LMFDB - Embedded newform 2744.2.a.h.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2744,2,Mod(1,2744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2744.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2744 = 2^{3} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2744.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$21.9109503146$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 29x^{10} + 304x^{8} - 1393x^{6} + 2574x^{4} - 1164x^{2} + 8 $ x^12 - 29x^10 + 304x^8 - 1393x^6 + 2574x^4 - 1164*x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.85982$ of defining polynomial
Character	$\chi$	$=$	2744.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85982 q^{3} +1.80700 q^{5} +0.458937 q^{9} +5.99266 q^{11} +4.33684 q^{13} -3.36070 q^{15} -7.43508 q^{17} +2.28797 q^{19} +3.07396 q^{23} -1.73475 q^{25} +4.72592 q^{27} +5.91158 q^{29} +3.61062 q^{31} -11.1453 q^{33} +8.67866 q^{37} -8.06575 q^{39} -11.6608 q^{41} -4.17484 q^{43} +0.829299 q^{45} +2.97955 q^{47} +13.8279 q^{51} -8.17484 q^{53} +10.8287 q^{55} -4.25522 q^{57} -13.5673 q^{59} +1.99594 q^{61} +7.83666 q^{65} +7.69990 q^{67} -5.71701 q^{69} +12.7339 q^{71} -6.60300 q^{73} +3.22633 q^{75} -0.873289 q^{79} -10.1662 q^{81} +10.4473 q^{83} -13.4352 q^{85} -10.9945 q^{87} +1.70795 q^{89} -6.71512 q^{93} +4.13437 q^{95} -7.23220 q^{97} +2.75025 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 22 q^{9} + 18 q^{11} + 28 q^{15} + 20 q^{23} + 14 q^{25} - 18 q^{29} - 18 q^{37} + 36 q^{39} + 10 q^{43} + 48 q^{51} - 38 q^{53} + 12 q^{57} + 8 q^{65} + 42 q^{67} + 56 q^{71} + 56 q^{79} + 52 q^{81}+ \cdots + 90 q^{99}+O(q^{100}) $ 12 * q + 22 * q^9 + 18 * q^11 + 28 * q^15 + 20 * q^23 + 14 * q^25 - 18 * q^29 - 18 * q^37 + 36 * q^39 + 10 * q^43 + 48 * q^51 - 38 * q^53 + 12 * q^57 + 8 * q^65 + 42 * q^67 + 56 * q^71 + 56 * q^79 + 52 * q^81 + 8 * q^85 - 48 * q^93 + 84 * q^95 + 90 * q^99

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
$3$	−1.85982	−1.07377	−0.536884	−	0.843656i	$-0.680399\pi$
$3$	−1.85982	−1.07377	−0.536884	+	0.843656i	$0.680399\pi$
$4$	0	0
$5$	1.80700	0.808114	0.404057	−	0.914734i	$-0.367600\pi$
$5$	1.80700	0.808114	0.404057	+	0.914734i	$0.367600\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.458937	0.152979
$10$	0	0
$11$	5.99266	1.80685	0.903427	−	0.428742i	$-0.141043\pi$
$11$	5.99266	1.80685	0.903427	+	0.428742i	$0.141043\pi$
$12$	0	0
$13$	4.33684	1.20282	0.601411	−	0.798939i	$-0.294605\pi$
$13$	4.33684	1.20282	0.601411	+	0.798939i	$0.294605\pi$
$14$	0	0
$15$	−3.36070	−0.867728
$16$	0	0
$17$	−7.43508	−1.80327	−0.901636	−	0.432496i	$-0.857633\pi$
$17$	−7.43508	−1.80327	−0.901636	+	0.432496i	$0.857633\pi$
$18$	0	0
$19$	2.28797	0.524897	0.262449	−	0.964946i	$-0.415470\pi$
$19$	2.28797	0.524897	0.262449	+	0.964946i	$0.415470\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.07396	0.640964	0.320482	−	0.947255i	$-0.396155\pi$
$23$	3.07396	0.640964	0.320482	+	0.947255i	$0.396155\pi$
$24$	0	0
$25$	−1.73475	−0.346951
$26$	0	0
$27$	4.72592	0.909505
$28$	0	0
$29$	5.91158	1.09775	0.548876	−	0.835904i	$-0.315056\pi$
$29$	5.91158	1.09775	0.548876	+	0.835904i	$0.315056\pi$
$30$	0	0
$31$	3.61062	0.648487	0.324244	−	0.945974i	$-0.394890\pi$
$31$	3.61062	0.648487	0.324244	+	0.945974i	$0.394890\pi$
$32$	0	0
$33$	−11.1453	−1.94014
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.67866	1.42676	0.713381	−	0.700776i	$-0.247163\pi$
$37$	8.67866	1.42676	0.713381	+	0.700776i	$0.247163\pi$
$38$	0	0
$39$	−8.06575	−1.29155
$40$	0	0
$41$	−11.6608	−1.82111	−0.910556	−	0.413385i	$-0.864346\pi$
$41$	−11.6608	−1.82111	−0.910556	+	0.413385i	$0.864346\pi$
$42$	0	0
$43$	−4.17484	−0.636657	−0.318328	−	0.947981i	$-0.603121\pi$
$43$	−4.17484	−0.636657	−0.318328	+	0.947981i	$0.603121\pi$
$44$	0	0
$45$	0.829299	0.123625
$46$	0	0
$47$	2.97955	0.434612	0.217306	−	0.976104i	$-0.430273\pi$
$47$	2.97955	0.434612	0.217306	+	0.976104i	$0.430273\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	13.8279	1.93630
$52$	0	0
$53$	−8.17484	−1.12290	−0.561450	−	0.827511i	$-0.689756\pi$
$53$	−8.17484	−1.12290	−0.561450	+	0.827511i	$0.689756\pi$
$54$	0	0
$55$	10.8287	1.46014
$56$	0	0
$57$	−4.25522	−0.563618
$58$	0	0
$59$	−13.5673	−1.76631	−0.883156	−	0.469080i	$-0.844586\pi$
$59$	−13.5673	−1.76631	−0.883156	+	0.469080i	$0.844586\pi$
$60$	0	0
$61$	1.99594	0.255554	0.127777	−	0.991803i	$-0.459216\pi$
$61$	1.99594	0.255554	0.127777	+	0.991803i	$0.459216\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.83666	0.972019
$66$	0	0
$67$	7.69990	0.940692	0.470346	−	0.882482i	$-0.344129\pi$
