LMFDB - Embedded newform 3072.2.a.k.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.874032$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.16228 q^{5} -1.74806 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.16228 q^{5} -1.74806 q^{7} +1.00000 q^{9} -6.47214 q^{11} -1.41421 q^{13} -3.16228 q^{15} +2.47214 q^{17} +6.47214 q^{19} +1.74806 q^{21} -5.65685 q^{23} +5.00000 q^{25} -1.00000 q^{27} +5.99070 q^{29} +3.90879 q^{31} +6.47214 q^{33} -5.52786 q^{35} -10.5672 q^{37} +1.41421 q^{39} -2.47214 q^{41} -1.52786 q^{43} +3.16228 q^{45} -3.94427 q^{49} -2.47214 q^{51} -11.6476 q^{53} -20.4667 q^{55} -6.47214 q^{57} -8.94427 q^{59} -2.08191 q^{61} -1.74806 q^{63} -4.47214 q^{65} -12.0000 q^{67} +5.65685 q^{69} +9.15298 q^{71} +2.94427 q^{73} -5.00000 q^{75} +11.3137 q^{77} +7.40492 q^{79} +1.00000 q^{81} +6.47214 q^{83} +7.81758 q^{85} -5.99070 q^{87} -10.0000 q^{89} +2.47214 q^{91} -3.90879 q^{93} +20.4667 q^{95} -12.9443 q^{97} -6.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} - 8 q^{17} + 8 q^{19} + 20 q^{25} - 4 q^{27} + 8 q^{33} - 40 q^{35} + 8 q^{41} - 24 q^{43} + 20 q^{49} + 8 q^{51} - 8 q^{57} - 48 q^{67} - 24 q^{73} - 20 q^{75} + 4 q^{81} + 8 q^{83} - 40 q^{89} - 8 q^{91} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 - 8 * q^17 + 8 * q^19 + 20 * q^25 - 4 * q^27 + 8 * q^33 - 40 * q^35 + 8 * q^41 - 24 * q^43 + 20 * q^49 + 8 * q^51 - 8 * q^57 - 48 * q^67 - 24 * q^73 - 20 * q^75 + 4 * q^81 + 8 * q^83 - 40 * q^89 - 8 * q^91 - 16 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.16228	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$5$	3.16228	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$6$	0	0
$7$	−1.74806	−0.660706	−0.330353	−	0.943857i	$-0.607168\pi$
$7$	−1.74806	−0.660706	−0.330353	+	0.943857i	$0.607168\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.47214	−1.95142	−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214	−1.95142	−0.975711	+	0.219061i	$0.929701\pi$
$12$	0	0
$13$	−1.41421	−0.392232	−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421	−0.392232	−0.196116	+	0.980581i	$0.562833\pi$
$14$	0	0
$15$	−3.16228	−0.816497
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	1.74806	0.381459
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.99070	1.11245	0.556223	−	0.831033i	$-0.312250\pi$
$29$	5.99070	1.11245	0.556223	+	0.831033i	$0.312250\pi$
$30$	0	0
$31$	3.90879	0.702039	0.351020	−	0.936368i	$-0.385835\pi$
$31$	3.90879	0.702039	0.351020	+	0.936368i	$0.385835\pi$
$32$	0	0
$33$	6.47214	1.12665
$34$	0	0
$35$	−5.52786	−0.934380
$36$	0	0
$37$	−10.5672	−1.73724	−0.868618	−	0.495482i	$-0.834991\pi$
$37$	−10.5672	−1.73724	−0.868618	+	0.495482i	$0.834991\pi$
$38$	0	0
$39$	1.41421	0.226455
$40$	0	0
$41$	−2.47214	−0.386083	−0.193041	−	0.981191i	$-0.561835\pi$
$41$	−2.47214	−0.386083	−0.193041	+	0.981191i	$0.561835\pi$
$42$	0	0
$43$	−1.52786	−0.232997	−0.116499	−	0.993191i	$-0.537167\pi$
$43$	−1.52786	−0.232997	−0.116499	+	0.993191i	$0.537167\pi$
$44$	0	0
$45$	3.16228	0.471405
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−3.94427	−0.563467
$50$	0	0
$51$	−2.47214	−0.346168
$52$	0	0
$53$	−11.6476	−1.59992	−0.799958	−	0.600056i	$-0.795145\pi$
$53$	−11.6476	−1.59992	−0.799958	+	0.600056i	$0.795145\pi$
$54$	0	0
$55$	−20.4667	−2.75973
$56$	0	0
$57$	−6.47214	−0.857255
$58$	0	0
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−2.08191	−0.266562	−0.133281	−	0.991078i	$-0.542551\pi$
$61$	−2.08191	−0.266562	−0.133281	+	0.991078i	$0.542551\pi$
$62$	0	0
$63$	−1.74806	−0.220235
$64$	0	0
$65$	−4.47214	−0.554700
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	9.15298	1.08626	0.543130	−	0.839649i	$-0.317239\pi$
$71$	9.15298	1.08626	0.543130	+	0.839649i	$0.317239\pi$
$72$	0	0
$73$	2.94427	0.344601	0.172300	−	0.985044i	$-0.444880\pi$
$73$	2.94427	0.344601	0.172300	+	0.985044i	$0.444880\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	11.3137	1.28932
$78$	0	0
$79$	7.40492	0.833118	0.416559	−	0.909109i	$-0.363236\pi$
$79$	7.40492	0.833118	0.416559	+	0.909109i	$0.363236\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.47214	0.710409	0.355205	−	0.934789i	$-0.384411\pi$
$83$	6.47214	0.710409	0.355205	+	0.934789i	$0.384411\pi$
$84$	0	0
$85$	7.81758	0.847936
$86$	0	0
$87$	−5.99070	−0.642271
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	2.47214	0.259150
$92$	0	0
$93$	−3.90879	−0.405323
$94$	0	0
$95$	20.4667	2.09984
$96$	0	0
$97$	−12.9443	−1.31429	−0.657146	−	0.753763i	$-0.728236\pi$
$97$	−12.9443	−1.31429	−0.657146	+	0.753763i	$0.728236\pi$
