LMFDB - Embedded newform 8036.2.a.q.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.72996$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.72996 q^{3} -0.138351 q^{5} -0.00725198 q^{9} +O(q^{10})$	$q-1.72996 q^{3} -0.138351 q^{5} -0.00725198 q^{9} +2.69315 q^{11} -5.05004 q^{13} +0.239341 q^{15} -4.38217 q^{17} +5.73074 q^{19} +8.09997 q^{23} -4.98086 q^{25} +5.20241 q^{27} +3.49382 q^{29} -2.84591 q^{31} -4.65904 q^{33} -4.04732 q^{37} +8.73635 q^{39} -1.00000 q^{41} -10.8473 q^{43} +0.00100332 q^{45} +0.549726 q^{47} +7.58096 q^{51} +11.8734 q^{53} -0.372600 q^{55} -9.91392 q^{57} -9.85476 q^{59} -7.09909 q^{61} +0.698677 q^{65} -2.80347 q^{67} -14.0126 q^{69} +14.5599 q^{71} -7.81173 q^{73} +8.61667 q^{75} -10.5320 q^{79} -8.97819 q^{81} -2.44863 q^{83} +0.606276 q^{85} -6.04415 q^{87} +1.78532 q^{89} +4.92331 q^{93} -0.792851 q^{95} -5.82664 q^{97} -0.0195307 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q - q^3 + 3 * q^5 + 30 * q^9	$ 15 q - q^{3} + 3 q^{5} + 30 q^{9} + 9 q^{11} + 7 q^{13} + 2 q^{15} + 3 q^{17} + 7 q^{19} - q^{23} + 32 q^{25} + 11 q^{27} + 18 q^{29} + 30 q^{31} - 16 q^{33} + 23 q^{37} + 5 q^{39} - 15 q^{41} + 12 q^{43} - 13 q^{45} - 16 q^{47} + 29 q^{51} + 33 q^{53} + 37 q^{55} + 16 q^{57} - 10 q^{59} + q^{61} + 16 q^{65} + 20 q^{67} + 21 q^{69} + 5 q^{71} - 3 q^{73} - 51 q^{75} + 25 q^{79} + 43 q^{81} + 18 q^{83} + 36 q^{85} - 53 q^{87} - 11 q^{89} + 65 q^{93} - 30 q^{95} + 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q - q^3 + 3 * q^5 + 30 * q^9 + 9 * q^11 + 7 * q^13 + 2 * q^15 + 3 * q^17 + 7 * q^19 - q^23 + 32 * q^25 + 11 * q^27 + 18 * q^29 + 30 * q^31 - 16 * q^33 + 23 * q^37 + 5 * q^39 - 15 * q^41 + 12 * q^43 - 13 * q^45 - 16 * q^47 + 29 * q^51 + 33 * q^53 + 37 * q^55 + 16 * q^57 - 10 * q^59 + q^61 + 16 * q^65 + 20 * q^67 + 21 * q^69 + 5 * q^71 - 3 * q^73 - 51 * q^75 + 25 * q^79 + 43 * q^81 + 18 * q^83 + 36 * q^85 - 53 * q^87 - 11 * q^89 + 65 * q^93 - 30 * q^95 + 16 * q^97 - 18 * 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
$3$	−1.72996	−0.998791	−0.499395	−	0.866374i	$-0.666444\pi$
$3$	−1.72996	−0.998791	−0.499395	+	0.866374i	$0.666444\pi$
$4$	0	0
$5$	−0.138351	−0.0618723	−0.0309362	−	0.999521i	$-0.509849\pi$
$5$	−0.138351	−0.0618723	−0.0309362	+	0.999521i	$0.509849\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.00725198	−0.00241733
$10$	0	0
$11$	2.69315	0.812016	0.406008	−	0.913869i	$-0.366920\pi$
$11$	2.69315	0.812016	0.406008	+	0.913869i	$0.366920\pi$
$12$	0	0
$13$	−5.05004	−1.40063	−0.700315	−	0.713834i	$-0.746957\pi$
$13$	−5.05004	−1.40063	−0.700315	+	0.713834i	$0.746957\pi$
$14$	0	0
$15$	0.239341	0.0617975
$16$	0	0
$17$	−4.38217	−1.06283	−0.531416	−	0.847111i	$-0.678340\pi$
$17$	−4.38217	−1.06283	−0.531416	+	0.847111i	$0.678340\pi$
$18$	0	0
$19$	5.73074	1.31472	0.657361	−	0.753576i	$-0.271673\pi$
$19$	5.73074	1.31472	0.657361	+	0.753576i	$0.271673\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.09997	1.68896	0.844480	−	0.535587i	$-0.179910\pi$
$23$	8.09997	1.68896	0.844480	+	0.535587i	$0.179910\pi$
$24$	0	0
$25$	−4.98086	−0.996172
$26$	0	0
$27$	5.20241	1.00121
$28$	0	0
$29$	3.49382	0.648785	0.324393	−	0.945923i	$-0.394840\pi$
$29$	3.49382	0.648785	0.324393	+	0.945923i	$0.394840\pi$
$30$	0	0
$31$	−2.84591	−0.511141	−0.255571	−	0.966790i	$-0.582263\pi$
$31$	−2.84591	−0.511141	−0.255571	+	0.966790i	$0.582263\pi$
$32$	0	0
$33$	−4.65904	−0.811034
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.04732	−0.665376	−0.332688	−	0.943037i	$-0.607956\pi$
$37$	−4.04732	−0.665376	−0.332688	+	0.943037i	$0.607956\pi$
$38$	0	0
$39$	8.73635	1.39894
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−10.8473	−1.65420	−0.827102	−	0.562051i	$-0.810012\pi$
$43$	−10.8473	−1.65420	−0.827102	+	0.562051i	$0.810012\pi$
$44$	0	0
$45$	0.00100332	0.000149566 0
$46$	0	0
$47$	0.549726	0.0801858	0.0400929	−	0.999196i	$-0.487235\pi$
$47$	0.549726	0.0801858	0.0400929	+	0.999196i	$0.487235\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.58096	1.06155
$52$	0	0
$53$	11.8734	1.63094	0.815471	−	0.578797i	$-0.196478\pi$
$53$	11.8734	1.63094	0.815471	+	0.578797i	$0.196478\pi$
$54$	0	0
$55$	−0.372600	−0.0502413
$56$	0	0
$57$	−9.91392	−1.31313
$58$	0	0
$59$	−9.85476	−1.28298	−0.641490	−	0.767131i	$-0.721684\pi$
$59$	−9.85476	−1.28298	−0.641490	+	0.767131i	$0.721684\pi$
$60$	0	0
$61$	−7.09909	−0.908945	−0.454473	−	0.890761i	$-0.650172\pi$
$61$	−7.09909	−0.908945	−0.454473	+	0.890761i	$0.650172\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.698677	0.0866602
$66$	0	0
$67$	−2.80347	−0.342498	−0.171249	−	0.985228i	$-0.554780\pi$
$67$	−2.80347	−0.342498	−0.171249	+	0.985228i	$0.554780\pi$
