LMFDB - Embedded newform 2744.2.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2744,2,Mod(1,2744)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 2744 = 2^{3} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2744.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:	$21.9109503146$
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			1.7
Root			$0.0835475$ 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)

$f(q)$	$=$	$q+0.0835475 q^{3} +3.81878 q^{5} -2.99302 q^{9} +O(q^{10})$	$q+0.0835475 q^{3} +3.81878 q^{5} -2.99302 q^{9} +1.01995 q^{11} -0.289833 q^{13} +0.319050 q^{15} -1.76652 q^{17} +6.29946 q^{19} -2.13396 q^{23} +9.58309 q^{25} -0.500702 q^{27} -7.26214 q^{29} +6.47176 q^{31} +0.0852143 q^{33} +7.89607 q^{37} -0.0242149 q^{39} +2.58764 q^{41} +7.62177 q^{43} -11.4297 q^{45} -1.07307 q^{47} -0.147588 q^{51} +3.62177 q^{53} +3.89496 q^{55} +0.526304 q^{57} +12.6864 q^{59} +7.39317 q^{61} -1.10681 q^{65} +10.1301 q^{67} -0.178287 q^{69} +3.55513 q^{71} -8.24913 q^{73} +0.800643 q^{75} +3.44126 q^{79} +8.93723 q^{81} -6.90920 q^{83} -6.74594 q^{85} -0.606734 q^{87} -17.8474 q^{89} +0.540699 q^{93} +24.0563 q^{95} -13.1744 q^{97} -3.05273 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 22 q^{9}+O(q^{10}) $ 12 * q + 22 * q^9	$ 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}+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

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
$3$	0.0835475	0.0482362	0.0241181	−	0.999709i	$-0.492322\pi$
$3$	0.0835475	0.0482362	0.0241181	+	0.999709i	$0.492322\pi$
$4$	0	0
$5$	3.81878	1.70781	0.853905	−	0.520428i	$-0.174228\pi$
$5$	3.81878	1.70781	0.853905	+	0.520428i	$0.174228\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.99302	−0.997673
$10$	0	0
$11$	1.01995	0.307526	0.153763	−	0.988108i	$-0.450861\pi$
$11$	1.01995	0.307526	0.153763	+	0.988108i	$0.450861\pi$
$12$	0	0
$13$	−0.289833	−0.0803853	−0.0401926	−	0.999192i	$-0.512797\pi$
$13$	−0.289833	−0.0803853	−0.0401926	+	0.999192i	$0.512797\pi$
$14$	0	0
$15$	0.319050	0.0823783
$16$	0	0
$17$	−1.76652	−0.428443	−0.214222	−	0.976785i	$-0.568722\pi$
$17$	−1.76652	−0.428443	−0.214222	+	0.976785i	$0.568722\pi$
$18$	0	0
$19$	6.29946	1.44520	0.722598	−	0.691269i	$-0.242948\pi$
$19$	6.29946	1.44520	0.722598	+	0.691269i	$0.242948\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.13396	−0.444961	−0.222480	−	0.974937i	$-0.571415\pi$
$23$	−2.13396	−0.444961	−0.222480	+	0.974937i	$0.571415\pi$
$24$	0	0
$25$	9.58309	1.91662
$26$	0	0
$27$	−0.500702	−0.0963601
$28$	0	0
$29$	−7.26214	−1.34855	−0.674273	−	0.738482i	$-0.735543\pi$
$29$	−7.26214	−1.34855	−0.674273	+	0.738482i	$0.735543\pi$
$30$	0	0
$31$	6.47176	1.16236	0.581181	−	0.813774i	$-0.302591\pi$