$98$	0	0
$99$	−6.47214	−0.650474
$100$	0	0
$101$	0.333851	0.0332194	0.0166097	−	0.999862i	$-0.494713\pi$
$101$	0.333851	0.0332194	0.0166097	+	0.999862i	$0.494713\pi$
$102$	0	0
$103$	1.74806	0.172242	0.0861209	−	0.996285i	$-0.472553\pi$
$103$	1.74806	0.172242	0.0861209	+	0.996285i	$0.472553\pi$
$104$	0	0
$105$	5.52786	0.539464
$106$	0	0
$107$	0.944272	0.0912862	0.0456431	−	0.998958i	$-0.485466\pi$
$107$	0.944272	0.0912862	0.0456431	+	0.998958i	$0.485466\pi$
$108$	0	0
$109$	−16.8918	−1.61794	−0.808968	−	0.587852i	$-0.799974\pi$
$109$	−16.8918	−1.61794	−0.808968	+	0.587852i	$0.799974\pi$
$110$	0	0
$111$	10.5672	1.00299
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−17.8885	−1.66812
$116$	0	0
$117$	−1.41421	−0.130744
$118$	0	0
$119$	−4.32145	−0.396147
$120$	0	0
$121$	30.8885	2.80805
$122$	0	0
$123$	2.47214	0.222905
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.9010	0.967311	0.483656	−	0.875258i	$-0.339309\pi$
$127$	10.9010	0.967311	0.483656	+	0.875258i	$0.339309\pi$
$128$	0	0
$129$	1.52786	0.134521
$130$	0	0
$131$	−16.9443	−1.48043	−0.740214	−	0.672371i	$-0.765276\pi$
$131$	−16.9443	−1.48043	−0.740214	+	0.672371i	$0.765276\pi$
$132$	0	0
$133$	−11.3137	−0.981023
$134$	0	0
$135$	−3.16228	−0.272166
$136$	0	0
$137$	2.47214	0.211209	0.105604	−	0.994408i	$-0.466322\pi$
$137$	2.47214	0.211209	0.105604	+	0.994408i	$0.466322\pi$
$138$	0	0
$139$	−16.9443	−1.43719	−0.718597	−	0.695427i	$-0.755215\pi$
$139$	−16.9443	−1.43719	−0.718597	+	0.695427i	$0.755215\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.15298	0.765411
$144$	0	0
$145$	18.9443	1.57324
$146$	0	0
$147$	3.94427	0.325318
$148$	0	0
$149$	3.82998	0.313764	0.156882	−	0.987617i	$-0.449856\pi$
$149$	3.82998	0.313764	0.156882	+	0.987617i	$0.449856\pi$
$150$	0	0
$151$	20.0540	1.63197	0.815987	−	0.578070i	$-0.196194\pi$
$151$	20.0540	1.63197	0.815987	+	0.578070i	$0.196194\pi$
$152$	0	0
$153$	2.47214	0.199860
$154$	0	0
$155$	12.3607	0.992834
$156$	0	0
$157$	−13.3956	−1.06909	−0.534544	−	0.845141i	$-0.679516\pi$
$157$	−13.3956	−1.06909	−0.534544	+	0.845141i	$0.679516\pi$
$158$	0	0
$159$	11.6476	0.923712
$160$	0	0
$161$	9.88854	0.779326
$162$	0	0
$163$	−14.4721	−1.13355	−0.566773	−	0.823874i	$-0.691808\pi$
$163$	−14.4721	−1.13355	−0.566773	+	0.823874i	$0.691808\pi$
$164$	0	0
$165$	20.4667	1.59333
$166$	0	0
$167$	−20.4667	−1.58376	−0.791880	−	0.610677i	$-0.790898\pi$
$167$	−20.4667	−1.58376	−0.791880	+	0.610677i	$0.790898\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	6.47214	0.494937
$172$	0	0
$173$	9.48683	0.721271	0.360635	−	0.932707i	$-0.382560\pi$
$173$	9.48683	0.721271	0.360635	+	0.932707i	$0.382560\pi$
$174$	0	0
$175$	−8.74032	−0.660706
$176$	0	0
$177$	8.94427	0.672293
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	15.5563	1.15629	0.578147	−	0.815933i	$-0.303776\pi$
$181$	15.5563	1.15629	0.578147	+	0.815933i	$0.303776\pi$
$182$	0	0
$183$	2.08191	0.153900
$184$	0	0
$185$	−33.4164	−2.45682
$186$	0	0
$187$	−16.0000	−1.17004
$188$	0	0
$189$	1.74806	0.127153
$190$	0	0
$191$	−14.8098	−1.07160	−0.535801	−	0.844344i	$-0.679990\pi$
$191$	−14.8098	−1.07160	−0.535801	+	0.844344i	$0.679990\pi$
$192$	0	0
$193$	4.94427	0.355896	0.177948	−	0.984040i	$-0.443054\pi$
$193$	4.94427	0.355896	0.177948	+	0.984040i	$0.443054\pi$
$194$	0	0
$195$	4.47214	0.320256
$196$	0	0
$197$	−7.32611	−0.521964	−0.260982	−	0.965344i	$-0.584046\pi$
$197$	−7.32611	−0.521964	−0.260982	+	0.965344i	$0.584046\pi$
$198$	0	0
$199$	5.24419	0.371751	0.185875	−	0.982573i	$-0.440488\pi$
$199$	5.24419	0.371751	0.185875	+	0.982573i	$0.440488\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−10.4721	−0.735000
$204$	0	0
$205$	−7.81758	−0.546003
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	−41.8885	−2.89749
$210$	0	0
$211$	−7.05573	−0.485736	−0.242868	−	0.970059i	$-0.578088\pi$
$211$	−7.05573	−0.485736	−0.242868	+	0.970059i	$0.578088\pi$
$212$	0	0
$213$	−9.15298	−0.627152
$214$	0	0
$215$	−4.83153	−0.329508
$216$	0	0
$217$	−6.83282	−0.463842
$218$	0	0
$219$	−2.94427	−0.198955
$220$	0	0
$221$	−3.49613	−0.235175
$222$	0	0
$223$	3.90879	0.261752	0.130876	−	0.991399i	$-0.458221\pi$
$223$	3.90879	0.261752	0.130876	+	0.991399i	$0.458221\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	14.4721	0.960549	0.480275	−	0.877118i	$-0.340537\pi$
$227$	14.4721	0.960549	0.480275	+	0.877118i	$0.340537\pi$
$228$	0	0
$229$	14.0633	0.929331	0.464665	−	0.885486i	$-0.346175\pi$
$229$	14.0633	0.929331	0.464665	+	0.885486i	$0.346175\pi$
$230$	0	0
$231$	−11.3137	−0.744387