$67$	7.69990	0.940692	0.470346	+	0.882482i	$0.344129\pi$
$68$	0	0
$69$	−5.71701	−0.688247
$70$	0	0
$71$	12.7339	1.51123	0.755617	−	0.655013i	$-0.227337\pi$
$71$	12.7339	1.51123	0.755617	+	0.655013i	$0.227337\pi$
$72$	0	0
$73$	−6.60300	−0.772822	−0.386411	−	0.922327i	$-0.626285\pi$
$73$	−6.60300	−0.772822	−0.386411	+	0.922327i	$0.626285\pi$
$74$	0	0
$75$	3.22633	0.372545
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.873289	−0.0982527	−0.0491263	−	0.998793i	$-0.515644\pi$
$79$	−0.873289	−0.0982527	−0.0491263	+	0.998793i	$0.515644\pi$
$80$	0	0
$81$	−10.1662	−1.12958
$82$	0	0
$83$	10.4473	1.14674	0.573371	−	0.819296i	$-0.305635\pi$
$83$	10.4473	1.14674	0.573371	+	0.819296i	$0.305635\pi$
$84$	0	0
$85$	−13.4352	−1.45725
$86$	0	0
$87$	−10.9945	−1.17873
$88$	0	0
$89$	1.70795	0.181042	0.0905211	−	0.995895i	$-0.471147\pi$
$89$	1.70795	0.181042	0.0905211	+	0.995895i	$0.471147\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.71512	−0.696325
$94$	0	0
$95$	4.13437	0.424177
$96$	0	0
$97$	−7.23220	−0.734319	−0.367159	−	0.930158i	$-0.619670\pi$
$97$	−7.23220	−0.734319	−0.367159	+	0.930158i	$0.619670\pi$
$98$	0	0
$99$	2.75025	0.276411
$100$	0	0
$101$	9.32343	0.927716	0.463858	−	0.885910i	$-0.346465\pi$
$101$	9.32343	0.927716	0.463858	+	0.885910i	$0.346465\pi$
$102$	0	0
$103$	−8.89097	−0.876053	−0.438027	−	0.898962i	$-0.644322\pi$
$103$	−8.89097	−0.876053	−0.438027	+	0.898962i	$0.644322\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.6375	1.12503	0.562517	−	0.826785i	$-0.309833\pi$
$107$	11.6375	1.12503	0.562517	+	0.826785i	$0.309833\pi$
$108$	0	0
$109$	9.06830	0.868586	0.434293	−	0.900772i	$-0.356998\pi$
$109$	9.06830	0.868586	0.434293	+	0.900772i	$0.356998\pi$
$110$	0	0
$111$	−16.1408	−1.53201
$112$	0	0
$113$	5.29983	0.498566	0.249283	−	0.968431i	$-0.419805\pi$
$113$	5.29983	0.498566	0.249283	+	0.968431i	$0.419805\pi$
$114$	0	0
$115$	5.55463	0.517972
$116$	0	0
$117$	1.99034	0.184007
$118$	0	0
$119$	0	0
$120$	0	0
$121$	24.9119	2.26472
$122$	0	0
$123$	21.6870	1.95545
$124$	0	0
$125$	−12.1697	−1.08849
$126$	0	0
$127$	−3.54943	−0.314961	−0.157480	−	0.987522i	$-0.550337\pi$
$127$	−3.54943	−0.314961	−0.157480	+	0.987522i	$0.550337\pi$
$128$	0	0
$129$	7.76445	0.683622
$130$	0	0
$131$	11.0014	0.961192	0.480596	−	0.876942i	$-0.340420\pi$
$131$	11.0014	0.961192	0.480596	+	0.876942i	$0.340420\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.53974	0.734984
$136$	0	0
$137$	−2.94330	−0.251463	−0.125731	−	0.992064i	$-0.540128\pi$
$137$	−2.94330	−0.251463	−0.125731	+	0.992064i	$0.540128\pi$
$138$	0	0
$139$	−6.99908	−0.593654	−0.296827	−	0.954931i	$-0.595928\pi$
$139$	−6.99908	−0.593654	−0.296827	+	0.954931i	$0.595928\pi$
$140$	0	0
$141$	−5.54143	−0.466673
$142$	0	0
$143$	25.9892	2.17333
$144$	0	0
$145$	10.6822	0.887110
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.9583	0.897738	0.448869	−	0.893598i	$-0.351827\pi$
$149$	10.9583	0.897738	0.448869	+	0.893598i	$0.351827\pi$
$150$	0	0
$151$	12.8927	1.04920	0.524598	−	0.851350i	$-0.324216\pi$
$151$	12.8927	1.04920	0.524598	+	0.851350i	$0.324216\pi$
$152$	0	0
$153$	−3.41223	−0.275863
$154$	0	0
$155$	6.52440	0.524052
$156$	0	0
$157$	8.38484	0.669183	0.334592	−	0.942363i	$-0.391402\pi$
$157$	8.38484	0.669183	0.334592	+	0.942363i	$0.391402\pi$
$158$	0	0
$159$	15.2037	1.20574
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.77875	−0.530953	−0.265477	−	0.964117i	$-0.585529\pi$
$163$	−6.77875	−0.530953	−0.265477	+	0.964117i	$0.585529\pi$
$164$	0	0
$165$	−20.1395	−1.56786
$166$	0	0
$167$	15.0494	1.16456	0.582279	−	0.812989i	$-0.302161\pi$
$167$	15.0494	1.16456	0.582279	+	0.812989i	$0.302161\pi$
$168$	0	0
$169$	5.80818	0.446783
$170$	0	0
$171$	1.05004	0.0802982
$172$	0	0
$173$	−6.85693	−0.521323	−0.260661	−	0.965430i	$-0.583941\pi$
$173$	−6.85693	−0.521323	−0.260661	+	0.965430i	$0.583941\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	25.2328	1.89661
$178$	0	0
$179$	2.99752	0.224045	0.112023	−	0.993706i	$-0.464267\pi$
$179$	2.99752	0.224045	0.112023	+	0.993706i	$0.464267\pi$
$180$	0	0
$181$	20.4597	1.52076	0.760380	−	0.649478i	$-0.225013\pi$
$181$	20.4597	1.52076	0.760380	+	0.649478i	$0.225013\pi$
$182$	0	0
$183$	−3.71210	−0.274406
$184$	0	0
$185$	15.6823	1.15299
$186$	0	0
$187$	−44.5559	−3.25825
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.1479	1.24078	0.620391	−	0.784293i	$-0.286974\pi$
$191$	17.1479	1.24078	0.620391	+	0.784293i	$0.286974\pi$
$192$	0	0
$193$	−3.36178	−0.241986	−0.120993	−	0.992653i	$-0.538608\pi$
$193$	−3.36178	−0.241986	−0.120993	+	0.992653i	$0.538608\pi$
$194$	0	0
$195$	−14.5748	−1.04372