$68$	0	0
$69$	−14.0126	−1.68692
$70$	0	0
$71$	14.5599	1.72794	0.863969	−	0.503545i	$-0.167971\pi$
$71$	14.5599	1.72794	0.863969	+	0.503545i	$0.167971\pi$
$72$	0	0
$73$	−7.81173	−0.914294	−0.457147	−	0.889391i	$-0.651129\pi$
$73$	−7.81173	−0.914294	−0.457147	+	0.889391i	$0.651129\pi$
$74$	0	0
$75$	8.61667	0.994967
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.5320	−1.18494	−0.592471	−	0.805592i	$-0.701847\pi$
$79$	−10.5320	−1.18494	−0.592471	+	0.805592i	$0.701847\pi$
$80$	0	0
$81$	−8.97819	−0.997577
$82$	0	0
$83$	−2.44863	−0.268772	−0.134386	−	0.990929i	$-0.542906\pi$
$83$	−2.44863	−0.268772	−0.134386	+	0.990929i	$0.542906\pi$
$84$	0	0
$85$	0.606276	0.0657598
$86$	0	0
$87$	−6.04415	−0.648001
$88$	0	0
$89$	1.78532	0.189243	0.0946216	−	0.995513i	$-0.469836\pi$
$89$	1.78532	0.189243	0.0946216	+	0.995513i	$0.469836\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.92331	0.510523
$94$	0	0
$95$	−0.792851	−0.0813448
$96$	0	0
$97$	−5.82664	−0.591605	−0.295803	−	0.955249i	$-0.595587\pi$
$97$	−5.82664	−0.591605	−0.295803	+	0.955249i	$0.595587\pi$
$98$	0	0
$99$	−0.0195307	−0.00196291
$100$	0	0
$101$	10.2817	1.02307	0.511533	−	0.859264i	$-0.329078\pi$
$101$	10.2817	1.02307	0.511533	+	0.859264i	$0.329078\pi$
$102$	0	0
$103$	14.5850	1.43710	0.718552	−	0.695473i	$-0.244805\pi$
$103$	14.5850	1.43710	0.718552	+	0.695473i	$0.244805\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.0163	1.25834	0.629168	−	0.777269i	$-0.283396\pi$
$107$	13.0163	1.25834	0.629168	+	0.777269i	$0.283396\pi$
$108$	0	0
$109$	4.37091	0.418658	0.209329	−	0.977845i	$-0.432872\pi$
$109$	4.37091	0.418658	0.209329	+	0.977845i	$0.432872\pi$
$110$	0	0
$111$	7.00169	0.664571
$112$	0	0
$113$	12.0509	1.13366	0.566828	−	0.823836i	$-0.308170\pi$
$113$	12.0509	1.13366	0.566828	+	0.823836i	$0.308170\pi$
$114$	0	0
$115$	−1.12064	−0.104500
$116$	0	0
$117$	0.0366228	0.00338578
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.74692	−0.340629
$122$	0	0
$123$	1.72996	0.155985
$124$	0	0
$125$	1.38086	0.123508
$126$	0	0
$127$	−5.44928	−0.483545	−0.241773	−	0.970333i	$-0.577729\pi$
$127$	−5.44928	−0.483545	−0.241773	+	0.970333i	$0.577729\pi$
$128$	0	0
$129$	18.7654	1.65220
$130$	0	0
$131$	−21.1505	−1.84793	−0.923966	−	0.382475i	$-0.875072\pi$
$131$	−21.1505	−1.84793	−0.923966	+	0.382475i	$0.875072\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.719757	−0.0619469
$136$	0	0
$137$	−11.3964	−0.973657	−0.486829	−	0.873498i	$-0.661846\pi$
$137$	−11.3964	−0.973657	−0.486829	+	0.873498i	$0.661846\pi$
$138$	0	0
$139$	20.9983	1.78106	0.890528	−	0.454929i	$-0.150335\pi$
$139$	20.9983	1.78106	0.890528	+	0.454929i	$0.150335\pi$
$140$	0	0
$141$	−0.951003	−0.0800889
$142$	0	0
$143$	−13.6005	−1.13733
$144$	0	0
$145$	−0.483372	−0.0401418
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.743622	−0.0609199	−0.0304599	−	0.999536i	$-0.509697\pi$
$149$	−0.743622	−0.0609199	−0.0304599	+	0.999536i	$0.509697\pi$
$150$	0	0
$151$	21.2384	1.72836	0.864178	−	0.503186i	$-0.167839\pi$
$151$	21.2384	1.72836	0.864178	+	0.503186i	$0.167839\pi$
$152$	0	0
$153$	0.0317794	0.00256921
$154$	0	0
$155$	0.393734	0.0316255
$156$	0	0
$157$	−16.9815	−1.35527	−0.677637	−	0.735396i	$-0.736996\pi$
$157$	−16.9815	−1.35527	−0.677637	+	0.735396i	$0.736996\pi$
$158$	0	0
$159$	−20.5405	−1.62897
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−24.9441	−1.95377	−0.976887	−	0.213755i	$-0.931430\pi$
$163$	−24.9441	−1.95377	−0.976887	+	0.213755i	$0.931430\pi$
$164$	0	0
$165$	0.644581	0.0501806
$166$	0	0
$167$	16.4889	1.27595	0.637973	−	0.770058i	$-0.279773\pi$
$167$	16.4889	1.27595	0.637973	+	0.770058i	$0.279773\pi$
$168$	0	0
$169$	12.5029	0.961763
$170$	0	0
$171$	−0.0415592	−0.00317811
$172$	0	0
$173$	4.58584	0.348655	0.174327	−	0.984688i	$-0.444225\pi$
$173$	4.58584	0.348655	0.174327	+	0.984688i	$0.444225\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.0483	1.28143
$178$	0	0
$179$	−22.6645	−1.69402	−0.847011	−	0.531576i	$-0.821600\pi$
$179$	−22.6645	−1.69402	−0.847011	+	0.531576i	$0.821600\pi$
$180$	0	0
$181$	−2.18126	−0.162132	−0.0810660	−	0.996709i	$-0.525832\pi$
$181$	−2.18126	−0.162132	−0.0810660	+	0.996709i	$0.525832\pi$
$182$	0	0
$183$	12.2811	0.907846
$184$	0	0
$185$	0.559950	0.0411683
$186$	0	0
$187$	−11.8018	−0.863037
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.1964	1.46136	0.730680	−	0.682720i	$-0.239203\pi$
$191$	20.1964	1.46136	0.730680	+	0.682720i	$0.239203\pi$
$192$	0	0
$193$	10.0768	0.725346	0.362673	−	0.931916i	$-0.381864\pi$
$193$	10.0768	0.725346	0.362673	+	0.931916i	$0.381864\pi$
$194$	0	0
$195$	−1.20868	−0.0865554
$196$	0	0
$197$	−2.87850	−0.205085	−0.102542	−	0.994729i	$-0.532698\pi$