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7.40492	−0.481001
$238$	0	0
$239$	26.1235	1.68979	0.844896	−	0.534931i	$-0.179662\pi$
$239$	26.1235	1.68979	0.844896	+	0.534931i	$0.179662\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−12.4729	−0.796863
$246$	0	0
$247$	−9.15298	−0.582390
$248$	0	0
$249$	−6.47214	−0.410155
$250$	0	0
$251$	−11.4164	−0.720597	−0.360299	−	0.932837i	$-0.617325\pi$
$251$	−11.4164	−0.720597	−0.360299	+	0.932837i	$0.617325\pi$
$252$	0	0
$253$	36.6119	2.30177
$254$	0	0
$255$	−7.81758	−0.489556
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	18.4721	1.14780
$260$	0	0
$261$	5.99070	0.370815
$262$	0	0
$263$	−12.6491	−0.779978	−0.389989	−	0.920820i	$-0.627521\pi$
$263$	−12.6491	−0.779978	−0.389989	+	0.920820i	$0.627521\pi$
$264$	0	0
$265$	−36.8328	−2.26262
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	5.32300	0.324549	0.162275	−	0.986746i	$-0.448117\pi$
$269$	5.32300	0.324549	0.162275	+	0.986746i	$0.448117\pi$
$270$	0	0
$271$	10.9010	0.662191	0.331096	−	0.943597i	$-0.392582\pi$
$271$	10.9010	0.662191	0.331096	+	0.943597i	$0.392582\pi$
$272$	0	0
$273$	−2.47214	−0.149620
$274$	0	0
$275$	−32.3607	−1.95142
$276$	0	0
$277$	18.3848	1.10463	0.552317	−	0.833634i	$-0.313744\pi$
$277$	18.3848	1.10463	0.552317	+	0.833634i	$0.313744\pi$
$278$	0	0
$279$	3.90879	0.234013
$280$	0	0
$281$	3.88854	0.231971	0.115986	−	0.993251i	$-0.462997\pi$
$281$	3.88854	0.231971	0.115986	+	0.993251i	$0.462997\pi$
$282$	0	0
$283$	−26.8328	−1.59505	−0.797523	−	0.603289i	$-0.793857\pi$
$283$	−26.8328	−1.59505	−0.797523	+	0.603289i	$0.793857\pi$
$284$	0	0
$285$	−20.4667	−1.21234
$286$	0	0
$287$	4.32145	0.255087
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	12.9443	0.758807
$292$	0	0
$293$	−22.9613	−1.34141	−0.670706	−	0.741723i	$-0.734009\pi$
$293$	−22.9613	−1.34141	−0.670706	+	0.741723i	$0.734009\pi$
$294$	0	0
$295$	−28.2843	−1.64677
$296$	0	0
$297$	6.47214	0.375551
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	2.67080	0.153943
$302$	0	0
$303$	−0.333851	−0.0191792
$304$	0	0
$305$	−6.58359	−0.376975
$306$	0	0
$307$	5.88854	0.336077	0.168038	−	0.985780i	$-0.446257\pi$
$307$	5.88854	0.336077	0.168038	+	0.985780i	$0.446257\pi$
$308$	0	0
$309$	−1.74806	−0.0994439
$310$	0	0
$311$	−19.6414	−1.11376	−0.556880	−	0.830593i	$-0.688002\pi$
$311$	−19.6414	−1.11376	−0.556880	+	0.830593i	$0.688002\pi$
$312$	0	0
$313$	20.9443	1.18384	0.591920	−	0.805997i	$-0.298370\pi$
$313$	20.9443	1.18384	0.591920	+	0.805997i	$0.298370\pi$
$314$	0	0
$315$	−5.52786	−0.311460
$316$	0	0
$317$	17.3044	0.971913	0.485956	−	0.873983i	$-0.338471\pi$
$317$	17.3044	0.971913	0.485956	+	0.873983i	$0.338471\pi$
$318$	0	0
$319$	−38.7727	−2.17085
$320$	0	0
$321$	−0.944272	−0.0527041
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−7.07107	−0.392232
$326$	0	0
$327$	16.8918	0.934116
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.9443	0.931341	0.465671	−	0.884958i	$-0.345813\pi$
$331$	16.9443	0.931341	0.465671	+	0.884958i	$0.345813\pi$
$332$	0	0
$333$	−10.5672	−0.579079
$334$	0	0
$335$	−37.9473	−2.07328
$336$	0	0
$337$	14.9443	0.814066	0.407033	−	0.913413i	$-0.366563\pi$
$337$	14.9443	0.814066	0.407033	+	0.913413i	$0.366563\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	−25.2982	−1.36998
$342$	0	0
$343$	19.1313	1.03299
$344$	0	0
$345$	17.8885	0.963087
$346$	0	0
$347$	−22.4721	−1.20637	−0.603184	−	0.797602i	$-0.706101\pi$
$347$	−22.4721	−1.20637	−0.603184	+	0.797602i	$0.706101\pi$
$348$	0	0
$349$	−9.23179	−0.494167	−0.247083	−	0.968994i	$-0.579472\pi$
$349$	−9.23179	−0.494167	−0.247083	+	0.968994i	$0.579472\pi$
$350$	0	0
$351$	1.41421	0.0754851
$352$	0	0
$353$	−11.8885	−0.632763	−0.316382	−	0.948632i	$-0.602468\pi$
$353$	−11.8885	−0.632763	−0.316382	+	0.948632i	$0.602468\pi$
$354$	0	0
$355$	28.9443	1.53620
$356$	0	0
$357$	4.32145	0.228715
$358$	0	0
$359$	−13.4744	−0.711153	−0.355577	−	0.934647i	$-0.615716\pi$
$359$	−13.4744	−0.711153	−0.355577	+	0.934647i	$0.615716\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	−30.8885	−1.62123
$364$	0	0
$365$	9.31061	0.487339
$366$	0	0
$367$	17.8933	0.934023	0.467011	−	0.884251i	$-0.345331\pi$
$367$	17.8933	0.934023	0.467011	+	0.884251i	$0.345331\pi$
$368$	0	0
$369$	−2.47214	−0.128694
$370$	0	0
$371$	20.3607	1.05707
$372$	0	0
$373$	21.8809	1.13295	0.566475	−	0.824079i	$-0.308307\pi$
$373$	21.8809	1.13295	0.566475	+	0.824079i	$0.308307\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.47214	−0.436337
$378$	0	0
$379$	27.4164	1.40829	0.704143	−	0.710058i	$-0.251331\pi$
$379$	27.4164	1.40829	0.704143	+	0.710058i	$0.251331\pi$
