi$
$798$	0	0
$799$	18.2306	0.644952
$800$	0	0
$801$	−6.31724	−0.223209
$802$	0	0
$803$	−9.89450	−0.349169
$804$	0	0
$805$	−17.8945	−0.630698
$806$	0	0
$807$	−29.1332	−1.02554
$808$	0	0
$809$	−10.9926	−0.386478	−0.193239	−	0.981152i	$-0.561899\pi$
$809$	−10.9926	−0.386478	−0.193239	+	0.981152i	$0.561899\pi$
$810$	0	0
$811$	−21.2498	−0.746182	−0.373091	−	0.927795i	$-0.621702\pi$
$811$	−21.2498	−0.746182	−0.373091	+	0.927795i	$0.621702\pi$
$812$	0	0
$813$	26.6506	0.934676
$814$	0	0
$815$	19.2574	0.674558
$816$	0	0
$817$	−3.55136	−0.124246
$818$	0	0
$819$	10.4455	0.364995
$820$	0	0
$821$	30.0572	1.04900	0.524501	−	0.851410i	$-0.324252\pi$
$821$	30.0572	1.04900	0.524501	+	0.851410i	$0.324252\pi$
$822$	0	0
$823$	55.0851	1.92015	0.960073	−	0.279751i	$-0.0902518\pi$
$823$	55.0851	1.92015	0.960073	+	0.279751i	$0.0902518\pi$
$824$	0	0
$825$	−0.749397	−0.0260907
$826$	0	0
$827$	34.5478	1.20135	0.600673	−	0.799495i	$-0.294899\pi$
$827$	34.5478	1.20135	0.600673	+	0.799495i	$0.294899\pi$
$828$	0	0
$829$	31.0046	1.07683	0.538417	−	0.842679i	$-0.319023\pi$
$829$	31.0046	1.07683	0.538417	+	0.842679i	$0.319023\pi$
$830$	0	0
$831$	17.1430	0.594685
$832$	0	0
$833$	−2.95487	−0.102380
$834$	0	0
$835$	49.9212	1.72759
$836$	0	0
$837$	6.55765	0.226665
$838$	0	0
$839$	−5.14195	−0.177520	−0.0887599	−	0.996053i	$-0.528290\pi$
$839$	−5.14195	−0.177520	−0.0887599	+	0.996053i	$0.528290\pi$
$840$	0	0
$841$	−10.0357	−0.346059
$842$	0	0
$843$	−2.76588	−0.0952620
$844$	0	0
$845$	9.10104	0.313085
$846$	0	0
$847$	−26.9867	−0.927272
$848$	0	0
$849$	6.34315	0.217696
$850$	0	0
$851$	10.9169	0.374226
$852$	0	0
$853$	−18.5060	−0.633632	−0.316816	−	0.948487i	$-0.602614\pi$
$853$	−18.5060	−0.633632	−0.316816	+	0.948487i	$0.602614\pi$
$854$	0	0
$855$	15.9437	0.545264
$856$	0	0
$857$	22.8878	0.781833	0.390916	−	0.920426i	$-0.372158\pi$
$857$	22.8878	0.781833	0.390916	+	0.920426i	$0.372158\pi$
$858$	0	0
$859$	35.5286	1.21222	0.606110	−	0.795381i	$-0.292729\pi$
$859$	35.5286	1.21222	0.606110	+	0.795381i	$0.292729\pi$
$860$	0	0
$861$	−2.01704	−0.0687407
$862$	0	0
$863$	43.9296	1.49538	0.747691	−	0.664047i	$-0.231163\pi$
$863$	43.9296	1.49538	0.747691	+	0.664047i	$0.231163\pi$
$864$	0	0
$865$	15.2286	0.517788
$866$	0	0
$867$	−24.5443	−0.833568
$868$	0	0
$869$	−4.23319	−0.143601
$870$	0	0
$871$	−12.2376	−0.414657
$872$	0	0
$873$	12.6533	0.428250
$874$	0	0
$875$	−24.5549	−0.830107
$876$	0	0
$877$	21.5773	0.728614	0.364307	−	0.931279i	$-0.381306\pi$