$31$	6.47176	1.16236	0.581181	+	0.813774i	$0.302591\pi$
$32$	0	0
$33$	0.0852143	0.0148339
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.89607	1.29811	0.649053	−	0.760743i	$-0.275165\pi$
$37$	7.89607	1.29811	0.649053	+	0.760743i	$0.275165\pi$
$38$	0	0
$39$	−0.0242149	−0.00387748
$40$	0	0
$41$	2.58764	0.404122	0.202061	−	0.979373i	$-0.435236\pi$
$41$	2.58764	0.404122	0.202061	+	0.979373i	$0.435236\pi$
$42$	0	0
$43$	7.62177	1.16231	0.581155	−	0.813793i	$-0.302601\pi$
$43$	7.62177	1.16231	0.581155	+	0.813793i	$0.302601\pi$
$44$	0	0
$45$	−11.4297	−1.70384
$46$	0	0
$47$	−1.07307	−0.156524	−0.0782620	−	0.996933i	$-0.524937\pi$
$47$	−1.07307	−0.156524	−0.0782620	+	0.996933i	$0.524937\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.147588	−0.0206665
$52$	0	0
$53$	3.62177	0.497489	0.248744	−	0.968569i	$-0.419982\pi$
$53$	3.62177	0.497489	0.248744	+	0.968569i	$0.419982\pi$
$54$	0	0
$55$	3.89496	0.525197
$56$	0	0
$57$	0.526304	0.0697107
$58$	0	0
$59$	12.6864	1.65163	0.825815	−	0.563942i	$-0.190716\pi$
$59$	12.6864	1.65163	0.825815	+	0.563942i	$0.190716\pi$
$60$	0	0
$61$	7.39317	0.946598	0.473299	−	0.880902i	$-0.343063\pi$
$61$	7.39317	0.946598	0.473299	+	0.880902i	$0.343063\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.10681	−0.137283
$66$	0	0
$67$	10.1301	1.23759	0.618793	−	0.785554i	$-0.287622\pi$
$67$	10.1301	1.23759	0.618793	+	0.785554i	$0.287622\pi$
$68$	0	0
$69$	−0.178287	−0.0214632
$70$	0	0
$71$	3.55513	0.421917	0.210958	−	0.977495i	$-0.432342\pi$
$71$	3.55513	0.421917	0.210958	+	0.977495i	$0.432342\pi$
$72$	0	0
$73$	−8.24913	−0.965487	−0.482744	−	0.875762i	$-0.660360\pi$
$73$	−8.24913	−0.965487	−0.482744	+	0.875762i	$0.660360\pi$
$74$	0	0
$75$	0.800643	0.0924503
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.44126	0.387173	0.193586	−	0.981083i	$-0.437988\pi$
$79$	3.44126	0.387173	0.193586	+	0.981083i	$0.437988\pi$
$80$	0	0
$81$	8.93723	0.993025
$82$	0	0
$83$	−6.90920	−0.758384	−0.379192	−	0.925318i	$-0.623798\pi$
$83$	−6.90920	−0.758384	−0.379192	+	0.925318i	$0.623798\pi$
$84$	0	0
$85$	−6.74594	−0.731700
$86$	0	0
$87$	−0.606734	−0.0650487
$88$	0	0
$89$	−17.8474	−1.89182	−0.945909	−	0.324432i	$-0.894827\pi$
$89$	−17.8474	−1.89182	−0.945909	+	0.324432i	$0.894827\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.540699	0.0560679
$94$	0	0
$95$	24.0563	2.46812
$96$	0	0
$97$	−13.1744	−1.33766	−0.668830	−	0.743416i	$-0.733204\pi$
$97$	−13.1744	−1.33766	−0.668830	+	0.743416i	$0.733204\pi$
$98$	0	0
$99$	−3.05273	−0.306811
$100$	0	0
$101$	8.75075	0.870732	0.435366	−	0.900253i	$-0.356619\pi$
$101$	8.75075	0.870732	0.435366	+	0.900253i	$0.356619\pi$
$102$	0	0
$103$	−14.5486	−1.43351	−0.716757	−	0.697323i	$-0.754374\pi$