$380$	0	0
$381$	−10.9010	−0.558478
$382$	0	0
$383$	−19.1313	−0.977563	−0.488782	−	0.872406i	$-0.662559\pi$
$383$	−19.1313	−0.977563	−0.488782	+	0.872406i	$0.662559\pi$
$384$	0	0
$385$	35.7771	1.82337
$386$	0	0
$387$	−1.52786	−0.0776657
$388$	0	0
$389$	33.4497	1.69596	0.847982	−	0.530024i	$-0.177817\pi$
$389$	33.4497	1.69596	0.847982	+	0.530024i	$0.177817\pi$
$390$	0	0
$391$	−13.9845	−0.707227
$392$	0	0
$393$	16.9443	0.854725
$394$	0	0
$395$	23.4164	1.17821
$396$	0	0
$397$	−9.07417	−0.455420	−0.227710	−	0.973729i	$-0.573124\pi$
$397$	−9.07417	−0.455420	−0.227710	+	0.973729i	$0.573124\pi$
$398$	0	0
$399$	11.3137	0.566394
$400$	0	0
$401$	13.5279	0.675549	0.337775	−	0.941227i	$-0.390326\pi$
$401$	13.5279	0.675549	0.337775	+	0.941227i	$0.390326\pi$
$402$	0	0
$403$	−5.52786	−0.275363
$404$	0	0
$405$	3.16228	0.157135
$406$	0	0
$407$	68.3923	3.39008
$408$	0	0
$409$	−12.9443	−0.640053	−0.320027	−	0.947409i	$-0.603692\pi$
$409$	−12.9443	−0.640053	−0.320027	+	0.947409i	$0.603692\pi$
$410$	0	0
$411$	−2.47214	−0.121941
$412$	0	0
$413$	15.6352	0.769356
$414$	0	0
$415$	20.4667	1.00467
$416$	0	0
$417$	16.9443	0.829765
$418$	0	0
$419$	1.52786	0.0746410	0.0373205	−	0.999303i	$-0.488118\pi$
$419$	1.52786	0.0746410	0.0373205	+	0.999303i	$0.488118\pi$
$420$	0	0
$421$	7.07107	0.344623	0.172311	−	0.985043i	$-0.444876\pi$
$421$	7.07107	0.344623	0.172311	+	0.985043i	$0.444876\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.3607	0.599581
$426$	0	0
$427$	3.63932	0.176119
$428$	0	0
$429$	−9.15298	−0.441910
$430$	0	0
$431$	−18.3060	−0.881767	−0.440884	−	0.897564i	$-0.645335\pi$
$431$	−18.3060	−0.881767	−0.440884	+	0.897564i	$0.645335\pi$
$432$	0	0
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	−18.9443	−0.908308
$436$	0	0
$437$	−36.6119	−1.75139
$438$	0	0
$439$	−6.06952	−0.289682	−0.144841	−	0.989455i	$-0.546267\pi$
$439$	−6.06952	−0.289682	−0.144841	+	0.989455i	$0.546267\pi$
$440$	0	0
$441$	−3.94427	−0.187822
$442$	0	0
$443$	11.4164	0.542410	0.271205	−	0.962522i	$-0.412578\pi$
$443$	11.4164	0.542410	0.271205	+	0.962522i	$0.412578\pi$
$444$	0	0
$445$	−31.6228	−1.49906
$446$	0	0
$447$	−3.82998	−0.181152
$448$	0	0
$449$	−23.4164	−1.10509	−0.552544	−	0.833484i	$-0.686343\pi$
$449$	−23.4164	−1.10509	−0.552544	+	0.833484i	$0.686343\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	−20.0540	−0.942220
$454$	0	0
$455$	7.81758	0.366494
$456$	0	0
$457$	3.05573	0.142941	0.0714705	−	0.997443i	$-0.477231\pi$
$457$	3.05573	0.142941	0.0714705	+	0.997443i	$0.477231\pi$
$458$	0	0
$459$	−2.47214	−0.115389
$460$	0	0
$461$	8.81913	0.410748	0.205374	−	0.978684i	$-0.434159\pi$
$461$	8.81913	0.410748	0.205374	+	0.978684i	$0.434159\pi$
$462$	0	0
$463$	−7.40492	−0.344136	−0.172068	−	0.985085i	$-0.555045\pi$
$463$	−7.40492	−0.344136	−0.172068	+	0.985085i	$0.555045\pi$
$464$	0	0
$465$	−12.3607	−0.573213
$466$	0	0
$467$	4.58359	0.212103	0.106052	−	0.994361i	$-0.466179\pi$
$467$	4.58359	0.212103	0.106052	+	0.994361i	$0.466179\pi$
$468$	0	0
$469$	20.9768	0.968617
$470$	0	0
$471$	13.3956	0.617238
$472$	0	0
$473$	9.88854	0.454676
$474$	0	0
$475$	32.3607	1.48481
$476$	0	0
$477$	−11.6476	−0.533305
$478$	0	0
$479$	3.49613	0.159742	0.0798711	−	0.996805i	$-0.474549\pi$
$479$	3.49613	0.159742	0.0798711	+	0.996805i	$0.474549\pi$
$480$	0	0
$481$	14.9443	0.681400
$482$	0	0
$483$	−9.88854	−0.449944
$484$	0	0
$485$	−40.9334	−1.85869
$486$	0	0
$487$	−16.5579	−0.750310	−0.375155	−	0.926962i	$-0.622411\pi$
$487$	−16.5579	−0.750310	−0.375155	+	0.926962i	$0.622411\pi$
$488$	0	0
$489$	14.4721	0.654453
$490$	0	0
$491$	−21.8885	−0.987816	−0.493908	−	0.869514i	$-0.664432\pi$
$491$	−21.8885	−0.987816	−0.493908	+	0.869514i	$0.664432\pi$
$492$	0	0
$493$	14.8098	0.667001
$494$	0	0
$495$	−20.4667	−0.919909
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	24.9443	1.11666	0.558329	−	0.829619i	$-0.311443\pi$
$499$	24.9443	1.11666	0.558329	+	0.829619i	$0.311443\pi$
$500$	0	0
$501$	20.4667	0.914384
$502$	0	0
$503$	4.83153	0.215427	0.107714	−	0.994182i	$-0.465647\pi$
$503$	4.83153	0.215427	0.107714	+	0.994182i	$0.465647\pi$
$504$	0	0
$505$	1.05573	0.0469793
$506$	0	0
$507$	11.0000	0.488527
$508$	0	0
$509$	−39.9318	−1.76995	−0.884974	−	0.465641i	$-0.845824\pi$
$509$	−39.9318	−1.76995	−0.884974	+	0.465641i	$0.845824\pi$
$510$	0	0
$511$	−5.14678	−0.227680
$512$	0	0
$513$	−6.47214	−0.285752
$514$	0	0
$515$	5.52786	0.243587
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−9.48683	−0.416426
$520$	0	0
$521$	−28.3607	−1.24250	−0.621252	−	0.783611i	$-0.713376\pi$