$197$	−2.87850	−0.205085	−0.102542	+	0.994729i	$0.532698\pi$
$198$	0	0
$199$	19.9273	1.41261	0.706304	−	0.707909i	$-0.250361\pi$
$199$	19.9273	1.41261	0.706304	+	0.707909i	$0.250361\pi$
$200$	0	0
$201$	4.84988	0.342084
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.138351	0.00966283
$206$	0	0
$207$	−0.0587408	−0.00408277
$208$	0	0
$209$	15.4338	1.06758
$210$	0	0
$211$	17.2509	1.18760	0.593802	−	0.804611i	$-0.297626\pi$
$211$	17.2509	1.18760	0.593802	+	0.804611i	$0.297626\pi$
$212$	0	0
$213$	−25.1879	−1.72585
$214$	0	0
$215$	1.50074	0.102349
$216$	0	0
$217$	0	0
$218$	0	0
$219$	13.5139	0.913188
$220$	0	0
$221$	22.1301	1.48863
$222$	0	0
$223$	−6.73906	−0.451281	−0.225640	−	0.974211i	$-0.572447\pi$
$223$	−6.73906	−0.451281	−0.225640	+	0.974211i	$0.572447\pi$
$224$	0	0
$225$	0.0361211	0.00240807
$226$	0	0
$227$	−1.93020	−0.128112	−0.0640560	−	0.997946i	$-0.520404\pi$
$227$	−1.93020	−0.128112	−0.0640560	+	0.997946i	$0.520404\pi$
$228$	0	0
$229$	−6.41374	−0.423832	−0.211916	−	0.977288i	$-0.567970\pi$
$229$	−6.41374	−0.423832	−0.211916	+	0.977288i	$0.567970\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.12913	−0.336021	−0.168010	−	0.985785i	$-0.553734\pi$
$233$	−5.12913	−0.336021	−0.168010	+	0.985785i	$0.553734\pi$
$234$	0	0
$235$	−0.0760550	−0.00496128
$236$	0	0
$237$	18.2199	1.18351
$238$	0	0
$239$	3.36794	0.217854	0.108927	−	0.994050i	$-0.465259\pi$
$239$	3.36794	0.217854	0.108927	+	0.994050i	$0.465259\pi$
$240$	0	0
$241$	2.39150	0.154050	0.0770250	−	0.997029i	$-0.475458\pi$
$241$	2.39150	0.154050	0.0770250	+	0.997029i	$0.475458\pi$
$242$	0	0
$243$	−0.0753646	−0.00483464
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−28.9405	−1.84144
$248$	0	0
$249$	4.23602	0.268447
$250$	0	0
$251$	8.70091	0.549197	0.274598	−	0.961559i	$-0.411455\pi$
$251$	8.70091	0.549197	0.274598	+	0.961559i	$0.411455\pi$
$252$	0	0
$253$	21.8145	1.37146
$254$	0	0
$255$	−1.04883	−0.0656803
$256$	0	0
$257$	2.25116	0.140424	0.0702118	−	0.997532i	$-0.477632\pi$
$257$	2.25116	0.140424	0.0702118	+	0.997532i	$0.477632\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.0253371	−0.00156833
$262$	0	0
$263$	4.82702	0.297647	0.148823	−	0.988864i	$-0.452451\pi$
$263$	4.82702	0.297647	0.148823	+	0.988864i	$0.452451\pi$
$264$	0	0
$265$	−1.64270	−0.100910
$266$	0	0
$267$	−3.08852	−0.189014
$268$	0	0
$269$	−9.07027	−0.553024	−0.276512	−	0.961010i	$-0.589179\pi$
$269$	−9.07027	−0.553024	−0.276512	+	0.961010i	$0.589179\pi$
$270$	0	0
$271$	−8.67279	−0.526834	−0.263417	−	0.964682i	$-0.584850\pi$
$271$	−8.67279	−0.526834	−0.263417	+	0.964682i	$0.584850\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.4142	−0.808908
$276$	0	0
$277$	26.9015	1.61635	0.808177	−	0.588940i	$-0.200454\pi$
$277$	26.9015	1.61635	0.808177	+	0.588940i	$0.200454\pi$
$278$	0	0
$279$	0.0206385	0.00123560
$280$	0	0
$281$	−15.2508	−0.909788	−0.454894	−	0.890546i	$-0.650323\pi$
$281$	−15.2508	−0.909788	−0.454894	+	0.890546i	$0.650323\pi$
$282$	0	0
$283$	−23.1544	−1.37639	−0.688194	−	0.725526i	$-0.741596\pi$
$283$	−23.1544	−1.37639	−0.688194	+	0.725526i	$0.741596\pi$
$284$	0	0
$285$	1.37160	0.0812465
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.20338	0.129611
$290$	0	0
$291$	10.0798	0.590890
$292$	0	0
$293$	1.68125	0.0982198	0.0491099	−	0.998793i	$-0.484362\pi$
$293$	1.68125	0.0982198	0.0491099	+	0.998793i	$0.484362\pi$
$294$	0	0
$295$	1.36341	0.0793810
$296$	0	0
$297$	14.0109	0.812995
$298$	0	0
$299$	−40.9052	−2.36561
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−17.7869	−1.02183
$304$	0	0
$305$	0.982164	0.0562385
$306$	0	0
$307$	21.2887	1.21501	0.607506	−	0.794315i	$-0.292170\pi$
$307$	21.2887	1.21501	0.607506	+	0.794315i	$0.292170\pi$
$308$	0	0
$309$	−25.2314	−1.43537
$310$	0	0
$311$	−24.0159	−1.36181	−0.680907	−	0.732369i	$-0.738414\pi$
$311$	−24.0159	−1.36181	−0.680907	+	0.732369i	$0.738414\pi$
$312$	0	0
$313$	18.9342	1.07022	0.535111	−	0.844782i	$-0.320270\pi$
$313$	18.9342	1.07022	0.535111	+	0.844782i	$0.320270\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.5827	1.77386	0.886930	−	0.461904i	$-0.152833\pi$
$317$	31.5827	1.77386	0.886930	+	0.461904i	$0.152833\pi$
$318$	0	0
$319$	9.40938	0.526824
$320$	0	0
$321$	−22.5177	−1.25681
$322$	0	0
$323$	−25.1130	−1.39733
$324$	0	0
$325$	25.1535	1.39527
$326$	0	0
$327$	−7.56149	−0.418151
$328$	0	0
$329$	0	0
$330$	0	0
$331$	35.9104	1.97382	0.986908	−	0.161286i	$-0.0515641\pi$
$331$	35.9104	1.97382	0.986908	+	0.161286i	$0.0515641\pi$
$332$	0	0
$333$	0.0293511	0.00160843
$334$	0	0
$335$	0.387862	0.0211912
$336$	0	0
$337$	6.58274	0.358585	0.179292	−	0.983796i	$-0.442619\pi$
$337$	6.58274	0.358585	0.179292	+	0.983796i	$0.442619\pi$
$338$	0	0