$103$	−14.5486	−1.43351	−0.716757	+	0.697323i	$0.754374\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3297	1.09528	0.547639	−	0.836714i	$-0.315527\pi$
$107$	11.3297	1.09528	0.547639	+	0.836714i	$0.315527\pi$
$108$	0	0
$109$	3.88772	0.372376	0.186188	−	0.982514i	$-0.440387\pi$
$109$	3.88772	0.372376	0.186188	+	0.982514i	$0.440387\pi$
$110$	0	0
$111$	0.659697	0.0626157
$112$	0	0
$113$	−12.3312	−1.16002	−0.580012	−	0.814608i	$-0.696952\pi$
$113$	−12.3312	−1.16002	−0.580012	+	0.814608i	$0.696952\pi$
$114$	0	0
$115$	−8.14911	−0.759909
$116$	0	0
$117$	0.867477	0.0801982
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.95970	−0.905428
$122$	0	0
$123$	0.216191	0.0194933
$124$	0	0
$125$	17.5018	1.56541
$126$	0	0
$127$	13.6869	1.21451	0.607256	−	0.794506i	$-0.292270\pi$
$127$	13.6869	1.21451	0.607256	+	0.794506i	$0.292270\pi$
$128$	0	0
$129$	0.636780	0.0560654
$130$	0	0
$131$	−14.7107	−1.28528	−0.642639	−	0.766169i	$-0.722160\pi$
$131$	−14.7107	−1.28528	−0.642639	+	0.766169i	$0.722160\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.91207	−0.164565
$136$	0	0
$137$	2.09678	0.179140	0.0895701	−	0.995981i	$-0.471451\pi$
$137$	2.09678	0.179140	0.0895701	+	0.995981i	$0.471451\pi$
$138$	0	0
$139$	17.8350	1.51274	0.756372	−	0.654141i	$-0.226970\pi$
$139$	17.8350	1.51274	0.756372	+	0.654141i	$0.226970\pi$
$140$	0	0
$141$	−0.0896527	−0.00755012
$142$	0	0
$143$	−0.295615	−0.0247206
$144$	0	0
$145$	−27.7325	−2.30306
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.12576	−0.665688	−0.332844	−	0.942982i	$-0.608008\pi$
$149$	−8.12576	−0.665688	−0.332844	+	0.942982i	$0.608008\pi$
$150$	0	0
$151$	2.69220	0.219088	0.109544	−	0.993982i	$-0.465061\pi$
$151$	2.69220	0.219088	0.109544	+	0.993982i	$0.465061\pi$
$152$	0	0
$153$	5.28722	0.427447
$154$	0	0
$155$	24.7142	1.98509
$156$	0	0
$157$	6.16593	0.492095	0.246047	−	0.969258i	$-0.420868\pi$
$157$	6.16593	0.492095	0.246047	+	0.969258i	$0.420868\pi$
$158$	0	0
$159$	0.302590	0.0239970
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−22.3261	−1.74872	−0.874359	−	0.485279i	$-0.838718\pi$
$163$	−22.3261	−1.74872	−0.874359	+	0.485279i	$0.838718\pi$
$164$	0	0
$165$	0.325415	0.0253335
$166$	0	0
$167$	−24.4048	−1.88850	−0.944250	−	0.329228i	$-0.893212\pi$
$167$	−24.4048	−1.88850	−0.944250	+	0.329228i	$0.893212\pi$
$168$	0	0
$169$	−12.9160	−0.993538
$170$	0	0
$171$	−18.8544	−1.44183
$172$	0	0
$173$	19.7608	1.50238	0.751192	−	0.660084i	$-0.229479\pi$
$173$	19.7608	1.50238	0.751192	+	0.660084i	$0.229479\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.05992	0.0796683
$178$	0	0
$179$	20.9162	1.56335	0.781674	−	0.623688i	$-0.214366\pi$
$179$	20.9162	1.56335	0.781674	+	0.623688i	$0.214366\pi$