$521$	−28.3607	−1.24250	−0.621252	+	0.783611i	$0.713376\pi$
$522$	0	0
$523$	16.3607	0.715403	0.357701	−	0.933836i	$-0.383561\pi$
$523$	16.3607	0.715403	0.357701	+	0.933836i	$0.383561\pi$
$524$	0	0
$525$	8.74032	0.381459
$526$	0	0
$527$	9.66306	0.420930
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	−8.94427	−0.388148
$532$	0	0
$533$	3.49613	0.151434
$534$	0	0
$535$	2.98605	0.129098
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	25.5279	1.09956
$540$	0	0
$541$	26.7124	1.14846	0.574229	−	0.818695i	$-0.305302\pi$
$541$	26.7124	1.14846	0.574229	+	0.818695i	$0.305302\pi$
$542$	0	0
$543$	−15.5563	−0.667587
$544$	0	0
$545$	−53.4164	−2.28811
$546$	0	0
$547$	−32.3607	−1.38364	−0.691821	−	0.722069i	$-0.743191\pi$
$547$	−32.3607	−1.38364	−0.691821	+	0.722069i	$0.743191\pi$
$548$	0	0
$549$	−2.08191	−0.0888540
$550$	0	0
$551$	38.7727	1.65177
$552$	0	0
$553$	−12.9443	−0.550446
$554$	0	0
$555$	33.4164	1.41845
$556$	0	0
$557$	−15.8114	−0.669950	−0.334975	−	0.942227i	$-0.608728\pi$
$557$	−15.8114	−0.669950	−0.334975	+	0.942227i	$0.608728\pi$
$558$	0	0
$559$	2.16073	0.0913890
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	−1.52786	−0.0643918	−0.0321959	−	0.999482i	$-0.510250\pi$
$563$	−1.52786	−0.0643918	−0.0321959	+	0.999482i	$0.510250\pi$
$564$	0	0
$565$	−6.32456	−0.266076
$566$	0	0
$567$	−1.74806	−0.0734118
$568$	0	0
$569$	7.41641	0.310912	0.155456	−	0.987843i	$-0.450315\pi$
$569$	7.41641	0.310912	0.155456	+	0.987843i	$0.450315\pi$
$570$	0	0
$571$	8.94427	0.374306	0.187153	−	0.982331i	$-0.440074\pi$
$571$	8.94427	0.374306	0.187153	+	0.982331i	$0.440074\pi$
$572$	0	0
$573$	14.8098	0.618690
$574$	0	0
$575$	−28.2843	−1.17954
$576$	0	0
$577$	26.9443	1.12170	0.560852	−	0.827916i	$-0.310474\pi$
$577$	26.9443	1.12170	0.560852	+	0.827916i	$0.310474\pi$
$578$	0	0
$579$	−4.94427	−0.205477
$580$	0	0
$581$	−11.3137	−0.469372
$582$	0	0
$583$	75.3846	3.12211
$584$	0	0
$585$	−4.47214	−0.184900
$586$	0	0
$587$	32.9443	1.35976	0.679878	−	0.733325i	$-0.262033\pi$
$587$	32.9443	1.35976	0.679878	+	0.733325i	$0.262033\pi$
$588$	0	0
$589$	25.2982	1.04240
$590$	0	0
$591$	7.32611	0.301356
$592$	0	0
$593$	7.88854	0.323944	0.161972	−	0.986795i	$-0.448215\pi$
$593$	7.88854	0.323944	0.161972	+	0.986795i	$0.448215\pi$
$594$	0	0
$595$	−13.6656	−0.560236
$596$	0	0
$597$	−5.24419	−0.214631
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	−6.94427	−0.283263	−0.141631	−	0.989919i	$-0.545235\pi$
$601$	−6.94427	−0.283263	−0.141631	+	0.989919i	$0.545235\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	97.6782	3.97118
$606$	0	0
$607$	44.8422	1.82009	0.910044	−	0.414512i	$-0.136048\pi$
$607$	44.8422	1.82009	0.910044	+	0.414512i	$0.136048\pi$
$608$	0	0
$609$	10.4721	0.424352
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.5672	0.426805	0.213403	−	0.976964i	$-0.431545\pi$
$613$	10.5672	0.426805	0.213403	+	0.976964i	$0.431545\pi$
$614$	0	0
$615$	7.81758	0.315235
$616$	0	0
$617$	−15.8885	−0.639649	−0.319824	−	0.947477i	$-0.603624\pi$
$617$	−15.8885	−0.639649	−0.319824	+	0.947477i	$0.603624\pi$
$618$	0	0
$619$	5.88854	0.236681	0.118340	−	0.992973i	$-0.462243\pi$
$619$	5.88854	0.236681	0.118340	+	0.992973i	$0.462243\pi$
$620$	0	0
$621$	5.65685	0.227002
$622$	0	0
$623$	17.4806	0.700347
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	41.8885	1.67287
$628$	0	0
$629$	−26.1235	−1.04161
$630$	0	0
$631$	27.0463	1.07670	0.538348	−	0.842723i	$-0.319049\pi$
$631$	27.0463	1.07670	0.538348	+	0.842723i	$0.319049\pi$
$632$	0	0
$633$	7.05573	0.280440
$634$	0	0
$635$	34.4721	1.36798
$636$	0	0
$637$	5.57804	0.221010
$638$	0	0
$639$	9.15298	0.362086
$640$	0	0
$641$	8.58359	0.339032	0.169516	−	0.985527i	$-0.445780\pi$
$641$	8.58359	0.339032	0.169516	+	0.985527i	$0.445780\pi$
$642$	0	0
$643$	8.36068	0.329713	0.164857	−	0.986318i	$-0.447284\pi$
$643$	8.36068	0.329713	0.164857	+	0.986318i	$0.447284\pi$
$644$	0	0
$645$	4.83153	0.190241
$646$	0	0
$647$	42.2688	1.66176	0.830879	−	0.556454i	$-0.187838\pi$
$647$	42.2688	1.66176	0.830879	+	0.556454i	$0.187838\pi$
$648$	0	0
$649$	57.8885	2.27232
$650$	0	0
$651$	6.83282	0.267799
$652$	0	0
$653$	38.4388	1.50423	0.752113	−	0.659034i	$-0.229035\pi$
$653$	38.4388	1.50423	0.752113	+	0.659034i	$0.229035\pi$
$654$	0	0
$655$	−53.5825	−2.09364
$656$	0	0
$657$	2.94427	0.114867
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−40.1869	−1.56309	−0.781544	−	0.623850i	$-0.785568\pi$
$661$	−40.1869	−1.56309	−0.781544	+	0.623850i	$0.785568\pi$
$662$	0	0
$663$	3.49613	0.135778
$664$	0	0
$665$	−35.7771	−1.38738
$666$	0	0