$339$	−20.8476	−1.13229
$340$	0	0
$341$	−7.66448	−0.415055
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.93865	0.104373
$346$	0	0
$347$	12.0202	0.645280	0.322640	−	0.946522i	$-0.395430\pi$
$347$	12.0202	0.645280	0.322640	+	0.946522i	$0.395430\pi$
$348$	0	0
$349$	36.7866	1.96914	0.984570	−	0.174993i	$-0.0559903\pi$
$349$	36.7866	1.96914	0.984570	+	0.174993i	$0.0559903\pi$
$350$	0	0
$351$	−26.2724	−1.40232
$352$	0	0
$353$	−2.41501	−0.128538	−0.0642691	−	0.997933i	$-0.520472\pi$
$353$	−2.41501	−0.128538	−0.0642691	+	0.997933i	$0.520472\pi$
$354$	0	0
$355$	−2.01437	−0.106911
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.93153	0.207498	0.103749	−	0.994604i	$-0.466916\pi$
$359$	3.93153	0.207498	0.103749	+	0.994604i	$0.466916\pi$
$360$	0	0
$361$	13.8413	0.728492
$362$	0	0
$363$	6.48201	0.340217
$364$	0	0
$365$	1.08076	0.0565695
$366$	0	0
$367$	−32.4268	−1.69266	−0.846332	−	0.532656i	$-0.821194\pi$
$367$	−32.4268	−1.69266	−0.846332	+	0.532656i	$0.821194\pi$
$368$	0	0
$369$	0.00725198	0.000377523 0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.7318	0.555673	0.277836	−	0.960628i	$-0.410383\pi$
$373$	10.7318	0.555673	0.277836	+	0.960628i	$0.410383\pi$
$374$	0	0
$375$	−2.38882	−0.123358
$376$	0	0
$377$	−17.6439	−0.908708
$378$	0	0
$379$	17.6723	0.907764	0.453882	−	0.891062i	$-0.350039\pi$
$379$	17.6723	0.907764	0.453882	+	0.891062i	$0.350039\pi$
$380$	0	0
$381$	9.42701	0.482960
$382$	0	0
$383$	−20.9037	−1.06813	−0.534066	−	0.845443i	$-0.679337\pi$
$383$	−20.9037	−1.06813	−0.534066	+	0.845443i	$0.679337\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.0786648	0.00399875
$388$	0	0
$389$	34.3177	1.73998	0.869989	−	0.493072i	$-0.164126\pi$
$389$	34.3177	1.73998	0.869989	+	0.493072i	$0.164126\pi$
$390$	0	0
$391$	−35.4954	−1.79508
$392$	0	0
$393$	36.5895	1.84570
$394$	0	0
$395$	1.45711	0.0733150
$396$	0	0
$397$	−13.8093	−0.693067	−0.346534	−	0.938038i	$-0.612641\pi$
$397$	−13.8093	−0.693067	−0.346534	+	0.938038i	$0.612641\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.27497	0.213482	0.106741	−	0.994287i	$-0.465958\pi$
$401$	4.27497	0.213482	0.106741	+	0.994287i	$0.465958\pi$
$402$	0	0
$403$	14.3720	0.715920
$404$	0	0
$405$	1.24214	0.0617224
$406$	0	0
$407$	−10.9001	−0.540296
$408$	0	0
$409$	29.5945	1.46336	0.731678	−	0.681650i	$-0.238738\pi$
$409$	29.5945	1.46336	0.731678	+	0.681650i	$0.238738\pi$
$410$	0	0
$411$	19.7152	0.972480
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.338769	0.0166295
$416$	0	0
$417$	−36.3262	−1.77890
$418$	0	0
$419$	24.8741	1.21518	0.607589	−	0.794251i	$-0.292137\pi$
$419$	24.8741	1.21518	0.607589	+	0.794251i	$0.292137\pi$
$420$	0	0
$421$	7.30826	0.356183	0.178091	−	0.984014i	$-0.443008\pi$
$421$	7.30826	0.356183	0.178091	+	0.984014i	$0.443008\pi$
$422$	0	0
$423$	−0.00398661	−0.000193835 0
$424$	0	0
$425$	21.8270	1.05876
$426$	0	0
$427$	0	0
$428$	0	0
$429$	23.5283	1.13596
$430$	0	0
$431$	−21.9690	−1.05821	−0.529105	−	0.848556i	$-0.677472\pi$
$431$	−21.9690	−1.05821	−0.529105	+	0.848556i	$0.677472\pi$
$432$	0	0
$433$	9.89063	0.475313	0.237657	−	0.971349i	$-0.423621\pi$
$433$	9.89063	0.475313	0.237657	+	0.971349i	$0.423621\pi$
$434$	0	0
$435$	0.836212	0.0400933
$436$	0	0
$437$	46.4188	2.22051
$438$	0	0
$439$	−6.10305	−0.291283	−0.145641	−	0.989337i	$-0.546525\pi$
$439$	−6.10305	−0.291283	−0.145641	+	0.989337i	$0.546525\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.3435	1.34664	0.673321	−	0.739350i	$-0.264867\pi$
$443$	28.3435	1.34664	0.673321	+	0.739350i	$0.264867\pi$
$444$	0	0
$445$	−0.247000	−0.0117089
$446$	0	0
$447$	1.28643	0.0608462
$448$	0	0
$449$	22.3983	1.05704	0.528521	−	0.848920i	$-0.322747\pi$
$449$	22.3983	1.05704	0.528521	+	0.848920i	$0.322747\pi$
$450$	0	0
$451$	−2.69315	−0.126816
$452$	0	0
$453$	−36.7415	−1.72627
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.90893	0.323186	0.161593	−	0.986857i	$-0.448337\pi$
$457$	6.90893	0.323186	0.161593	+	0.986857i	$0.448337\pi$
$458$	0	0
$459$	−22.7978	−1.06411
$460$	0	0
$461$	16.8884	0.786573	0.393286	−	0.919416i	$-0.371338\pi$
$461$	16.8884	0.786573	0.393286	+	0.919416i	$0.371338\pi$
$462$	0	0
$463$	−28.3956	−1.31966	−0.659828	−	0.751417i	$-0.729371\pi$
$463$	−28.3956	−1.31966	−0.659828	+	0.751417i	$0.729371\pi$
$464$	0	0
$465$	−0.681143	−0.0315872
$466$	0	0
$467$	19.9414	0.922780	0.461390	−	0.887197i	$-0.347351\pi$
$467$	19.9414	0.922780	0.461390	+	0.887197i	$0.347351\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	29.3773	1.35364
$472$	0	0
$473$	−29.2136	−1.34324
$474$	0	0
$475$	−28.5440	−1.30969
$476$	0	0
$477$	−0.0861060	−0.00394252
$478$	0	0
$479$	34.5557	1.57889	0.789445	−	0.613822i	$-0.210369\pi$