$180$	0	0
$181$	−22.5330	−1.67487	−0.837433	−	0.546539i	$-0.815945\pi$
$181$	−22.5330	−1.67487	−0.837433	+	0.546539i	$0.815945\pi$
$182$	0	0
$183$	0.617681	0.0456603
$184$	0	0
$185$	30.1534	2.21692
$186$	0	0
$187$	−1.80176	−0.131758
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.16589	0.156718	0.0783592	−	0.996925i	$-0.475032\pi$
$191$	2.16589	0.156718	0.0783592	+	0.996925i	$0.475032\pi$
$192$	0	0
$193$	18.2959	1.31697	0.658485	−	0.752594i	$-0.271197\pi$
$193$	18.2959	1.31697	0.658485	+	0.752594i	$0.271197\pi$
$194$	0	0
$195$	−0.0924712	−0.00662200
$196$	0	0
$197$	0.0803297	0.00572325	0.00286163	−	0.999996i	$-0.499089\pi$
$197$	0.0803297	0.00572325	0.00286163	+	0.999996i	$0.499089\pi$
$198$	0	0
$199$	−20.5594	−1.45741	−0.728707	−	0.684825i	$-0.759879\pi$
$199$	−20.5594	−1.45741	−0.728707	+	0.684825i	$0.759879\pi$
$200$	0	0
$201$	0.846343	0.0596965
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.88164	0.690164
$206$	0	0
$207$	6.38698	0.443925
$208$	0	0
$209$	6.42513	0.444436
$210$	0	0
$211$	8.85784	0.609799	0.304899	−	0.952385i	$-0.401377\pi$
$211$	8.85784	0.609799	0.304899	+	0.952385i	$0.401377\pi$
$212$	0	0
$213$	0.297023	0.0203516
$214$	0	0
$215$	29.1059	1.98500
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.689194	−0.0465714
$220$	0	0
$221$	0.511995	0.0344405
$222$	0	0
$223$	17.9612	1.20277	0.601384	−	0.798960i	$-0.294616\pi$
$223$	17.9612	1.20277	0.601384	+	0.798960i	$0.294616\pi$
$224$	0	0
$225$	−28.6824	−1.91216
$226$	0	0
$227$	15.8927	1.05484	0.527418	−	0.849606i	$-0.323160\pi$
$227$	15.8927	1.05484	0.527418	+	0.849606i	$0.323160\pi$
$228$	0	0
$229$	27.1956	1.79714	0.898569	−	0.438831i	$-0.144607\pi$
$229$	27.1956	1.79714	0.898569	+	0.438831i	$0.144607\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.96727	0.652978	0.326489	−	0.945201i	$-0.394134\pi$
$233$	9.96727	0.652978	0.326489	+	0.945201i	$0.394134\pi$
$234$	0	0
$235$	−4.09783	−0.267313
$236$	0	0
$237$	0.287509	0.0186757
$238$	0	0
$239$	−10.7296	−0.694039	−0.347020	−	0.937858i	$-0.612806\pi$
$239$	−10.7296	−0.694039	−0.347020	+	0.937858i	$0.612806\pi$
$240$	0	0
$241$	−11.8736	−0.764844	−0.382422	−	0.923988i	$-0.624910\pi$
$241$	−11.8736	−0.764844	−0.382422	+	0.923988i	$0.624910\pi$
$242$	0	0
$243$	2.24879	0.144260
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.82579	−0.116172
$248$	0	0
$249$	−0.577247	−0.0365815
$250$	0	0
$251$	0.531997	0.0335793	0.0167897	−	0.999859i	$-0.494655\pi$
$251$	0.531997	0.0335793	0.0167897	+	0.999859i	$0.494655\pi$
$252$	0	0
$253$	−2.17653	−0.136837
$254$	0	0
$255$	−0.563607	−0.0352944
$256$	0	0
$257$	4.05224	0.252772	0.126386	−	0.991981i	$-0.459662\pi$
$257$	4.05224	0.252772	0.126386	+	0.991981i	$0.459662\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	21.7357	1.34541