$877$	21.5773	0.728614	0.364307	+	0.931279i	$0.381306\pi$
$878$	0	0
$879$	11.6078	0.391520
$880$	0	0
$881$	−21.6686	−0.730035	−0.365018	−	0.931001i	$-0.618937\pi$
$881$	−21.6686	−0.730035	−0.365018	+	0.931001i	$0.618937\pi$
$882$	0	0
$883$	0.0834930	0.00280976	0.00140488	−	0.999999i	$-0.499553\pi$
$883$	0.0834930	0.00280976	0.00140488	+	0.999999i	$0.499553\pi$
$884$	0	0
$885$	13.9929	0.470368
$886$	0	0
$887$	−30.8043	−1.03431	−0.517154	−	0.855892i	$-0.673009\pi$
$887$	−30.8043	−1.03431	−0.517154	+	0.855892i	$0.673009\pi$
$888$	0	0
$889$	−31.2404	−1.04777
$890$	0	0
$891$	0.669808	0.0224394
$892$	0	0
$893$	18.2306	0.610063
$894$	0	0
$895$	46.4682	1.55326
$896$	0	0
$897$	11.5514	0.385689
$898$	0	0
$899$	−28.5573	−0.952438
$900$	0	0
$901$	23.4951	0.782735
$902$	0	0
$903$	1.40922	0.0468960
$904$	0	0
$905$	−22.2008	−0.737979
$906$	0	0
$907$	49.5215	1.64434	0.822168	−	0.569245i	$-0.192764\pi$
$907$	49.5215	1.64434	0.822168	+	0.569245i	$0.192764\pi$
$908$	0	0
$909$	10.6417	0.352962
$910$	0	0
$911$	0.0829331	0.00274770	0.00137385	−	0.999999i	$-0.499563\pi$
$911$	0.0829331	0.00274770	0.00137385	+	0.999999i	$0.499563\pi$
$912$	0	0
$913$	−0.607822	−0.0201160
$914$	0	0
$915$	−15.3435	−0.507241
$916$	0	0
$917$	−13.6569	−0.450989
$918$	0	0
$919$	−20.1161	−0.663568	−0.331784	−	0.943355i	$-0.607650\pi$
$919$	−20.1161	−0.663568	−0.331784	+	0.943355i	$0.607650\pi$
$920$	0	0
$921$	14.7855	0.487198
$922$	0	0
$923$	−20.8910	−0.687635
$924$	0	0
$925$	−4.31833	−0.141986
$926$	0	0
$927$	−3.33686	−0.109597
$928$	0	0
$929$	8.55098	0.280549	0.140274	−	0.990113i	$-0.455202\pi$
$929$	8.55098	0.280549	0.140274	+	0.990113i	$0.455202\pi$
$930$	0	0
$931$	−2.95487	−0.0968420
$932$	0	0
$933$	−15.0761	−0.493568
$934$	0	0
$935$	10.6792	0.349248
$936$	0	0
$937$	33.5780	1.09695	0.548473	−	0.836168i	$-0.315209\pi$
$937$	33.5780	1.09695	0.548473	+	0.836168i	$0.315209\pi$
$938$	0	0
$939$	−23.0027	−0.750666
$940$	0	0
$941$	−11.9991	−0.391159	−0.195579	−	0.980688i	$-0.562659\pi$
$941$	−11.9991	−0.391159	−0.195579	+	0.980688i	$0.562659\pi$
$942$	0	0
$943$	−2.23059	−0.0726380
$944$	0	0
$945$	−6.32666	−0.205806
$946$	0	0
$947$	−25.2537	−0.820635	−0.410318	−	0.911943i	$-0.634582\pi$
$947$	−25.2537	−0.820635	−0.410318	+	0.911943i	$0.634582\pi$
$948$	0	0
$949$	−60.3298	−1.95839
$950$	0	0
$951$	−9.55855	−0.309957
$952$	0	0
$953$	3.86469	0.125190	0.0625948	−	0.998039i	$-0.480062\pi$
$953$	3.86469	0.125190	0.0625948	+	0.998039i	$0.480062\pi$
$954$	0	0
$955$	−13.8537	−0.448296
$956$	0	0