$196$	0	0
$197$	2.14378	0.152738	0.0763690	−	0.997080i	$-0.475667\pi$
$197$	2.14378	0.152738	0.0763690	+	0.997080i	$0.475667\pi$
$198$	0	0
$199$	24.3084	1.72317	0.861587	−	0.507609i	$-0.169471\pi$
$199$	24.3084	1.72317	0.861587	+	0.507609i	$0.169471\pi$
$200$	0	0
$201$	−14.3204	−1.01009
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−21.0711	−1.47167
$206$	0	0
$207$	1.41075	0.0980540
$208$	0	0
$209$	13.7110	0.948413
$210$	0	0
$211$	4.51994	0.311165	0.155583	−	0.987823i	$-0.450275\pi$
$211$	4.51994	0.311165	0.155583	+	0.987823i	$0.450275\pi$
$212$	0	0
$213$	−23.6828	−1.62272
$214$	0	0
$215$	−7.54393	−0.514492
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.2804	0.829833
$220$	0	0
$221$	−32.2448	−2.16902
$222$	0	0
$223$	−2.21440	−0.148287	−0.0741437	−	0.997248i	$-0.523622\pi$
$223$	−2.21440	−0.148287	−0.0741437	+	0.997248i	$0.523622\pi$
$224$	0	0
$225$	−0.796143	−0.0530762
$226$	0	0
$227$	−22.1957	−1.47318	−0.736589	−	0.676341i	$-0.763565\pi$
$227$	−22.1957	−1.47318	−0.736589	+	0.676341i	$0.763565\pi$
$228$	0	0
$229$	14.5988	0.964716	0.482358	−	0.875974i	$-0.339780\pi$
$229$	14.5988	0.964716	0.482358	+	0.875974i	$0.339780\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.6948	−1.68332	−0.841660	−	0.540008i	$-0.818421\pi$
$233$	−25.6948	−1.68332	−0.841660	+	0.540008i	$0.818421\pi$
$234$	0	0
$235$	5.38404	0.351216
$236$	0	0
$237$	1.62416	0.105501
$238$	0	0
$239$	−16.0732	−1.03969	−0.519844	−	0.854261i	$-0.674010\pi$
$239$	−16.0732	−1.03969	−0.519844	+	0.854261i	$0.674010\pi$
$240$	0	0
$241$	−2.09340	−0.134848	−0.0674239	−	0.997724i	$-0.521478\pi$
$241$	−2.09340	−0.134848	−0.0674239	+	0.997724i	$0.521478\pi$
$242$	0	0
$243$	4.72952	0.303399
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.92257	0.631358
$248$	0	0
$249$	−19.4301	−1.23134
$250$	0	0
$251$	4.64927	0.293459	0.146730	−	0.989177i	$-0.453125\pi$
$251$	4.64927	0.293459	0.146730	+	0.989177i	$0.453125\pi$
$252$	0	0
$253$	18.4212	1.15813
$254$	0	0
$255$	24.9870	1.56475
$256$	0	0
$257$	27.9240	1.74185	0.870926	−	0.491414i	$-0.163520\pi$
$257$	27.9240	1.74185	0.870926	+	0.491414i	$0.163520\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.71304	0.167933
$262$	0	0
$263$	−3.43195	−0.211623	−0.105812	−	0.994386i	$-0.533744\pi$
$263$	−3.43195	−0.211623	−0.105812	+	0.994386i	$0.533744\pi$
$264$	0	0
$265$	−14.7719	−0.907432
$266$	0	0
$267$	−3.17648	−0.194397
$268$	0	0
$269$	−5.79142	−0.353109	−0.176555	−	0.984291i	$-0.556495\pi$
$269$	−5.79142	−0.353109	−0.176555	+	0.984291i	$0.556495\pi$
$270$	0	0
$271$	5.58288	0.339136	0.169568	−	0.985518i	$-0.445763\pi$
$271$	5.58288	0.339136	0.169568	+	0.985518i	$0.445763\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.3958	−0.626890
$276$	0	0
$277$	2.16895	0.130320	0.0651598	−	0.997875i	$-0.479244\pi$
$277$	2.16895	0.130320	0.0651598	+	0.997875i	$0.479244\pi$
$278$	0	0
$279$	1.65705	0.0992050
$280$	0	0
$281$	−8.58262	−0.511996	−0.255998	−	0.966677i	$-0.582404\pi$
$281$	−8.58262	−0.511996	−0.255998	+	0.966677i	$0.582404\pi$
$282$	0	0
$283$	0.259317	0.0154148	0.00770741	−	0.999970i	$-0.497547\pi$
$283$	0.259317	0.0154148	0.00770741	+	0.999970i	$0.497547\pi$
$284$	0	0
$285$	−7.68918	−0.455468
$286$	0	0
$287$	0	0
$288$	0	0
$289$	38.2804	2.25179
$290$	0	0
$291$	13.4506	0.788488
$292$	0	0
$293$	29.5313	1.72524	0.862620	−	0.505853i	$-0.168822\pi$
$293$	29.5313	1.72524	0.862620	+	0.505853i	$0.168822\pi$
$294$	0	0
$295$	−24.5161	−1.42738
$296$	0	0
$297$	28.3208	1.64334
$298$	0	0
$299$	13.3313	0.770966
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−17.3399	−0.996152
$304$	0	0
$305$	3.60667	0.206517
$306$	0	0
$307$	21.9179	1.25092	0.625459	−	0.780257i	$-0.284912\pi$
$307$	21.9179	1.25092	0.625459	+	0.780257i	$0.284912\pi$
$308$	0	0
$309$	16.5356	0.940679
$310$	0	0
$311$	−10.8680	−0.616267	−0.308134	−	0.951343i	$-0.599704\pi$
$311$	−10.8680	−0.616267	−0.308134	+	0.951343i	$0.599704\pi$
$312$	0	0
$313$	−8.22083	−0.464669	−0.232334	−	0.972636i	$-0.574636\pi$
$313$	−8.22083	−0.464669	−0.232334	+	0.972636i	$0.574636\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.3888	−1.14515	−0.572574	−	0.819853i	$-0.694055\pi$
$317$	−20.3888	−1.14515	−0.572574	+	0.819853i	$0.694055\pi$
$318$	0	0
$319$	35.4261	1.98348
$320$	0	0
$321$	−21.6436	−1.20803
$322$	0	0
$323$	−17.0113	−0.946532
$324$	0	0
$325$	−7.52335	−0.417321
$326$	0	0
$327$	−16.8654	−0.932660
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−23.0167	−1.26511	−0.632556	−	0.774514i	$-0.717994\pi$
$331$	−23.0167	−1.26511	−0.632556	+	0.774514i	$0.717994\pi$
$332$	0	0
$333$	3.98296	0.218265