$479$	34.5557	1.57889	0.789445	+	0.613822i	$0.210369\pi$
$480$	0	0
$481$	20.4392	0.931945
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.806119	0.0366040
$486$	0	0
$487$	13.9840	0.633674	0.316837	−	0.948480i	$-0.397379\pi$
$487$	13.9840	0.633674	0.316837	+	0.948480i	$0.397379\pi$
$488$	0	0
$489$	43.1522	1.95141
$490$	0	0
$491$	25.3725	1.14504	0.572522	−	0.819890i	$-0.305965\pi$
$491$	25.3725	1.14504	0.572522	+	0.819890i	$0.305965\pi$
$492$	0	0
$493$	−15.3105	−0.689549
$494$	0	0
$495$	0.00270209	0.000121450 0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.7146	−0.569185	−0.284593	−	0.958649i	$-0.591858\pi$
$499$	−12.7146	−0.569185	−0.284593	+	0.958649i	$0.591858\pi$
$500$	0	0
$501$	−28.5250	−1.27440
$502$	0	0
$503$	21.7738	0.970844	0.485422	−	0.874280i	$-0.338666\pi$
$503$	21.7738	0.970844	0.485422	+	0.874280i	$0.338666\pi$
$504$	0	0
$505$	−1.42248	−0.0632994
$506$	0	0
$507$	−21.6295	−0.960600
$508$	0	0
$509$	38.5413	1.70831	0.854156	−	0.520017i	$-0.174074\pi$
$509$	38.5413	1.70831	0.854156	+	0.520017i	$0.174074\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	29.8137	1.31631
$514$	0	0
$515$	−2.01785	−0.0889169
$516$	0	0
$517$	1.48050	0.0651122
$518$	0	0
$519$	−7.93330	−0.348233
$520$	0	0
$521$	9.28852	0.406938	0.203469	−	0.979081i	$-0.434778\pi$
$521$	9.28852	0.406938	0.203469	+	0.979081i	$0.434778\pi$
$522$	0	0
$523$	−7.16308	−0.313220	−0.156610	−	0.987661i	$-0.550057\pi$
$523$	−7.16308	−0.313220	−0.156610	+	0.987661i	$0.550057\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.4713	0.543257
$528$	0	0
$529$	42.6095	1.85259
$530$	0	0
$531$	0.0714666	0.00310138
$532$	0	0
$533$	5.05004	0.218742
$534$	0	0
$535$	−1.80082	−0.0778562
$536$	0	0
$537$	39.2085	1.69197
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.5380	0.797012	0.398506	−	0.917166i	$-0.369529\pi$
$541$	18.5380	0.797012	0.398506	+	0.917166i	$0.369529\pi$
$542$	0	0
$543$	3.77349	0.161936
$544$	0	0
$545$	−0.604719	−0.0259033
$546$	0	0
$547$	−24.6673	−1.05470	−0.527348	−	0.849649i	$-0.676814\pi$
$547$	−24.6673	−1.05470	−0.527348	+	0.849649i	$0.676814\pi$
$548$	0	0
$549$	0.0514825	0.00219722
$550$	0	0
$551$	20.0221	0.852972
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.968689	−0.0411186
$556$	0	0
$557$	−18.8686	−0.799488	−0.399744	−	0.916627i	$-0.630901\pi$
$557$	−18.8686	−0.799488	−0.399744	+	0.916627i	$0.630901\pi$
$558$	0	0
$559$	54.7796	2.31693
$560$	0	0
$561$	20.4167	0.861993
$562$	0	0
$563$	12.0948	0.509736	0.254868	−	0.966976i	$-0.417968\pi$
$563$	12.0948	0.509736	0.254868	+	0.966976i	$0.417968\pi$
$564$	0	0
$565$	−1.66725	−0.0701419
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.0776	1.59630	0.798149	−	0.602460i	$-0.205813\pi$
$569$	38.0776	1.59630	0.798149	+	0.602460i	$0.205813\pi$
$570$	0	0
$571$	35.3021	1.47735	0.738673	−	0.674064i	$-0.235453\pi$
$571$	35.3021	1.47735	0.738673	+	0.674064i	$0.235453\pi$
$572$	0	0
$573$	−34.9389	−1.45959
$574$	0	0
$575$	−40.3448	−1.68249
$576$	0	0
$577$	23.1430	0.963455	0.481728	−	0.876321i	$-0.340009\pi$
$577$	23.1430	0.963455	0.481728	+	0.876321i	$0.340009\pi$
$578$	0	0
$579$	−17.4325	−0.724469
$580$	0	0
$581$	0	0
$582$	0	0
$583$	31.9770	1.32435
$584$	0	0
$585$	−0.00506679	−0.000209486 0
$586$	0	0
$587$	8.02365	0.331172	0.165586	−	0.986195i	$-0.447049\pi$
$587$	8.02365	0.331172	0.165586	+	0.986195i	$0.447049\pi$
$588$	0	0
$589$	−16.3092	−0.672008
$590$	0	0
$591$	4.97968	0.204837
$592$	0	0
$593$	−23.8724	−0.980323	−0.490162	−	0.871632i	$-0.663062\pi$
$593$	−23.8724	−0.980323	−0.490162	+	0.871632i	$0.663062\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−34.4733	−1.41090
$598$	0	0
$599$	−36.0214	−1.47179	−0.735896	−	0.677094i	$-0.763239\pi$
$599$	−36.0214	−1.47179	−0.735896	+	0.677094i	$0.763239\pi$
$600$	0	0
$601$	−3.67996	−0.150109	−0.0750543	−	0.997179i	$-0.523913\pi$
$601$	−3.67996	−0.150109	−0.0750543	+	0.997179i	$0.523913\pi$
$602$	0	0
$603$	0.0203307	0.000827930 0
$604$	0	0
$605$	0.518389	0.0210755
$606$	0	0
$607$	30.8608	1.25260	0.626300	−	0.779582i	$-0.284569\pi$
$607$	30.8608	1.25260	0.626300	+	0.779582i	$0.284569\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.77614	−0.112311
$612$	0	0
$613$	6.98088	0.281955	0.140978	−	0.990013i	$-0.454975\pi$
$613$	6.98088	0.281955	0.140978	+	0.990013i	$0.454975\pi$
$614$	0	0
$615$	−0.239341	−0.00965114
$616$	0	0
$617$	29.9068	1.20400	0.602002	−	0.798495i	$-0.294370\pi$
$617$	29.9068	1.20400	0.602002	+	0.798495i	$0.294370\pi$
$618$	0	0
$619$	22.2912	0.895960	0.447980	−	0.894044i	$-0.352144\pi$
$619$	22.2912	0.895960	0.447980	+	0.894044i	$0.352144\pi$
$620$	0	0
$621$	42.1394	1.69100
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.7133	0.988530
$626$	0	0
$627$	−26.6997	−1.06628