$957$	−2.91688	−0.0942894
$958$	0	0
$959$	−13.0590	−0.421698
$960$	0	0
$961$	12.0027	0.387185
$962$	0	0
$963$	−19.8874	−0.640864
$964$	0	0
$965$	−48.1154	−1.54889
$966$	0	0
$967$	37.8714	1.21786	0.608930	−	0.793224i	$-0.291599\pi$
$967$	37.8714	1.21786	0.608930	+	0.793224i	$0.291599\pi$
$968$	0	0
$969$	−41.5443	−1.33460
$970$	0	0
$971$	3.74587	0.120211	0.0601053	−	0.998192i	$-0.480856\pi$
$971$	3.74587	0.120211	0.0601053	+	0.998192i	$0.480856\pi$
$972$	0	0
$973$	42.5933	1.36548
$974$	0	0
$975$	−4.56930	−0.146335
$976$	0	0
$977$	−17.6530	−0.564768	−0.282384	−	0.959301i	$-0.591125\pi$
$977$	−17.6530	−0.564768	−0.282384	+	0.959301i	$0.591125\pi$
$978$	0	0
$979$	−4.23134	−0.135234
$980$	0	0
$981$	3.91598	0.125028
$982$	0	0
$983$	−22.3557	−0.713035	−0.356518	−	0.934289i	$-0.616036\pi$
$983$	−22.3557	−0.713035	−0.356518	+	0.934289i	$0.616036\pi$
$984$	0	0
$985$	4.33057	0.137983
$986$	0	0
$987$	−7.23412	−0.230265
$988$	0	0
$989$	1.55842	0.0495548
$990$	0	0
$991$	−17.8769	−0.567878	−0.283939	−	0.958842i	$-0.591641\pi$
$991$	−17.8769	−0.567878	−0.283939	+	0.958842i	$0.591641\pi$
$992$	0	0
$993$	27.8079	0.882456
$994$	0	0
$995$	2.45807	0.0779260
$996$	0	0
$997$	6.06146	0.191968	0.0959841	−	0.995383i	$-0.469400\pi$
$997$	6.06146	0.191968	0.0959841	+	0.995383i	$0.469400\pi$
$998$	0	0
$999$	3.85970	0.122116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.n.1.3		4
3.2	odd	2		9216.2.a.x.1.2		4
4.3	odd	2		3072.2.a.t.1.3		4
8.3	odd	2		3072.2.a.i.1.2		4
8.5	even	2		3072.2.a.o.1.2		4
12.11	even	2		9216.2.a.y.1.2		4
16.3	odd	4		3072.2.d.f.1537.6		8
16.5	even	4		3072.2.d.i.1537.7		8
16.11	odd	4		3072.2.d.f.1537.3		8
16.13	even	4		3072.2.d.i.1537.2		8
24.5	odd	2		9216.2.a.bn.1.3		4
24.11	even	2		9216.2.a.bo.1.3		4
32.3	odd	8		384.2.j.b.289.3		8
32.5	even	8		192.2.j.a.49.4		8
32.11	odd	8		384.2.j.b.97.3		8
32.13	even	8		192.2.j.a.145.4		8
32.19	odd	8		48.2.j.a.13.2	✓	8
32.21	even	8		384.2.j.a.97.1		8
32.27	odd	8		48.2.j.a.37.2	yes	8
32.29	even	8		384.2.j.a.289.1		8
96.5	odd	8		576.2.k.b.433.1		8
96.11	even	8		1152.2.k.c.865.4		8
96.29	odd	8		1152.2.k.f.289.4		8
96.35	even	8		1152.2.k.c.289.4		8
96.53	odd	8		1152.2.k.f.865.4		8
96.59	even	8		144.2.k.b.37.3		8
96.77	odd	8		576.2.k.b.145.1		8
96.83	even	8		144.2.k.b.109.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.2	✓	8	32.19	odd	8
48.2.j.a.37.2	yes	8	32.27	odd	8
144.2.k.b.37.3		8	96.59	even	8
144.2.k.b.109.3		8	96.83	even	8
192.2.j.a.49.4		8	32.5	even	8
192.2.j.a.145.4		8	32.13	even	8