$667$	−33.8885	−1.31217
$668$	0	0
$669$	−3.90879	−0.151123
$670$	0	0
$671$	13.4744	0.520175
$672$	0	0
$673$	−17.8885	−0.689553	−0.344776	−	0.938685i	$-0.612045\pi$
$673$	−17.8885	−0.689553	−0.344776	+	0.938685i	$0.612045\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−0.491473	−0.0188889	−0.00944443	−	0.999955i	$-0.503006\pi$
$677$	−0.491473	−0.0188889	−0.00944443	+	0.999955i	$0.503006\pi$
$678$	0	0
$679$	22.6274	0.868361
$680$	0	0
$681$	−14.4721	−0.554573
$682$	0	0
$683$	−3.41641	−0.130725	−0.0653626	−	0.997862i	$-0.520820\pi$
$683$	−3.41641	−0.130725	−0.0653626	+	0.997862i	$0.520820\pi$
$684$	0	0
$685$	7.81758	0.298694
$686$	0	0
$687$	−14.0633	−0.536549
$688$	0	0
$689$	16.4721	0.627538
$690$	0	0
$691$	25.5279	0.971126	0.485563	−	0.874202i	$-0.338615\pi$
$691$	25.5279	0.971126	0.485563	+	0.874202i	$0.338615\pi$
$692$	0	0
$693$	11.3137	0.429772
$694$	0	0
$695$	−53.5825	−2.03250
$696$	0	0
$697$	−6.11146	−0.231488
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	−35.6104	−1.34499	−0.672493	−	0.740104i	$-0.734776\pi$
$701$	−35.6104	−1.34499	−0.672493	+	0.740104i	$0.734776\pi$
$702$	0	0
$703$	−68.3923	−2.57947
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.583592	−0.0219482
$708$	0	0
$709$	33.8623	1.27173	0.635863	−	0.771802i	$-0.280644\pi$
$709$	33.8623	1.27173	0.635863	+	0.771802i	$0.280644\pi$
$710$	0	0
$711$	7.40492	0.277706
$712$	0	0
$713$	−22.1115	−0.828081
$714$	0	0
$715$	28.9443	1.08245
$716$	0	0
$717$	−26.1235	−0.975602
$718$	0	0
$719$	21.8021	0.813081	0.406540	−	0.913633i	$-0.366735\pi$
$719$	21.8021	0.813081	0.406540	+	0.913633i	$0.366735\pi$
$720$	0	0
$721$	−3.05573	−0.113801
$722$	0	0
$723$	0	0
$724$	0	0
$725$	29.9535	1.11245
$726$	0	0
$727$	−20.0540	−0.743763	−0.371881	−	0.928280i	$-0.621287\pi$
$727$	−20.0540	−0.743763	−0.371881	+	0.928280i	$0.621287\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.77709	−0.139701
$732$	0	0
$733$	−31.0339	−1.14626	−0.573131	−	0.819463i	$-0.694272\pi$
$733$	−31.0339	−1.14626	−0.573131	+	0.819463i	$0.694272\pi$
$734$	0	0
$735$	12.4729	0.460069
$736$	0	0
$737$	77.6656	2.86085
$738$	0	0
$739$	32.9443	1.21187	0.605937	−	0.795512i	$-0.292798\pi$
$739$	32.9443	1.21187	0.605937	+	0.795512i	$0.292798\pi$
$740$	0	0
$741$	9.15298	0.336243
$742$	0	0
$743$	2.16073	0.0792694	0.0396347	−	0.999214i	$-0.487381\pi$
$743$	2.16073	0.0792694	0.0396347	+	0.999214i	$0.487381\pi$
$744$	0	0
$745$	12.1115	0.443729
$746$	0	0
$747$	6.47214	0.236803
$748$	0	0
$749$	−1.65065	−0.0603134
$750$	0	0
$751$	−7.40492	−0.270209	−0.135105	−	0.990831i	$-0.543137\pi$
$751$	−7.40492	−0.270209	−0.135105	+	0.990831i	$0.543137\pi$
$752$	0	0
$753$	11.4164	0.416037
$754$	0	0
$755$	63.4164	2.30796
$756$	0	0
$757$	−14.0633	−0.511140	−0.255570	−	0.966791i	$-0.582263\pi$
$757$	−14.0633	−0.511140	−0.255570	+	0.966791i	$0.582263\pi$
$758$	0	0
$759$	−36.6119	−1.32893
$760$	0	0
$761$	−7.41641	−0.268845	−0.134422	−	0.990924i	$-0.542918\pi$
$761$	−7.41641	−0.268845	−0.134422	+	0.990924i	$0.542918\pi$
$762$	0	0
$763$	29.5279	1.06898
$764$	0	0
$765$	7.81758	0.282645
$766$	0	0
$767$	12.6491	0.456733
$768$	0	0
$769$	−30.8328	−1.11186	−0.555930	−	0.831229i	$-0.687638\pi$
$769$	−30.8328	−1.11186	−0.555930	+	0.831229i	$0.687638\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	0	0
$773$	−6.65841	−0.239486	−0.119743	−	0.992805i	$-0.538207\pi$
$773$	−6.65841	−0.239486	−0.119743	+	0.992805i	$0.538207\pi$
$774$	0	0
$775$	19.5440	0.702039
$776$	0	0
$777$	−18.4721	−0.662684
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	−59.2393	−2.11975
$782$	0	0
$783$	−5.99070	−0.214090
$784$	0	0
$785$	−42.3607	−1.51192
$786$	0	0
$787$	14.4721	0.515876	0.257938	−	0.966161i	$-0.416957\pi$
$787$	14.4721	0.515876	0.257938	+	0.966161i	$0.416957\pi$
$788$	0	0
$789$	12.6491	0.450320
$790$	0	0
$791$	3.49613	0.124308
$792$	0	0
$793$	2.94427	0.104554
$794$	0	0
$795$	36.8328	1.30633
$796$	0	0
$797$	−5.16538	−0.182967	−0.0914836	−	0.995807i	$-0.529161\pi$
$797$	−5.16538	−0.182967	−0.0914836	+	0.995807i	$0.529161\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	−19.0557	−0.672462
$804$	0	0
$805$	31.2703	1.10213
$806$	0	0
$807$	−5.32300	−0.187379
$808$	0	0
$809$	−23.4164	−0.823277	−0.411639	−	0.911347i	$-0.635043\pi$
$809$	−23.4164	−0.823277	−0.411639	+	0.911347i	$0.635043\pi$
$810$	0	0
$811$	−29.3050	−1.02904	−0.514518	−	0.857480i	$-0.672029\pi$
$811$	−29.3050	−1.02904	−0.514518	+	0.857480i	$0.672029\pi$
$812$	0	0
$813$	−10.9010	−0.382316
$814$	0	0
$815$	−45.7649	−1.60307
$816$	0	0
$817$	−9.88854	−0.345956
$818$	0	0
$819$	2.47214	0.0863834