$334$	0	0
$335$	13.9137	0.760187
$336$	0	0
$337$	15.5469	0.846892	0.423446	−	0.905921i	$-0.360820\pi$
$337$	15.5469	0.846892	0.423446	+	0.905921i	$0.360820\pi$
$338$	0	0
$339$	−9.85674	−0.535344
$340$	0	0
$341$	21.6372	1.17172
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−10.3306	−0.556182
$346$	0	0
$347$	−31.0992	−1.66949	−0.834746	−	0.550635i	$-0.814385\pi$
$347$	−31.0992	−1.66949	−0.834746	+	0.550635i	$0.814385\pi$
$348$	0	0
$349$	27.4632	1.47007	0.735035	−	0.678029i	$-0.237166\pi$
$349$	27.4632	1.47007	0.735035	+	0.678029i	$0.237166\pi$
$350$	0	0
$351$	20.4956	1.09397
$352$	0	0
$353$	9.02473	0.480338	0.240169	−	0.970731i	$-0.422797\pi$
$353$	9.02473	0.480338	0.240169	+	0.970731i	$0.422797\pi$
$354$	0	0
$355$	23.0101	1.22125
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.04097	−0.266052	−0.133026	−	0.991113i	$-0.542469\pi$
$359$	−5.04097	−0.266052	−0.133026	+	0.991113i	$0.542469\pi$
$360$	0	0
$361$	−13.7652	−0.724483
$362$	0	0
$363$	−46.3318	−2.43179
$364$	0	0
$365$	−11.9316	−0.624529
$366$	0	0
$367$	−9.21277	−0.480902	−0.240451	−	0.970661i	$-0.577295\pi$
$367$	−9.21277	−0.480902	−0.240451	+	0.970661i	$0.577295\pi$
$368$	0	0
$369$	−5.35158	−0.278592
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4.84388	−0.250806	−0.125403	−	0.992106i	$-0.540022\pi$
$373$	−4.84388	−0.250806	−0.125403	+	0.992106i	$0.540022\pi$
$374$	0	0
$375$	22.6335	1.16879
$376$	0	0
$377$	25.6376	1.32040
$378$	0	0
$379$	15.2149	0.781537	0.390769	−	0.920489i	$-0.372209\pi$
$379$	15.2149	0.781537	0.390769	+	0.920489i	$0.372209\pi$
$380$	0	0
$381$	6.60130	0.338195
$382$	0	0
$383$	−2.81716	−0.143950	−0.0719752	−	0.997406i	$-0.522930\pi$
$383$	−2.81716	−0.143950	−0.0719752	+	0.997406i	$0.522930\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.91599	−0.0973951
$388$	0	0
$389$	24.9136	1.26317	0.631584	−	0.775308i	$-0.282405\pi$
$389$	24.9136	1.26317	0.631584	+	0.775308i	$0.282405\pi$
$390$	0	0
$391$	−22.8551	−1.15583
$392$	0	0
$393$	−20.4606	−1.03210
$394$	0	0
$395$	−1.57803	−0.0793994
$396$	0	0
$397$	−35.1079	−1.76202	−0.881008	−	0.473101i	$-0.843135\pi$
$397$	−35.1079	−1.76202	−0.881008	+	0.473101i	$0.843135\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.7806	0.937858	0.468929	−	0.883236i	$-0.344640\pi$
$401$	18.7806	0.937858	0.468929	+	0.883236i	$0.344640\pi$
$402$	0	0
$403$	15.6587	0.780015
$404$	0	0
$405$	−18.3703	−0.912827
$406$	0	0
$407$	52.0082	2.57795
$408$	0	0
$409$	−24.0529	−1.18934	−0.594671	−	0.803969i	$-0.702718\pi$
$409$	−24.0529	−1.18934	−0.594671	+	0.803969i	$0.702718\pi$
$410$	0	0
$411$	5.47401	0.270013
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.8783	0.926699
$416$	0	0
$417$	13.0170	0.637447
$418$	0	0
$419$	9.26529	0.452639	0.226320	−	0.974053i	$-0.427331\pi$
$419$	9.26529	0.452639	0.226320	+	0.974053i	$0.427331\pi$
$420$	0	0
$421$	−1.53694	−0.0749059	−0.0374529	−	0.999298i	$-0.511924\pi$
$421$	−1.53694	−0.0749059	−0.0374529	+	0.999298i	$0.511924\pi$
$422$	0	0
$423$	1.36743	0.0664865
$424$	0	0
$425$	12.8980	0.625647
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−48.3353	−2.33365
$430$	0	0
$431$	24.8089	1.19500	0.597501	−	0.801868i	$-0.296160\pi$
$431$	24.8089	1.19500	0.597501	+	0.801868i	$0.296160\pi$
$432$	0	0
$433$	−3.13130	−0.150481	−0.0752404	−	0.997165i	$-0.523972\pi$
$433$	−3.13130	−0.150481	−0.0752404	+	0.997165i	$0.523972\pi$
$434$	0	0
$435$	−19.8670	−0.952551
$436$	0	0
$437$	7.03313	0.336440
$438$	0	0
$439$	−16.1630	−0.771417	−0.385709	−	0.922621i	$-0.626043\pi$
$439$	−16.1630	−0.771417	−0.385709	+	0.922621i	$0.626043\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.72367	0.0818940	0.0409470	−	0.999161i	$-0.486963\pi$
$443$	1.72367	0.0818940	0.0409470	+	0.999161i	$0.486963\pi$
$444$	0	0
$445$	3.08626	0.146303
$446$	0	0
$447$	−20.3805	−0.963963
$448$	0	0
$449$	−40.1162	−1.89320	−0.946600	−	0.322409i	$-0.895507\pi$
$449$	−40.1162	−1.89320	−0.946600	+	0.322409i	$0.895507\pi$
$450$	0	0
$451$	−69.8792	−3.29048
$452$	0	0
$453$	−23.9782	−1.12659
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.0053	−1.21648	−0.608238	−	0.793754i	$-0.708124\pi$
$457$	−26.0053	−1.21648	−0.608238	+	0.793754i	$0.708124\pi$
$458$	0	0
$459$	−35.1376	−1.64008
$460$	0	0
$461$	15.1701	0.706544	0.353272	−	0.935521i	$-0.385069\pi$
$461$	15.1701	0.706544	0.353272	+	0.935521i	$0.385069\pi$
$462$	0	0
$463$	−10.2124	−0.474611	−0.237305	−	0.971435i	$-0.576264\pi$
$463$	−10.2124	−0.474611	−0.237305	+	0.971435i	$0.576264\pi$
$464$	0	0
$465$	−12.1342	−0.562711
$466$	0	0
$467$	−25.0606	−1.15967	−0.579834	−	0.814734i	$-0.696883\pi$