$628$	0	0
$629$	17.7360	0.707183
$630$	0	0
$631$	−22.9844	−0.914994	−0.457497	−	0.889211i	$-0.651254\pi$
$631$	−22.9844	−0.914994	−0.457497	+	0.889211i	$0.651254\pi$
$632$	0	0
$633$	−29.8434	−1.18617
$634$	0	0
$635$	0.753911	0.0299180
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.105588	−0.00417699
$640$	0	0
$641$	−28.7896	−1.13712	−0.568560	−	0.822642i	$-0.692499\pi$
$641$	−28.7896	−1.13712	−0.568560	+	0.822642i	$0.692499\pi$
$642$	0	0
$643$	−22.1980	−0.875402	−0.437701	−	0.899121i	$-0.644207\pi$
$643$	−22.1980	−0.875402	−0.437701	+	0.899121i	$0.644207\pi$
$644$	0	0
$645$	−2.59621	−0.102226
$646$	0	0
$647$	−20.8413	−0.819356	−0.409678	−	0.912230i	$-0.634359\pi$
$647$	−20.8413	−0.819356	−0.409678	+	0.912230i	$0.634359\pi$
$648$	0	0
$649$	−26.5404	−1.04180
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−47.8337	−1.87188	−0.935938	−	0.352164i	$-0.885446\pi$
$653$	−47.8337	−1.87188	−0.935938	+	0.352164i	$0.885446\pi$
$654$	0	0
$655$	2.92619	0.114336
$656$	0	0
$657$	0.0566505	0.00221015
$658$	0	0
$659$	−3.61599	−0.140859	−0.0704296	−	0.997517i	$-0.522437\pi$
$659$	−3.61599	−0.140859	−0.0704296	+	0.997517i	$0.522437\pi$
$660$	0	0
$661$	44.0109	1.71183	0.855913	−	0.517119i	$-0.172996\pi$
$661$	44.0109	1.71183	0.855913	+	0.517119i	$0.172996\pi$
$662$	0	0
$663$	−38.2841	−1.48683
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.2998	1.09577
$668$	0	0
$669$	11.6583	0.450735
$670$	0	0
$671$	−19.1189	−0.738079
$672$	0	0
$673$	−5.20547	−0.200656	−0.100328	−	0.994954i	$-0.531989\pi$
$673$	−5.20547	−0.200656	−0.100328	+	0.994954i	$0.531989\pi$
$674$	0	0
$675$	−25.9125	−0.997372
$676$	0	0
$677$	17.8560	0.686261	0.343131	−	0.939288i	$-0.388513\pi$
$677$	17.8560	0.686261	0.343131	+	0.939288i	$0.388513\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.33917	0.127957
$682$	0	0
$683$	26.6454	1.01956	0.509779	−	0.860305i	$-0.329727\pi$
$683$	26.6454	1.01956	0.509779	+	0.860305i	$0.329727\pi$
$684$	0	0
$685$	1.57669	0.0602424
$686$	0	0
$687$	11.0955	0.423319
$688$	0	0
$689$	−59.9614	−2.28435
$690$	0	0
$691$	34.2765	1.30394	0.651969	−	0.758245i	$-0.273943\pi$
$691$	34.2765	1.30394	0.651969	+	0.758245i	$0.273943\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.90513	−0.110198
$696$	0	0
$697$	4.38217	0.165986
$698$	0	0
$699$	8.87317	0.335614
$700$	0	0
$701$	−33.0947	−1.24997	−0.624986	−	0.780636i	$-0.714895\pi$
$701$	−33.0947	−1.24997	−0.624986	+	0.780636i	$0.714895\pi$
$702$	0	0
$703$	−23.1941	−0.874784
$704$	0	0
$705$	0.131572	0.00495528
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−19.8549	−0.745666	−0.372833	−	0.927898i	$-0.621614\pi$
$709$	−19.8549	−0.745666	−0.372833	+	0.927898i	$0.621614\pi$
$710$	0	0
$711$	0.0763778	0.00286439
$712$	0	0
$713$	−23.0518	−0.863297
$714$	0	0
$715$	1.88164	0.0703695
$716$	0	0
$717$	−5.82639	−0.217590
$718$	0	0
$719$	42.4164	1.58187	0.790933	−	0.611903i	$-0.209596\pi$
$719$	42.4164	1.58187	0.790933	+	0.611903i	$0.209596\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.13719	−0.153864
$724$	0	0
$725$	−17.4022	−0.646302
$726$	0	0
$727$	−26.0078	−0.964576	−0.482288	−	0.876013i	$-0.660194\pi$
$727$	−26.0078	−0.964576	−0.482288	+	0.876013i	$0.660194\pi$
$728$	0	0
$729$	27.0650	1.00241
$730$	0	0
$731$	47.5349	1.75814
$732$	0	0
$733$	−28.9065	−1.06769	−0.533843	−	0.845583i	$-0.679253\pi$
$733$	−28.9065	−1.06769	−0.533843	+	0.845583i	$0.679253\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.55017	−0.278114
$738$	0	0
$739$	−3.80880	−0.140109	−0.0700546	−	0.997543i	$-0.522317\pi$
$739$	−3.80880	−0.140109	−0.0700546	+	0.997543i	$0.522317\pi$
$740$	0	0
$741$	50.0657	1.83921
$742$	0	0
$743$	−11.2071	−0.411148	−0.205574	−	0.978642i	$-0.565906\pi$
$743$	−11.2071	−0.411148	−0.205574	+	0.978642i	$0.565906\pi$
$744$	0	0
$745$	0.102881	0.00376925
$746$	0	0
$747$	0.0177574	0.000649709 0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0.718156	0.0262059	0.0131029	−	0.999914i	$-0.495829\pi$
$751$	0.718156	0.0262059	0.0131029	+	0.999914i	$0.495829\pi$
$752$	0	0
$753$	−15.0522	−0.548533
$754$	0	0
$755$	−2.93835	−0.106937
$756$	0	0
$757$	−14.6357	−0.531942	−0.265971	−	0.963981i	$-0.585693\pi$
$757$	−14.6357	−0.531942	−0.265971	+	0.963981i	$0.585693\pi$
$758$	0	0
$759$	−37.7381	−1.36980
$760$	0	0
$761$	8.40732	0.304765	0.152383	−	0.988322i	$-0.451305\pi$
$761$	8.40732	0.304765	0.152383	+	0.988322i	$0.451305\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.00439670	−0.000158963 0
$766$	0	0
$767$	49.7670	1.79698
$768$	0	0
$769$	−6.66726	−0.240428	−0.120214	−	0.992748i	$-0.538358\pi$
$769$	−6.66726	−0.240428	−0.120214	+	0.992748i	$0.538358\pi$
$770$	0	0
$771$	−3.89441	−0.140254
$772$	0	0