$820$	0	0
$821$	50.2626	1.75418	0.877088	−	0.480329i	$-0.159483\pi$
$821$	50.2626	1.75418	0.877088	+	0.480329i	$0.159483\pi$
$822$	0	0
$823$	1.74806	0.0609337	0.0304668	−	0.999536i	$-0.490301\pi$
$823$	1.74806	0.0609337	0.0304668	+	0.999536i	$0.490301\pi$
$824$	0	0
$825$	32.3607	1.12665
$826$	0	0
$827$	−18.1115	−0.629797	−0.314899	−	0.949125i	$-0.601970\pi$
$827$	−18.1115	−0.629797	−0.314899	+	0.949125i	$0.601970\pi$
$828$	0	0
$829$	−5.57804	−0.193733	−0.0968667	−	0.995297i	$-0.530882\pi$
$829$	−5.57804	−0.193733	−0.0968667	+	0.995297i	$0.530882\pi$
$830$	0	0
$831$	−18.3848	−0.637761
$832$	0	0
$833$	−9.75078	−0.337844
$834$	0	0
$835$	−64.7214	−2.23978
$836$	0	0
$837$	−3.90879	−0.135108
$838$	0	0
$839$	6.48218	0.223790	0.111895	−	0.993720i	$-0.464308\pi$
$839$	6.48218	0.223790	0.111895	+	0.993720i	$0.464308\pi$
$840$	0	0
$841$	6.88854	0.237536
$842$	0	0
$843$	−3.88854	−0.133929
$844$	0	0
$845$	−34.7851	−1.19664
$846$	0	0
$847$	−53.9952	−1.85530
$848$	0	0
$849$	26.8328	0.920900
$850$	0	0
$851$	59.7771	2.04913
$852$	0	0
$853$	−30.3662	−1.03972	−0.519859	−	0.854252i	$-0.674016\pi$
$853$	−30.3662	−1.03972	−0.519859	+	0.854252i	$0.674016\pi$
$854$	0	0
$855$	20.4667	0.699946
$856$	0	0
$857$	55.4164	1.89299	0.946494	−	0.322721i	$-0.104597\pi$
$857$	55.4164	1.89299	0.946494	+	0.322721i	$0.104597\pi$
$858$	0	0
$859$	54.4721	1.85857	0.929283	−	0.369369i	$-0.120426\pi$
$859$	54.4721	1.85857	0.929283	+	0.369369i	$0.120426\pi$
$860$	0	0
$861$	−4.32145	−0.147275
$862$	0	0
$863$	28.7943	0.980171	0.490086	−	0.871674i	$-0.336966\pi$
$863$	28.7943	0.980171	0.490086	+	0.871674i	$0.336966\pi$
$864$	0	0
$865$	30.0000	1.02003
$866$	0	0
$867$	10.8885	0.369794
$868$	0	0
$869$	−47.9256	−1.62577
$870$	0	0
$871$	16.9706	0.575026
$872$	0	0
$873$	−12.9443	−0.438097
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.3801	−0.924561	−0.462281	−	0.886734i	$-0.652969\pi$
$877$	−27.3801	−0.924561	−0.462281	+	0.886734i	$0.652969\pi$
$878$	0	0
$879$	22.9613	0.774464
$880$	0	0
$881$	−7.88854	−0.265772	−0.132886	−	0.991131i	$-0.542424\pi$
$881$	−7.88854	−0.265772	−0.132886	+	0.991131i	$0.542424\pi$
$882$	0	0
$883$	−9.52786	−0.320638	−0.160319	−	0.987065i	$-0.551252\pi$
$883$	−9.52786	−0.320638	−0.160319	+	0.987065i	$0.551252\pi$
$884$	0	0
$885$	28.2843	0.950765
$886$	0	0
$887$	8.32766	0.279615	0.139808	−	0.990179i	$-0.455352\pi$
$887$	8.32766	0.279615	0.139808	+	0.990179i	$0.455352\pi$
$888$	0	0
$889$	−19.0557	−0.639109
$890$	0	0
$891$	−6.47214	−0.216825
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−37.9473	−1.26844
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	23.4164	0.780981
$900$	0	0
$901$	−28.7943	−0.959279
$902$	0	0
$903$	−2.67080	−0.0888788
$904$	0	0
$905$	49.1935	1.63525
$906$	0	0
$907$	−47.1935	−1.56703	−0.783517	−	0.621370i	$-0.786576\pi$
$907$	−47.1935	−1.56703	−0.783517	+	0.621370i	$0.786576\pi$
$908$	0	0
$909$	0.333851	0.0110731
$910$	0	0
$911$	−14.8098	−0.490672	−0.245336	−	0.969438i	$-0.578898\pi$
$911$	−14.8098	−0.490672	−0.245336	+	0.969438i	$0.578898\pi$
$912$	0	0
$913$	−41.8885	−1.38631
$914$	0	0
$915$	6.58359	0.217647
$916$	0	0
$917$	29.6197	0.978128
$918$	0	0
$919$	−2.57339	−0.0848882	−0.0424441	−	0.999099i	$-0.513514\pi$
$919$	−2.57339	−0.0848882	−0.0424441	+	0.999099i	$0.513514\pi$
$920$	0	0
$921$	−5.88854	−0.194034
$922$	0	0
$923$	−12.9443	−0.426066
$924$	0	0
$925$	−52.8360	−1.73724
$926$	0	0
$927$	1.74806	0.0574140
$928$	0	0
$929$	−2.47214	−0.0811081	−0.0405541	−	0.999177i	$-0.512912\pi$
$929$	−2.47214	−0.0811081	−0.0405541	+	0.999177i	$0.512912\pi$
$930$	0	0
$931$	−25.5279	−0.836642
$932$	0	0
$933$	19.6414	0.643029
$934$	0	0
$935$	−50.5964	−1.65468
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	−20.9443	−0.683490
$940$	0	0
$941$	31.4465	1.02513	0.512564	−	0.858649i	$-0.328696\pi$
$941$	31.4465	1.02513	0.512564	+	0.858649i	$0.328696\pi$
$942$	0	0
$943$	13.9845	0.455398
$944$	0	0
$945$	5.52786	0.179821
$946$	0	0
$947$	−23.7771	−0.772652	−0.386326	−	0.922362i	$-0.626256\pi$
$947$	−23.7771	−0.772652	−0.386326	+	0.922362i	$0.626256\pi$
$948$	0	0
$949$	−4.16383	−0.135164
$950$	0	0
$951$	−17.3044	−0.561134
$952$	0	0
$953$	−18.4721	−0.598371	−0.299186	−	0.954195i	$-0.596715\pi$
$953$	−18.4721	−0.598371	−0.299186	+	0.954195i	$0.596715\pi$
$954$	0	0
$955$	−46.8328	−1.51547
$956$	0	0
$957$	38.7727	1.25334
$958$	0	0
$959$	−4.32145	−0.139547
$960$	0	0
$961$	−15.7214	−0.507141
$962$	0	0
$963$	0.944272	0.0304287
$964$	0	0
$965$	15.6352	0.503314
$966$	0	0
$967$	−38.3600	−1.23357	−0.616787	−	0.787130i	$-0.711566\pi$