$467$	−25.0606	−1.15967	−0.579834	+	0.814734i	$0.696883\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15.5943	−0.718548
$472$	0	0
$473$	−25.0184	−1.15035
$474$	0	0
$475$	−3.96907	−0.182114
$476$	0	0
$477$	−3.75174	−0.171780
$478$	0	0
$479$	7.02203	0.320845	0.160422	−	0.987048i	$-0.448714\pi$
$479$	7.02203	0.320845	0.160422	+	0.987048i	$0.448714\pi$
$480$	0	0
$481$	37.6380	1.71614
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.0686	−0.593414
$486$	0	0
$487$	14.1674	0.641985	0.320992	−	0.947082i	$-0.395984\pi$
$487$	14.1674	0.641985	0.320992	+	0.947082i	$0.395984\pi$
$488$	0	0
$489$	12.6073	0.570121
$490$	0	0
$491$	1.41302	0.0637689	0.0318845	−	0.999492i	$-0.489849\pi$
$491$	1.41302	0.0637689	0.0318845	+	0.999492i	$0.489849\pi$
$492$	0	0
$493$	−43.9531	−1.97955
$494$	0	0
$495$	4.96970	0.223372
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−27.1799	−1.21674	−0.608369	−	0.793654i	$-0.708176\pi$
$499$	−27.1799	−1.21674	−0.608369	+	0.793654i	$0.708176\pi$
$500$	0	0
$501$	−27.9892	−1.25047
$502$	0	0
$503$	−35.6790	−1.59085	−0.795425	−	0.606053i	$-0.792752\pi$
$503$	−35.6790	−1.59085	−0.795425	+	0.606053i	$0.792752\pi$
$504$	0	0
$505$	16.8474	0.749701
$506$	0	0
$507$	−10.8022	−0.479741
$508$	0	0
$509$	−32.3391	−1.43341	−0.716703	−	0.697378i	$-0.754350\pi$
$509$	−32.3391	−1.43341	−0.716703	+	0.697378i	$0.754350\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	10.8128	0.477396
$514$	0	0
$515$	−16.0660	−0.707951
$516$	0	0
$517$	17.8554	0.785280
$518$	0	0
$519$	12.7527	0.559780
$520$	0	0
$521$	−34.1031	−1.49408	−0.747042	−	0.664777i	$-0.768527\pi$
$521$	−34.1031	−1.49408	−0.747042	+	0.664777i	$0.768527\pi$
$522$	0	0
$523$	16.9588	0.741555	0.370777	−	0.928722i	$-0.379091\pi$
$523$	16.9588	0.741555	0.370777	+	0.928722i	$0.379091\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.8453	−1.16940
$528$	0	0
$529$	−13.5508	−0.589165
$530$	0	0
$531$	−6.22653	−0.270209
$532$	0	0
$533$	−50.5711	−2.19048
$534$	0	0
$535$	21.0289	0.909157
$536$	0	0
$537$	−5.57486	−0.240573
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.8210	−0.465233	−0.232616	−	0.972569i	$-0.574729\pi$
$541$	−10.8210	−0.465233	−0.232616	+	0.972569i	$0.574729\pi$
$542$	0	0
$543$	−38.0515	−1.63294
$544$	0	0
$545$	16.3864	0.701917
$546$	0	0
$547$	24.0339	1.02761	0.513807	−	0.857906i	$-0.328235\pi$
$547$	24.0339	1.02761	0.513807	+	0.857906i	$0.328235\pi$
$548$	0	0
$549$	0.916013	0.0390945
$550$	0	0
$551$	13.5255	0.576207
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−29.1663	−1.23804
$556$	0	0
$557$	−41.5717	−1.76145	−0.880725	−	0.473628i	$-0.842944\pi$
$557$	−41.5717	−1.76145	−0.880725	+	0.473628i	$0.842944\pi$
$558$	0	0
$559$	−18.1056	−0.765785
$560$	0	0
$561$	82.8660	3.49861
$562$	0	0
$563$	−2.36405	−0.0996329	−0.0498165	−	0.998758i	$-0.515864\pi$
$563$	−2.36405	−0.0996329	−0.0498165	+	0.998758i	$0.515864\pi$
$564$	0	0
$565$	9.57678	0.402898
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.86653	0.162093	0.0810467	−	0.996710i	$-0.474174\pi$
$569$	3.86653	0.162093	0.0810467	+	0.996710i	$0.474174\pi$
$570$	0	0
$571$	22.8952	0.958133	0.479066	−	0.877779i	$-0.340975\pi$
$571$	22.8952	0.958133	0.479066	+	0.877779i	$0.340975\pi$
$572$	0	0
$573$	−31.8921	−1.33231
$574$	0	0
$575$	−5.33256	−0.222383
$576$	0	0
$577$	4.70723	0.195965	0.0979823	−	0.995188i	$-0.468761\pi$
$577$	4.70723	0.195965	0.0979823	+	0.995188i	$0.468761\pi$
$578$	0	0
$579$	6.25231	0.259837
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−48.9890	−2.02892
$584$	0	0
$585$	3.59654	0.148698
$586$	0	0
$587$	−41.8909	−1.72902	−0.864512	−	0.502613i	$-0.832372\pi$
$587$	−41.8909	−1.72902	−0.864512	+	0.502613i	$0.832372\pi$
$588$	0	0
$589$	8.26101	0.340389
$590$	0	0
$591$	−3.98705	−0.164005
$592$	0	0
$593$	10.2053	0.419082	0.209541	−	0.977800i	$-0.432803\pi$
$593$	10.2053	0.419082	0.209541	+	0.977800i	$0.432803\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−45.2092	−1.85029
$598$	0	0
$599$	32.4856	1.32732	0.663662	−	0.748033i	$-0.269001\pi$
$599$	32.4856	1.32732	0.663662	+	0.748033i	$0.269001\pi$
$600$	0	0
$601$	10.0289	0.409086	0.204543	−	0.978858i	$-0.434429\pi$
$601$	10.0289	0.409086	0.204543	+	0.978858i	$0.434429\pi$
$602$	0	0
$603$	3.53377	0.143906
$604$	0	0
$605$	45.0158	1.83015
$606$	0	0
$607$	27.6186	1.12100	0.560502	−	0.828153i	$-0.310608\pi$
$607$	27.6186	1.12100	0.560502	+	0.828153i	$0.310608\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.9218	0.522761
$612$	0	0
$613$	30.1649	1.21835	0.609174	−	0.793036i	$-0.291501\pi$
$613$	30.1649	1.21835	0.609174	+	0.793036i	$0.291501\pi$
$614$	0	0