$262$	0	0
$263$	−20.8992	−1.28870	−0.644349	−	0.764732i	$-0.722871\pi$
$263$	−20.8992	−1.28870	−0.644349	+	0.764732i	$0.722871\pi$
$264$	0	0
$265$	13.8308	0.849617
$266$	0	0
$267$	−1.49110	−0.0912541
$268$	0	0
$269$	14.0472	0.856472	0.428236	−	0.903667i	$-0.359135\pi$
$269$	14.0472	0.856472	0.428236	+	0.903667i	$0.359135\pi$
$270$	0	0
$271$	13.4849	0.819148	0.409574	−	0.912277i	$-0.365677\pi$
$271$	13.4849	0.819148	0.409574	+	0.912277i	$0.365677\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.77427	0.589411
$276$	0	0
$277$	−6.85805	−0.412060	−0.206030	−	0.978546i	$-0.566055\pi$
$277$	−6.85805	−0.412060	−0.206030	+	0.978546i	$0.566055\pi$
$278$	0	0
$279$	−19.3701	−1.15966
$280$	0	0
$281$	−12.9162	−0.770517	−0.385259	−	0.922809i	$-0.625888\pi$
$281$	−12.9162	−0.770517	−0.385259	+	0.922809i	$0.625888\pi$
$282$	0	0
$283$	−21.7350	−1.29201	−0.646005	−	0.763333i	$-0.723561\pi$
$283$	−21.7350	−1.29201	−0.646005	+	0.763333i	$0.723561\pi$
$284$	0	0
$285$	2.00984	0.119053
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.8794	−0.816436
$290$	0	0
$291$	−1.10069	−0.0645236
$292$	0	0
$293$	−10.2888	−0.601077	−0.300538	−	0.953770i	$-0.597166\pi$
$293$	−10.2888	−0.601077	−0.300538	+	0.953770i	$0.597166\pi$
$294$	0	0
$295$	48.4466	2.82067
$296$	0	0
$297$	−0.510691	−0.0296333
$298$	0	0
$299$	0.618492	0.0357683
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.731104	0.0420008
$304$	0	0
$305$	28.2329	1.61661
$306$	0	0
$307$	−16.1036	−0.919080	−0.459540	−	0.888157i	$-0.651986\pi$
$307$	−16.1036	−0.919080	−0.459540	+	0.888157i	$0.651986\pi$
$308$	0	0
$309$	−1.21550	−0.0691473
$310$	0	0
$311$	9.67143	0.548416	0.274208	−	0.961670i	$-0.411584\pi$
$311$	9.67143	0.548416	0.274208	+	0.961670i	$0.411584\pi$
$312$	0	0
$313$	−0.409877	−0.0231676	−0.0115838	−	0.999933i	$-0.503687\pi$
$313$	−0.409877	−0.0231676	−0.0115838	+	0.999933i	$0.503687\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.1299	−1.01827	−0.509137	−	0.860685i	$-0.670035\pi$
$317$	−18.1299	−1.01827	−0.509137	+	0.860685i	$0.670035\pi$
$318$	0	0
$319$	−7.40702	−0.414713
$320$	0	0
$321$	0.946564	0.0528321
$322$	0	0
$323$	−11.1281	−0.619184
$324$	0	0
$325$	−2.77750	−0.154068
$326$	0	0
$327$	0.324809	0.0179620
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.7333	1.35946	0.679732	−	0.733461i	$-0.262096\pi$
$331$	24.7333	1.35946	0.679732	+	0.733461i	$0.262096\pi$
$332$	0	0
$333$	−23.6331	−1.29509
$334$	0	0
$335$	38.6846	2.11356
$336$	0	0
$337$	30.6186	1.66790	0.833951	−	0.551838i	$-0.186073\pi$
$337$	30.6186	1.66790	0.833951	+	0.551838i	$0.186073\pi$
$338$	0	0
$339$	−1.03024	−0.0559551
$340$	0	0
$341$	6.60087	0.357457