$967$	−38.3600	−1.23357	−0.616787	+	0.787130i	$0.711566\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	−24.3607	−0.781771	−0.390886	−	0.920439i	$-0.627831\pi$
$971$	−24.3607	−0.781771	−0.390886	+	0.920439i	$0.627831\pi$
$972$	0	0
$973$	29.6197	0.949563
$974$	0	0
$975$	7.07107	0.226455
$976$	0	0
$977$	39.4164	1.26104	0.630521	−	0.776172i	$-0.282841\pi$
$977$	39.4164	1.26104	0.630521	+	0.776172i	$0.282841\pi$
$978$	0	0
$979$	64.7214	2.06850
$980$	0	0
$981$	−16.8918	−0.539312
$982$	0	0
$983$	2.98605	0.0952402	0.0476201	−	0.998866i	$-0.484836\pi$
$983$	2.98605	0.0952402	0.0476201	+	0.998866i	$0.484836\pi$
$984$	0	0
$985$	−23.1672	−0.738168
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.64290	0.274828
$990$	0	0
$991$	−14.3972	−0.457341	−0.228671	−	0.973504i	$-0.573438\pi$
$991$	−14.3972	−0.457341	−0.228671	+	0.973504i	$0.573438\pi$
$992$	0	0
$993$	−16.9443	−0.537710
$994$	0	0
$995$	16.5836	0.525735
$996$	0	0
$997$	41.6799	1.32002	0.660008	−	0.751259i	$-0.270553\pi$
$997$	41.6799	1.32002	0.660008	+	0.751259i	$0.270553\pi$
$998$	0	0
$999$	10.5672	0.334331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.k.1.4		4
3.2	odd	2		9216.2.a.bj.1.2		4
4.3	odd	2		3072.2.a.q.1.3		4
8.3	odd	2	inner	3072.2.a.k.1.1		4
8.5	even	2		3072.2.a.q.1.2		4
12.11	even	2		9216.2.a.bd.1.1		4
16.3	odd	4		3072.2.d.g.1537.6		8
16.5	even	4		3072.2.d.g.1537.7		8
16.11	odd	4		3072.2.d.g.1537.4		8
16.13	even	4		3072.2.d.g.1537.1		8
24.5	odd	2		9216.2.a.bd.1.4		4
24.11	even	2		9216.2.a.bj.1.3		4
32.3	odd	8		1536.2.j.h.1153.3	yes	8
32.5	even	8		1536.2.j.g.385.4	yes	8
32.11	odd	8		1536.2.j.h.385.3	yes	8
32.13	even	8		1536.2.j.g.1153.4	yes	8
32.19	odd	8		1536.2.j.g.1153.2	yes	8
32.21	even	8		1536.2.j.h.385.1	yes	8
32.27	odd	8		1536.2.j.g.385.2	✓	8
32.29	even	8		1536.2.j.h.1153.1	yes	8
96.5	odd	8		4608.2.k.bf.3457.1		8
96.11	even	8		4608.2.k.bg.3457.4		8
96.29	odd	8		4608.2.k.bg.1153.4		8
96.35	even	8		4608.2.k.bg.1153.3		8
96.53	odd	8		4608.2.k.bg.3457.3		8
96.59	even	8		4608.2.k.bf.3457.2		8
96.77	odd	8		4608.2.k.bf.1153.2		8
96.83	even	8		4608.2.k.bf.1153.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.g.385.2	✓	8	32.27	odd	8
1536.2.j.g.385.4	yes	8	32.5	even	8
1536.2.j.g.1153.2	yes	8	32.19	odd	8
1536.2.j.g.1153.4	yes	8	32.13	even	8
1536.2.j.h.385.1	yes	8	32.21	even	8
1536.2.j.h.385.3	yes	8	32.11	odd	8
1536.2.j.h.1153.1	yes	8	32.29	even	8
1536.2.j.h.1153.3	yes	8	32.3	odd	8
3072.2.a.k.1.1		4	8.3	odd	2	inner
3072.2.a.k.1.4		4	1.1	even	1	trivial
3072.2.a.q.1.2		4	8.5	even	2
3072.2.a.q.1.3		4	4.3	odd	2
3072.2.d.g.1537.1		8	16.13	even	4
3072.2.d.g.1537.4		8	16.11	odd	4
3072.2.d.g.1537.6		8	16.3	odd	4
3072.2.d.g.1537.7		8	16.5	even	4
4608.2.k.bf.1153.1		8	96.83	even	8
4608.2.k.bf.1153.2		8	96.77	odd	8
4608.2.k.bf.3457.1		8	96.5	odd	8
4608.2.k.bf.3457.2		8	96.59	even	8
4608.2.k.bg.1153.3		8	96.35	even	8
4608.2.k.bg.1153.4		8	96.29	odd	8
4608.2.k.bg.3457.3		8	96.53	odd	8
4608.2.k.bg.3457.4		8	96.11	even	8
9216.2.a.bd.1.1		4	12.11	even	2
9216.2.a.bd.1.4		4	24.5	odd	2
9216.2.a.bj.1.2		4	3.2	odd	2
9216.2.a.bj.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.16228 q^{5} -1.74806 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.16228 q^{5} -1.74806 q^{7} +1.00000 q^{9} -6.47214 q^{11} -1.41421 q^{13} -3.16228 q^{15} +2.47214 q^{17} +6.47214 q^{19} +1.74806 q^{21} -5.65685 q^{23} +5.00000 q^{25} -1.00000 q^{27} +5.99070 q^{29} +3.90879 q^{31} +6.47214 q^{33} -5.52786 q^{35} -10.5672 q^{37} +1.41421 q^{39} -2.47214 q^{41} -1.52786 q^{43} +3.16228 q^{45} -3.94427 q^{49} -2.47214 q^{51} -11.6476 q^{53} -20.4667 q^{55} -6.47214 q^{57} -8.94427 q^{59} -2.08191 q^{61} -1.74806 q^{63} -4.47214 q^{65} -12.0000 q^{67} +5.65685 q^{69} +9.15298 q^{71} +2.94427 q^{73} -5.00000 q^{75} +11.3137 q^{77} +7.40492 q^{79} +1.00000 q^{81} +6.47214 q^{83} +7.81758 q^{85} -5.99070 q^{87} -10.0000 q^{89} +2.47214 q^{91} -3.90879 q^{93} +20.4667 q^{95} -12.9443 q^{97} -6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} - 8 q^{17} + 8 q^{19} + 20 q^{25} - 4 q^{27} + 8 q^{33} - 40 q^{35} + 8 q^{41} - 24 q^{43} + 20 q^{49} + 8 q^{51} - 8 q^{57} - 48 q^{67} - 24 q^{73} - 20 q^{75} + 4 q^{81} + 8 q^{83} - 40 q^{89} - 8 q^{91} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 - 8 * q^17 + 8 * q^19 + 20 * q^25 - 4 * q^27 + 8 * q^33 - 40 * q^35 + 8 * q^41 - 24 * q^43 + 20 * q^49 + 8 * q^51 - 8 * q^57 - 48 * q^67 - 24 * q^73 - 20 * q^75 + 4 * q^81 + 8 * q^83 - 40 * q^89 - 8 * q^91 - 16 * q^97 - 8 * q^99

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