$615$	39.1884	1.58023
$616$	0	0
$617$	−29.7251	−1.19669	−0.598343	−	0.801240i	$-0.704174\pi$
$617$	−29.7251	−1.19669	−0.598343	+	0.801240i	$0.704174\pi$
$618$	0	0
$619$	38.4722	1.54633	0.773164	−	0.634206i	$-0.218673\pi$
$619$	38.4722	1.54633	0.773164	+	0.634206i	$0.218673\pi$
$620$	0	0
$621$	14.5273	0.582960
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.3169	−0.532674
$626$	0	0
$627$	−25.5001	−1.01838
$628$	0	0
$629$	−64.5265	−2.57284
$630$	0	0
$631$	−35.3108	−1.40570	−0.702851	−	0.711337i	$-0.748090\pi$
$631$	−35.3108	−1.40570	−0.702851	+	0.711337i	$0.748090\pi$
$632$	0	0
$633$	−8.40628	−0.334119
$634$	0	0
$635$	−6.41381	−0.254524
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.84405	0.231187
$640$	0	0
$641$	−5.36875	−0.212053	−0.106026	−	0.994363i	$-0.533813\pi$
$641$	−5.36875	−0.212053	−0.106026	+	0.994363i	$0.533813\pi$
$642$	0	0
$643$	36.6161	1.44400	0.721999	−	0.691894i	$-0.243224\pi$
$643$	36.6161	1.44400	0.721999	+	0.691894i	$0.243224\pi$
$644$	0	0
$645$	14.0304	0.552445
$646$	0	0
$647$	4.38291	0.172310	0.0861551	−	0.996282i	$-0.472542\pi$
$647$	4.38291	0.172310	0.0861551	+	0.996282i	$0.472542\pi$
$648$	0	0
$649$	−81.3041	−3.19147
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.4512	−1.19165	−0.595824	−	0.803115i	$-0.703175\pi$
$653$	−30.4512	−1.19165	−0.595824	+	0.803115i	$0.703175\pi$
$654$	0	0
$655$	19.8794	0.776754
$656$	0	0
$657$	−3.03036	−0.118226
$658$	0	0
$659$	−25.6971	−1.00102	−0.500509	−	0.865731i	$-0.666854\pi$
$659$	−25.6971	−1.00102	−0.500509	+	0.865731i	$0.666854\pi$
$660$	0	0
$661$	22.4626	0.873695	0.436848	−	0.899536i	$-0.356095\pi$
$661$	22.4626	0.873695	0.436848	+	0.899536i	$0.356095\pi$
$662$	0	0
$663$	59.9695	2.32902
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.1719	0.703620
$668$	0	0
$669$	4.11839	0.159226
$670$	0	0
$671$	11.9610	0.461750
$672$	0	0
$673$	−50.3231	−1.93981	−0.969906	−	0.243481i	$-0.921711\pi$
$673$	−50.3231	−1.93981	−0.969906	+	0.243481i	$0.921711\pi$
$674$	0	0
$675$	−8.19832	−0.315553
$676$	0	0
$677$	4.17199	0.160343	0.0801714	−	0.996781i	$-0.474453\pi$
$677$	4.17199	0.160343	0.0801714	+	0.996781i	$0.474453\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	41.2800	1.58185
$682$	0	0
$683$	−13.8619	−0.530412	−0.265206	−	0.964192i	$-0.585440\pi$
$683$	−13.8619	−0.530412	−0.265206	+	0.964192i	$0.585440\pi$
$684$	0	0
$685$	−5.31853	−0.203211
$686$	0	0
$687$	−27.1512	−1.03588
$688$	0	0
$689$	−35.4530	−1.35065
$690$	0	0
$691$	9.04806	0.344205	0.172102	−	0.985079i	$-0.444944\pi$
$691$	9.04806	0.344205	0.172102	+	0.985079i	$0.444944\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.6473	−0.479740
$696$	0	0
$697$	86.6991	3.28396
$698$	0	0
$699$	47.7877	1.80750
$700$	0	0
$701$	21.7922	0.823080	0.411540	−	0.911392i	$-0.364991\pi$
$701$	21.7922	0.823080	0.411540	+	0.911392i	$0.364991\pi$
$702$	0	0
$703$	19.8565	0.748904
$704$	0	0
$705$	−10.0134	−0.377125
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−45.0178	−1.69068	−0.845340	−	0.534229i	$-0.820602\pi$
$709$	−45.0178	−1.69068	−0.845340	+	0.534229i	$0.820602\pi$
$710$	0	0
$711$	−0.400785	−0.0150306
$712$	0	0
$713$	11.0989	0.415657
$714$	0	0
$715$	46.9624	1.75630
$716$	0	0
$717$	29.8933	1.11638
$718$	0	0
$719$	−50.0423	−1.86626	−0.933132	−	0.359535i	$-0.882935\pi$
$719$	−50.0423	−1.86626	−0.933132	+	0.359535i	$0.882935\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.89335	0.144795
$724$	0	0
$725$	−10.2551	−0.380866
$726$	0	0
$727$	21.8461	0.810229	0.405114	−	0.914266i	$-0.367232\pi$
$727$	21.8461	0.810229	0.405114	+	0.914266i	$0.367232\pi$
$728$	0	0
$729$	21.7025	0.803796
$730$	0	0
$731$	31.0403	1.14807
$732$	0	0
$733$	43.9598	1.62369	0.811847	−	0.583871i	$-0.198463\pi$
$733$	43.9598	1.62369	0.811847	+	0.583871i	$0.198463\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.1428	1.69969
$738$	0	0
$739$	−8.13544	−0.299267	−0.149634	−	0.988742i	$-0.547809\pi$
$739$	−8.13544	−0.299267	−0.149634	+	0.988742i	$0.547809\pi$
$740$	0	0
$741$	−18.4542	−0.677933
$742$	0	0
$743$	13.5187	0.495953	0.247976	−	0.968766i	$-0.420234\pi$
$743$	13.5187	0.495953	0.247976	+	0.968766i	$0.420234\pi$
$744$	0	0
$745$	19.8016	0.725475
$746$	0	0
$747$	4.79466	0.175427
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.0653	−0.732194	−0.366097	−	0.930577i	$-0.619306\pi$
$751$	−20.0653	−0.732194	−0.366097	+	0.930577i	$0.619306\pi$
$752$	0	0
$753$	−8.64682	−0.315107
$754$	0	0
$755$	23.2971	0.847870
$756$	0	0
$757$	50.2545	1.82653	0.913265	−	0.407367i	$-0.133553\pi$
$757$	50.2545	1.82653	0.913265	+	0.407367i	$0.133553\pi$
$758$	0	0
$759$	−34.2601	−1.24356