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.680838	−0.0366551
$346$	0	0
$347$	−18.8309	−1.01090	−0.505448	−	0.862857i	$-0.668673\pi$
$347$	−18.8309	−1.01090	−0.505448	+	0.862857i	$0.668673\pi$
$348$	0	0
$349$	−10.6663	−0.570955	−0.285477	−	0.958385i	$-0.592152\pi$
$349$	−10.6663	−0.570955	−0.285477	+	0.958385i	$0.592152\pi$
$350$	0	0
$351$	0.145120	0.00774594
$352$	0	0
$353$	13.1069	0.697610	0.348805	−	0.937195i	$-0.386588\pi$
$353$	13.1069	0.697610	0.348805	+	0.937195i	$0.386588\pi$
$354$	0	0
$355$	13.5763	0.720554
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.9129	1.05096	0.525481	−	0.850805i	$-0.323885\pi$
$359$	19.9129	1.05096	0.525481	+	0.850805i	$0.323885\pi$
$360$	0	0
$361$	20.6832	1.08859
$362$	0	0
$363$	−0.832109	−0.0436744
$364$	0	0
$365$	−31.5016	−1.64887
$366$	0	0
$367$	−14.3538	−0.749260	−0.374630	−	0.927174i	$-0.622230\pi$
$367$	−14.3538	−0.749260	−0.374630	+	0.927174i	$0.622230\pi$
$368$	0	0
$369$	−7.74487	−0.403182
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7.03456	−0.364236	−0.182118	−	0.983277i	$-0.558295\pi$
$373$	−7.03456	−0.364236	−0.182118	+	0.983277i	$0.558295\pi$
$374$	0	0
$375$	1.46223	0.0755094
$376$	0	0
$377$	2.10481	0.108403
$378$	0	0
$379$	18.6300	0.956960	0.478480	−	0.878098i	$-0.341188\pi$
$379$	18.6300	0.956960	0.478480	+	0.878098i	$0.341188\pi$
$380$	0	0
$381$	1.14350	0.0585835
$382$	0	0
$383$	−3.38096	−0.172759	−0.0863795	−	0.996262i	$-0.527530\pi$
$383$	−3.38096	−0.172759	−0.0863795	+	0.996262i	$0.527530\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−22.8121	−1.15960
$388$	0	0
$389$	−25.7376	−1.30495	−0.652475	−	0.757810i	$-0.726269\pi$
$389$	−25.7376	−1.30495	−0.652475	+	0.757810i	$0.726269\pi$
$390$	0	0
$391$	3.76967	0.190641
$392$	0	0
$393$	−1.22904	−0.0619969
$394$	0	0
$395$	13.1414	0.661217
$396$	0	0
$397$	5.40636	0.271337	0.135669	−	0.990754i	$-0.456682\pi$
$397$	5.40636	0.271337	0.135669	+	0.990754i	$0.456682\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.1122	0.504979	0.252490	−	0.967600i	$-0.418751\pi$
$401$	10.1122	0.504979	0.252490	+	0.967600i	$0.418751\pi$
$402$	0	0
$403$	−1.87573	−0.0934368
$404$	0	0
$405$	34.1293	1.69590
$406$	0	0
$407$	8.05360	0.399202
$408$	0	0
$409$	−8.90427	−0.440288	−0.220144	−	0.975467i	$-0.570653\pi$
$409$	−8.90427	−0.440288	−0.220144	+	0.975467i	$0.570653\pi$
$410$	0	0
$411$	0.175181	0.00864104
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−26.3847	−1.29518
$416$	0	0
$417$	1.49007	0.0729690
$418$	0	0
$419$	−31.9459	−1.56066	−0.780330	−	0.625368i	$-0.784949\pi$
$419$	−31.9459	−1.56066	−0.780330	+	0.625368i	$0.784949\pi$
$420$	0	0
$421$	−26.1355	−1.27376	−0.636882	−	0.770961i	$-0.719776\pi$
$421$	−26.1355	−1.27376	−0.636882	+	0.770961i	$0.719776\pi$