$773$	19.7134	0.709040	0.354520	−	0.935048i	$-0.384644\pi$
$773$	19.7134	0.709040	0.354520	+	0.935048i	$0.384644\pi$
$774$	0	0
$775$	14.1751	0.509185
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.73074	−0.205325
$780$	0	0
$781$	39.2119	1.40311
$782$	0	0
$783$	18.1763	0.649567
$784$	0	0
$785$	2.34941	0.0838540
$786$	0	0
$787$	16.2381	0.578825	0.289413	−	0.957204i	$-0.406540\pi$
$787$	16.2381	0.578825	0.289413	+	0.957204i	$0.406540\pi$
$788$	0	0
$789$	−8.35053	−0.297287
$790$	0	0
$791$	0	0
$792$	0	0
$793$	35.8507	1.27310
$794$	0	0
$795$	2.84180	0.100788
$796$	0	0
$797$	−15.2774	−0.541152	−0.270576	−	0.962699i	$-0.587214\pi$
$797$	−15.2774	−0.541152	−0.270576	+	0.962699i	$0.587214\pi$
$798$	0	0
$799$	−2.40899	−0.0852240
$800$	0	0
$801$	−0.0129471	−0.000457463 0
$802$	0	0
$803$	−21.0382	−0.742421
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.6912	0.552356
$808$	0	0
$809$	−3.25997	−0.114614	−0.0573072	−	0.998357i	$-0.518251\pi$
$809$	−3.25997	−0.114614	−0.0573072	+	0.998357i	$0.518251\pi$
$810$	0	0
$811$	−7.78233	−0.273275	−0.136637	−	0.990621i	$-0.543629\pi$
$811$	−7.78233	−0.273275	−0.136637	+	0.990621i	$0.543629\pi$
$812$	0	0
$813$	15.0035	0.526197
$814$	0	0
$815$	3.45104	0.120885
$816$	0	0
$817$	−62.1633	−2.17482
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.51951	0.157732	0.0788660	−	0.996885i	$-0.474870\pi$
$821$	4.51951	0.157732	0.0788660	+	0.996885i	$0.474870\pi$
$822$	0	0
$823$	−3.68068	−0.128301	−0.0641503	−	0.997940i	$-0.520434\pi$
$823$	−3.68068	−0.128301	−0.0641503	+	0.997940i	$0.520434\pi$
$824$	0	0
$825$	23.2060	0.807930
$826$	0	0
$827$	−9.84848	−0.342465	−0.171233	−	0.985231i	$-0.554775\pi$
$827$	−9.84848	−0.342465	−0.171233	+	0.985231i	$0.554775\pi$
$828$	0	0
$829$	8.84388	0.307161	0.153580	−	0.988136i	$-0.450920\pi$
$829$	8.84388	0.307161	0.153580	+	0.988136i	$0.450920\pi$
$830$	0	0
$831$	−46.5384	−1.61440
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.28125	−0.0789458
$836$	0	0
$837$	−14.8056	−0.511757
$838$	0	0
$839$	−3.82105	−0.131917	−0.0659587	−	0.997822i	$-0.521011\pi$
$839$	−3.82105	−0.131917	−0.0659587	+	0.997822i	$0.521011\pi$
$840$	0	0
$841$	−16.7933	−0.579078
$842$	0	0
$843$	26.3833	0.908688
$844$	0	0
$845$	−1.72979	−0.0595065
$846$	0	0
$847$	0	0
$848$	0	0
$849$	40.0562	1.37472
$850$	0	0
$851$	−32.7832	−1.12379
$852$	0	0
$853$	47.3338	1.62068	0.810340	−	0.585960i	$-0.199283\pi$
$853$	47.3338	1.62068	0.810340	+	0.585960i	$0.199283\pi$
$854$	0	0
$855$	0.00574974	0.000196637 0
$856$	0	0
$857$	−15.6208	−0.533597	−0.266798	−	0.963752i	$-0.585966\pi$
$857$	−15.6208	−0.533597	−0.266798	+	0.963752i	$0.585966\pi$
$858$	0	0
$859$	49.2078	1.67895	0.839474	−	0.543400i	$-0.182863\pi$
$859$	49.2078	1.67895	0.839474	+	0.543400i	$0.182863\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34.6153	−1.17832	−0.589160	−	0.808017i	$-0.700541\pi$
$863$	−34.6153	−1.17832	−0.589160	+	0.808017i	$0.700541\pi$
$864$	0	0
$865$	−0.634454	−0.0215721
$866$	0	0
$867$	−3.81176	−0.129454
$868$	0	0
$869$	−28.3643	−0.962192
$870$	0	0
$871$	14.1576	0.479713
$872$	0	0
$873$	0.0422547	0.00143010
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.7040	−1.30694	−0.653471	−	0.756952i	$-0.726688\pi$
$877$	−38.7040	−1.30694	−0.653471	+	0.756952i	$0.726688\pi$
$878$	0	0
$879$	−2.90849	−0.0981010
$880$	0	0
$881$	40.7943	1.37440	0.687198	−	0.726470i	$-0.258841\pi$
$881$	40.7943	1.37440	0.687198	+	0.726470i	$0.258841\pi$
$882$	0	0
$883$	−20.4213	−0.687232	−0.343616	−	0.939110i	$-0.611652\pi$
$883$	−20.4213	−0.687232	−0.343616	+	0.939110i	$0.611652\pi$
$884$	0	0
$885$	−2.35864	−0.0792850
$886$	0	0
$887$	−14.9970	−0.503552	−0.251776	−	0.967786i	$-0.581015\pi$
$887$	−14.9970	−0.503552	−0.251776	+	0.967786i	$0.581015\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−24.1796	−0.810049
$892$	0	0
$893$	3.15034	0.105422
$894$	0	0
$895$	3.13564	0.104813
$896$	0	0
$897$	70.7642	2.36275
$898$	0	0
$899$	−9.94310	−0.331621
$900$	0	0
$901$	−52.0314	−1.73342
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.301779	0.0100315
$906$	0	0
$907$	−24.7950	−0.823305	−0.411652	−	0.911341i	$-0.635048\pi$
$907$	−24.7950	−0.823305	−0.411652	+	0.911341i	$0.635048\pi$
$908$	0	0
$909$	−0.0745626	−0.00247308
$910$	0	0
$911$	−18.8479	−0.624458	−0.312229	−	0.950007i	$-0.601076\pi$
$911$	−18.8479	−0.624458	−0.312229	+	0.950007i	$0.601076\pi$
$912$	0	0
$913$	−6.59453	−0.218247
$914$	0	0
$915$	−1.69910	−0.0561705
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.65808	−0.186643	−0.0933215	−	0.995636i	$-0.529748\pi$
$919$	−5.65808	−0.186643	−0.0933215	+	0.995636i	$0.529748\pi$
$920$	0	0
$921$	−36.8286	−1.21354
$922$	0	0
$923$	−73.5279	−2.42020
$924$	0	0