$760$	0	0
$761$	6.35552	0.230387	0.115194	−	0.993343i	$-0.463251\pi$
$761$	6.35552	0.230387	0.115194	+	0.993343i	$0.463251\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.16590	−0.222929
$766$	0	0
$767$	−58.8392	−2.12456
$768$	0	0
$769$	8.69774	0.313649	0.156824	−	0.987627i	$-0.449874\pi$
$769$	8.69774	0.313649	0.156824	+	0.987627i	$0.449874\pi$
$770$	0	0
$771$	−51.9337	−1.87035
$772$	0	0
$773$	−6.44577	−0.231838	−0.115919	−	0.993259i	$-0.536981\pi$
$773$	−6.44577	−0.231838	−0.115919	+	0.993259i	$0.536981\pi$
$774$	0	0
$775$	−6.26355	−0.224993
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26.6796	−0.955897
$780$	0	0
$781$	76.3098	2.73058
$782$	0	0
$783$	27.9377	0.998411
$784$	0	0
$785$	15.1514	0.540777
$786$	0	0
$787$	7.45869	0.265874	0.132937	−	0.991125i	$-0.457559\pi$
$787$	7.45869	0.265874	0.132937	+	0.991125i	$0.457559\pi$
$788$	0	0
$789$	6.38282	0.227234
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8.65609	0.307387
$794$	0	0
$795$	27.4731	0.974372
$796$	0	0
$797$	4.38199	0.155218	0.0776090	−	0.996984i	$-0.475271\pi$
$797$	4.38199	0.155218	0.0776090	+	0.996984i	$0.475271\pi$
$798$	0	0
$799$	−22.1532	−0.783724
$800$	0	0
$801$	0.783841	0.0276956
$802$	0	0
$803$	−39.5695	−1.39638
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.7710	0.379158
$808$	0	0
$809$	18.5226	0.651221	0.325610	−	0.945504i	$-0.394430\pi$
$809$	18.5226	0.651221	0.325610	+	0.945504i	$0.394430\pi$
$810$	0	0
$811$	11.8480	0.416040	0.208020	−	0.978125i	$-0.433298\pi$
$811$	11.8480	0.416040	0.208020	+	0.978125i	$0.433298\pi$
$812$	0	0
$813$	−10.3832	−0.364154
$814$	0	0
$815$	−12.2492	−0.429071
$816$	0	0
$817$	−9.55192	−0.334179
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.0377	−1.78123	−0.890615	−	0.454759i	$-0.849726\pi$
$821$	−51.0377	−1.78123	−0.890615	+	0.454759i	$0.849726\pi$
$822$	0	0
$823$	−26.4789	−0.922997	−0.461498	−	0.887141i	$-0.652688\pi$
$823$	−26.4789	−0.922997	−0.461498	+	0.887141i	$0.652688\pi$
$824$	0	0
$825$	19.3343	0.673135
$826$	0	0
$827$	1.94521	0.0676417	0.0338209	−	0.999428i	$-0.489232\pi$
$827$	1.94521	0.0676417	0.0338209	+	0.999428i	$0.489232\pi$
$828$	0	0
$829$	−45.0435	−1.56443	−0.782213	−	0.623011i	$-0.785909\pi$
$829$	−45.0435	−1.56443	−0.782213	+	0.623011i	$0.785909\pi$
$830$	0	0
$831$	−4.03386	−0.139933
$832$	0	0
$833$	0	0
$834$	0	0
$835$	27.1943	0.941096
$836$	0	0
$837$	17.0635	0.589802
$838$	0	0
$839$	−20.5499	−0.709461	−0.354731	−	0.934969i	$-0.615427\pi$
$839$	−20.5499	−0.709461	−0.354731	+	0.934969i	$0.615427\pi$
$840$	0	0
$841$	5.94677	0.205061
$842$	0	0
$843$	15.9621	0.549765
$844$	0	0
$845$	10.4954	0.361052
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−0.482284	−0.0165520
$850$	0	0
$851$	26.6778	0.914504
$852$	0	0
$853$	−4.22587	−0.144691	−0.0723455	−	0.997380i	$-0.523048\pi$
$853$	−4.22587	−0.144691	−0.0723455	+	0.997380i	$0.523048\pi$
$854$	0	0
$855$	1.89741	0.0648902
$856$	0	0
$857$	−16.2188	−0.554024	−0.277012	−	0.960867i	$-0.589344\pi$
$857$	−16.2188	−0.554024	−0.277012	+	0.960867i	$0.589344\pi$
$858$	0	0
$859$	−15.8641	−0.541277	−0.270639	−	0.962681i	$-0.587235\pi$
$859$	−15.8641	−0.541277	−0.270639	+	0.962681i	$0.587235\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.6467	−1.28151	−0.640754	−	0.767747i	$-0.721378\pi$
$863$	−37.6467	−1.28151	−0.640754	+	0.767747i	$0.721378\pi$
$864$	0	0
$865$	−12.3905	−0.421288
$866$	0	0
$867$	−71.1948	−2.41790
$868$	0	0
$869$	−5.23332	−0.177528
$870$	0	0
$871$	33.3932	1.13149
$872$	0	0
$873$	−3.31913	−0.112335
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.4564	−1.26481	−0.632406	−	0.774637i	$-0.717933\pi$
$877$	−37.4564	−1.26481	−0.632406	+	0.774637i	$0.717933\pi$
$878$	0	0
$879$	−54.9230	−1.85251
$880$	0	0
$881$	−11.0901	−0.373634	−0.186817	−	0.982395i	$-0.559817\pi$
$881$	−11.0901	−0.373634	−0.186817	+	0.982395i	$0.559817\pi$
$882$	0	0
$883$	−30.4606	−1.02508	−0.512540	−	0.858663i	$-0.671295\pi$
$883$	−30.4606	−1.02508	−0.512540	+	0.858663i	$0.671295\pi$
$884$	0	0
$885$	45.5956	1.53268
$886$	0	0
$887$	−33.6377	−1.12944	−0.564721	−	0.825282i	$-0.691016\pi$
$887$	−33.6377	−1.12944	−0.564721	+	0.825282i	$0.691016\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−60.9225	−2.04098
$892$	0	0
$893$	6.81713	0.228127
$894$	0	0
$895$	5.41652	0.181054
$896$	0	0
$897$	−24.7938	−0.827839
$898$	0	0
$899$	21.3445	0.711879
$900$	0	0
$901$	60.7806	2.02489
$902$	0	0
$903$	0	0
$904$	0	0
$905$	36.9707	1.22895
$906$	0	0
$907$	25.0860	0.832967	0.416484	−	0.909143i	$-0.363262\pi$
$907$	25.0860	0.832967	0.416484	+	0.909143i	$0.363262\pi$
$908$	0	0