384.2.j.a.97.1		8	32.21	even	8
384.2.j.a.289.1		8	32.29	even	8
384.2.j.b.97.3		8	32.11	odd	8
384.2.j.b.289.3		8	32.3	odd	8
576.2.k.b.145.1		8	96.77	odd	8
576.2.k.b.433.1		8	96.5	odd	8
1152.2.k.c.289.4		8	96.35	even	8
1152.2.k.c.865.4		8	96.11	even	8
1152.2.k.f.289.4		8	96.29	odd	8
1152.2.k.f.865.4		8	96.53	odd	8
3072.2.a.i.1.2		4	8.3	odd	2
3072.2.a.n.1.3		4	1.1	even	1	trivial
3072.2.a.o.1.2		4	8.5	even	2
3072.2.a.t.1.3		4	4.3	odd	2
3072.2.d.f.1537.3		8	16.11	odd	4
3072.2.d.f.1537.6		8	16.3	odd	4
3072.2.d.i.1537.2		8	16.13	even	4
3072.2.d.i.1537.7		8	16.5	even	4
9216.2.a.x.1.2		4	3.2	odd	2
9216.2.a.y.1.2		4	12.11	even	2
9216.2.a.bn.1.3		4	24.5	odd	2
9216.2.a.bo.1.3		4	24.11	even	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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.47363 q^{5} +2.55765 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.47363 q^{5} +2.55765 q^{7} +1.00000 q^{9} +0.669808 q^{11} +4.08402 q^{13} -2.47363 q^{15} +6.44549 q^{17} +6.44549 q^{19} -2.55765 q^{21} -2.82843 q^{23} +1.11882 q^{25} -1.00000 q^{27} +4.35480 q^{29} -6.55765 q^{31} -0.669808 q^{33} +6.32666 q^{35} -3.85970 q^{37} -4.08402 q^{39} +0.788632 q^{41} -0.550984 q^{43} +2.47363 q^{45} +2.82843 q^{47} -0.458440 q^{49} -6.44549 q^{51} +3.64520 q^{53} +1.65685 q^{55} -6.44549 q^{57} -5.65685 q^{59} +6.20285 q^{61} +2.55765 q^{63} +10.1023 q^{65} -2.99647 q^{67} +2.82843 q^{69} -5.11529 q^{71} -14.7721 q^{73} -1.11882 q^{75} +1.71313 q^{77} -6.32000 q^{79} +1.00000 q^{81} -0.907457 q^{83} +15.9437 q^{85} -4.35480 q^{87} -6.31724 q^{89} +10.4455 q^{91} +6.55765 q^{93} +15.9437 q^{95} +12.6533 q^{97} +0.669808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 8 q^{13} - 4 q^{15} + 4 q^{21} + 4 q^{25} - 4 q^{27} + 12 q^{29} - 12 q^{31} + 16 q^{37} - 8 q^{39} + 4 q^{45} + 4 q^{49} + 20 q^{53} - 16 q^{55} + 16 q^{61} - 4 q^{63} - 8 q^{65} + 16 q^{67} + 8 q^{71} - 8 q^{73} - 4 q^{75} + 24 q^{77} - 12 q^{79} + 4 q^{81} + 24 q^{85} - 12 q^{87} - 8 q^{89} + 16 q^{91} + 12 q^{93} + 24 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 8 * q^13 - 4 * q^15 + 4 * q^21 + 4 * q^25 - 4 * q^27 + 12 * q^29 - 12 * q^31 + 16 * q^37 - 8 * q^39 + 4 * q^45 + 4 * q^49 + 20 * q^53 - 16 * q^55 + 16 * q^61 - 4 * q^63 - 8 * q^65 + 16 * q^67 + 8 * q^71 - 8 * q^73 - 4 * q^75 + 24 * q^77 - 12 * q^79 + 4 * q^81 + 24 * q^85 - 12 * q^87 - 8 * q^89 + 16 * q^91 + 12 * q^93 + 24 * q^95

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