$925$	20.1592	0.662829
$926$	0	0
$927$	−0.105770	−0.00347395
$928$	0	0
$929$	−50.7518	−1.66511	−0.832556	−	0.553941i	$-0.813123\pi$
$929$	−50.7518	−1.66511	−0.832556	+	0.553941i	$0.813123\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	41.5464	1.36017
$934$	0	0
$935$	1.63279	0.0533981
$936$	0	0
$937$	−25.6871	−0.839162	−0.419581	−	0.907718i	$-0.637823\pi$
$937$	−25.6871	−0.839162	−0.419581	+	0.907718i	$0.637823\pi$
$938$	0	0
$939$	−32.7553	−1.06893
$940$	0	0
$941$	−11.9819	−0.390599	−0.195299	−	0.980744i	$-0.562568\pi$
$941$	−11.9819	−0.390599	−0.195299	+	0.980744i	$0.562568\pi$
$942$	0	0
$943$	−8.09997	−0.263771
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.5029	1.02371	0.511854	−	0.859073i	$-0.328959\pi$
$947$	31.5029	1.02371	0.511854	+	0.859073i	$0.328959\pi$
$948$	0	0
$949$	39.4496	1.28059
$950$	0	0
$951$	−54.6367	−1.77171
$952$	0	0
$953$	30.0320	0.972833	0.486416	−	0.873727i	$-0.338304\pi$
$953$	30.0320	0.972833	0.486416	+	0.873727i	$0.338304\pi$
$954$	0	0
$955$	−2.79419	−0.0904177
$956$	0	0
$957$	−16.2778	−0.526187
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−22.9008	−0.738735
$962$	0	0
$963$	−0.0943942	−0.00304181
$964$	0	0
$965$	−1.39414	−0.0448788
$966$	0	0
$967$	−20.7206	−0.666328	−0.333164	−	0.942869i	$-0.608116\pi$
$967$	−20.7206	−0.666328	−0.333164	+	0.942869i	$0.608116\pi$
$968$	0	0
$969$	43.4445	1.39564
$970$	0	0
$971$	47.6630	1.52958	0.764789	−	0.644281i	$-0.222843\pi$
$971$	47.6630	1.52958	0.764789	+	0.644281i	$0.222843\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−43.5145	−1.39358
$976$	0	0
$977$	47.2389	1.51131	0.755653	−	0.654973i	$-0.227320\pi$
$977$	47.2389	1.51131	0.755653	+	0.654973i	$0.227320\pi$
$978$	0	0
$979$	4.80813	0.153669
$980$	0	0
$981$	−0.0316978	−0.00101203
$982$	0	0
$983$	44.7066	1.42592	0.712960	−	0.701204i	$-0.247354\pi$
$983$	44.7066	1.42592	0.712960	+	0.701204i	$0.247354\pi$
$984$	0	0
$985$	0.398243	0.0126891
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−87.8632	−2.79389
$990$	0	0
$991$	−3.17283	−0.100788	−0.0503941	−	0.998729i	$-0.516048\pi$
$991$	−3.17283	−0.100788	−0.0503941	+	0.998729i	$0.516048\pi$
$992$	0	0
$993$	−62.1235	−1.97143
$994$	0	0
$995$	−2.75695	−0.0874013
$996$	0	0
$997$	−1.92328	−0.0609110	−0.0304555	−	0.999536i	$-0.509696\pi$
$997$	−1.92328	−0.0609110	−0.0304555	+	0.999536i	$0.509696\pi$
$998$	0	0
$999$	−21.0559	−0.666178

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.q.1.5		15
7.2	even	3		1148.2.i.e.165.11	✓	30
7.4	even	3		1148.2.i.e.821.11	yes	30
7.6	odd	2		8036.2.a.r.1.11		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.11	✓	30	7.2	even	3
1148.2.i.e.821.11	yes	30	7.4	even	3
8036.2.a.q.1.5		15	1.1	even	1	trivial
8036.2.a.r.1.11		15	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.72996 q^{3} -0.138351 q^{5} -0.00725198 q^{9} +O(q^{10})\)	\(q-1.72996 q^{3} -0.138351 q^{5} -0.00725198 q^{9} +2.69315 q^{11} -5.05004 q^{13} +0.239341 q^{15} -4.38217 q^{17} +5.73074 q^{19} +8.09997 q^{23} -4.98086 q^{25} +5.20241 q^{27} +3.49382 q^{29} -2.84591 q^{31} -4.65904 q^{33} -4.04732 q^{37} +8.73635 q^{39} -1.00000 q^{41} -10.8473 q^{43} +0.00100332 q^{45} +0.549726 q^{47} +7.58096 q^{51} +11.8734 q^{53} -0.372600 q^{55} -9.91392 q^{57} -9.85476 q^{59} -7.09909 q^{61} +0.698677 q^{65} -2.80347 q^{67} -14.0126 q^{69} +14.5599 q^{71} -7.81173 q^{73} +8.61667 q^{75} -10.5320 q^{79} -8.97819 q^{81} -2.44863 q^{83} +0.606276 q^{85} -6.04415 q^{87} +1.78532 q^{89} +4.92331 q^{93} -0.792851 q^{95} -5.82664 q^{97} -0.0195307 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q - q^3 + 3 * q^5 + 30 * q^9	\( 15 q - q^{3} + 3 q^{5} + 30 q^{9} + 9 q^{11} + 7 q^{13} + 2 q^{15} + 3 q^{17} + 7 q^{19} - q^{23} + 32 q^{25} + 11 q^{27} + 18 q^{29} + 30 q^{31} - 16 q^{33} + 23 q^{37} + 5 q^{39} - 15 q^{41} + 12 q^{43} - 13 q^{45} - 16 q^{47} + 29 q^{51} + 33 q^{53} + 37 q^{55} + 16 q^{57} - 10 q^{59} + q^{61} + 16 q^{65} + 20 q^{67} + 21 q^{69} + 5 q^{71} - 3 q^{73} - 51 q^{75} + 25 q^{79} + 43 q^{81} + 18 q^{83} + 36 q^{85} - 53 q^{87} - 11 q^{89} + 65 q^{93} - 30 q^{95} + 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q - q^3 + 3 * q^5 + 30 * q^9 + 9 * q^11 + 7 * q^13 + 2 * q^15 + 3 * q^17 + 7 * q^19 - q^23 + 32 * q^25 + 11 * q^27 + 18 * q^29 + 30 * q^31 - 16 * q^33 + 23 * q^37 + 5 * q^39 - 15 * q^41 + 12 * q^43 - 13 * q^45 - 16 * q^47 + 29 * q^51 + 33 * q^53 + 37 * q^55 + 16 * q^57 - 10 * q^59 + q^61 + 16 * q^65 + 20 * q^67 + 21 * q^69 + 5 * q^71 - 3 * q^73 - 51 * q^75 + 25 * q^79 + 43 * q^81 + 18 * q^83 + 36 * q^85 - 53 * q^87 - 11 * q^89 + 65 * q^93 - 30 * q^95 + 16 * q^97 - 18 * q^99

Introduction

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