$909$	4.27887	0.141921
$910$	0	0
$911$	59.4501	1.96967	0.984834	−	0.173498i	$-0.0555069\pi$
$911$	59.4501	1.96967	0.984834	+	0.173498i	$0.0555069\pi$
$912$	0	0
$913$	62.6072	2.07200
$914$	0	0
$915$	−6.70776	−0.221752
$916$	0	0
$917$	0	0
$918$	0	0
$919$	53.6904	1.77108	0.885541	−	0.464562i	$-0.153788\pi$
$919$	53.6904	1.77108	0.885541	+	0.464562i	$0.153788\pi$
$920$	0	0
$921$	−40.7633	−1.34320
$922$	0	0
$923$	55.2248	1.81775
$924$	0	0
$925$	−15.0553	−0.495017
$926$	0	0
$927$	−4.08040	−0.134018
$928$	0	0
$929$	−21.7460	−0.713462	−0.356731	−	0.934207i	$-0.616109\pi$
$929$	−21.7460	−0.713462	−0.356731	+	0.934207i	$0.616109\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	20.2125	0.661728
$934$	0	0
$935$	−80.5124	−2.63304
$936$	0	0
$937$	−54.1910	−1.77034	−0.885171	−	0.465266i	$-0.845959\pi$
$937$	−54.1910	−1.77034	−0.885171	+	0.465266i	$0.845959\pi$
$938$	0	0
$939$	15.2893	0.498947
$940$	0	0
$941$	52.4405	1.70951	0.854755	−	0.519031i	$-0.173707\pi$
$941$	52.4405	1.70951	0.854755	+	0.519031i	$0.173707\pi$
$942$	0	0
$943$	−35.8448	−1.16727
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.74171	−0.0565979	−0.0282990	−	0.999600i	$-0.509009\pi$
$947$	−1.74171	−0.0565979	−0.0282990	+	0.999600i	$0.509009\pi$
$948$	0	0
$949$	−28.6361	−0.929569
$950$	0	0
$951$	37.9195	1.22962
$952$	0	0
$953$	21.6701	0.701965	0.350982	−	0.936382i	$-0.385848\pi$
$953$	21.6701	0.701965	0.350982	+	0.936382i	$0.385848\pi$
$954$	0	0
$955$	30.9863	1.00269
$956$	0	0
$957$	−65.8862	−2.12980
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−17.9634	−0.579464
$962$	0	0
$963$	5.34086	0.172107
$964$	0	0
$965$	−6.07473	−0.195552
$966$	0	0
$967$	−0.774832	−0.0249169	−0.0124585	−	0.999922i	$-0.503966\pi$
$967$	−0.774832	−0.0249169	−0.0124585	+	0.999922i	$0.503966\pi$
$968$	0	0
$969$	31.6379	1.01636
$970$	0	0
$971$	50.9273	1.63434	0.817168	−	0.576400i	$-0.195543\pi$
$971$	50.9273	1.63434	0.817168	+	0.576400i	$0.195543\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	13.9921	0.448106
$976$	0	0
$977$	10.1187	0.323727	0.161864	−	0.986813i	$-0.448249\pi$
$977$	10.1187	0.323727	0.161864	+	0.986813i	$0.448249\pi$
$978$	0	0
$979$	10.2351	0.327117
$980$	0	0
$981$	4.16178	0.132875
$982$	0	0
$983$	−46.4718	−1.48222	−0.741111	−	0.671383i	$-0.765701\pi$
$983$	−46.4718	−1.48222	−0.741111	+	0.671383i	$0.765701\pi$
$984$	0	0
$985$	3.87381	0.123430
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.8333	−0.408074
$990$	0	0
$991$	12.3035	0.390833	0.195416	−	0.980720i	$-0.437394\pi$
$991$	12.3035	0.390833	0.195416	+	0.980720i	$0.437394\pi$
$992$	0	0
$993$	42.8070	1.35844
$994$	0	0
$995$	43.9252	1.39252
$996$	0	0
$997$	27.3745	0.866958	0.433479	−	0.901164i	$-0.357286\pi$
$997$	27.3745	0.866958	0.433479	+	0.901164i	$0.357286\pi$
$998$	0	0
$999$	41.0147	1.29765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2744.2.a.h.1.4	✓	12
4.3	odd	2		5488.2.a.w.1.9		12
7.6	odd	2	inner	2744.2.a.h.1.9	yes	12
28.27	even	2		5488.2.a.w.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2744.2.a.h.1.4	✓	12	1.1	even	1	trivial
2744.2.a.h.1.9	yes	12	7.6	odd	2	inner
5488.2.a.w.1.4		12	28.27	even	2
5488.2.a.w.1.9		12	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2744 = 2^{3} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2744.a (trivial)

\(f(q)\)	\(=\)	\(q-1.85982 q^{3} +1.80700 q^{5} +0.458937 q^{9} +5.99266 q^{11} +4.33684 q^{13} -3.36070 q^{15} -7.43508 q^{17} +2.28797 q^{19} +3.07396 q^{23} -1.73475 q^{25} +4.72592 q^{27} +5.91158 q^{29} +3.61062 q^{31} -11.1453 q^{33} +8.67866 q^{37} -8.06575 q^{39} -11.6608 q^{41} -4.17484 q^{43} +0.829299 q^{45} +2.97955 q^{47} +13.8279 q^{51} -8.17484 q^{53} +10.8287 q^{55} -4.25522 q^{57} -13.5673 q^{59} +1.99594 q^{61} +7.83666 q^{65} +7.69990 q^{67} -5.71701 q^{69} +12.7339 q^{71} -6.60300 q^{73} +3.22633 q^{75} -0.873289 q^{79} -10.1662 q^{81} +10.4473 q^{83} -13.4352 q^{85} -10.9945 q^{87} +1.70795 q^{89} -6.71512 q^{93} +4.13437 q^{95} -7.23220 q^{97} +2.75025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 22 q^{9} + 18 q^{11} + 28 q^{15} + 20 q^{23} + 14 q^{25} - 18 q^{29} - 18 q^{37} + 36 q^{39} + 10 q^{43} + 48 q^{51} - 38 q^{53} + 12 q^{57} + 8 q^{65} + 42 q^{67} + 56 q^{71} + 56 q^{79} + 52 q^{81}+ \cdots + 90 q^{99}+O(q^{100}) \) 12 * q + 22 * q^9 + 18 * q^11 + 28 * q^15 + 20 * q^23 + 14 * q^25 - 18 * q^29 - 18 * q^37 + 36 * q^39 + 10 * q^43 + 48 * q^51 - 38 * q^53 + 12 * q^57 + 8 * q^65 + 42 * q^67 + 56 * q^71 + 56 * q^79 + 52 * q^81 + 8 * q^85 - 48 * q^93 + 84 * q^95 + 90 * q^99

Introduction

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