$422$	0	0
$423$	3.21173	0.156160
$424$	0	0
$425$	−16.9287	−0.821162
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.0246979	−0.00119243
$430$	0	0
$431$	−25.4989	−1.22824	−0.614119	−	0.789213i	$-0.710489\pi$
$431$	−25.4989	−1.22824	−0.614119	+	0.789213i	$0.710489\pi$
$432$	0	0
$433$	−34.3398	−1.65026	−0.825132	−	0.564940i	$-0.808899\pi$
$433$	−34.3398	−1.65026	−0.825132	+	0.564940i	$0.808899\pi$
$434$	0	0
$435$	−2.31698	−0.111091
$436$	0	0
$437$	−13.4428	−0.643055
$438$	0	0
$439$	−5.91092	−0.282113	−0.141056	−	0.990002i	$-0.545050\pi$
$439$	−5.91092	−0.282113	−0.141056	+	0.990002i	$0.545050\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.0904	−1.42964	−0.714819	−	0.699310i	$-0.753491\pi$
$443$	−30.0904	−1.42964	−0.714819	+	0.699310i	$0.753491\pi$
$444$	0	0
$445$	−68.1552	−3.23087
$446$	0	0
$447$	−0.678887	−0.0321103
$448$	0	0
$449$	−10.6270	−0.501520	−0.250760	−	0.968049i	$-0.580681\pi$
$449$	−10.6270	−0.501520	−0.250760	+	0.968049i	$0.580681\pi$
$450$	0	0
$451$	2.63927	0.124278
$452$	0	0
$453$	0.224926	0.0105680
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.1092	1.17456	0.587279	−	0.809384i	$-0.300199\pi$
$457$	25.1092	1.17456	0.587279	+	0.809384i	$0.300199\pi$
$458$	0	0
$459$	0.884499	0.0412849
$460$	0	0
$461$	5.57770	0.259779	0.129890	−	0.991528i	$-0.458538\pi$
$461$	5.57770	0.259779	0.129890	+	0.991528i	$0.458538\pi$
$462$	0	0
$463$	−13.0580	−0.606858	−0.303429	−	0.952854i	$-0.598132\pi$
$463$	−13.0580	−0.606858	−0.303429	+	0.952854i	$0.598132\pi$
$464$	0	0
$465$	2.06481	0.0957534
$466$	0	0
$467$	−17.4611	−0.808002	−0.404001	−	0.914759i	$-0.632381\pi$
$467$	−17.4611	−0.808002	−0.404001	+	0.914759i	$0.632381\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.515148	0.0237368
$472$	0	0
$473$	7.77382	0.357441
$474$	0	0
$475$	60.3683	2.76989
$476$	0	0
$477$	−10.8400	−0.496331
$478$	0	0
$479$	4.45105	0.203374	0.101687	−	0.994816i	$-0.467576\pi$
$479$	4.45105	0.203374	0.101687	+	0.994816i	$0.467576\pi$
$480$	0	0
$481$	−2.28854	−0.104349
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−50.3102	−2.28447
$486$	0	0
$487$	16.3537	0.741057	0.370528	−	0.928821i	$-0.379177\pi$
$487$	16.3537	0.741057	0.370528	+	0.928821i	$0.379177\pi$
$488$	0	0
$489$	−1.86529	−0.0843515
$490$	0	0
$491$	8.47650	0.382539	0.191269	−	0.981538i	$-0.438740\pi$
$491$	8.47650	0.382539	0.191269	+	0.981538i	$0.438740\pi$
$492$	0	0
$493$	12.8287	0.577776
$494$	0	0
$495$	−11.6577	−0.523975
$496$	0	0
$497$	0	0
$498$	0	0
$499$	35.5230	1.59023	0.795113	−	0.606461i	$-0.207411\pi$
$499$	35.5230	1.59023	0.795113	+	0.606461i	$0.207411\pi$
$500$	0	0
$501$	−2.03896	−0.0910941
$502$	0	0
$503$	−32.3172	−1.44095	−0.720477	−	0.693479i	$-0.756077\pi$
$503$	−32.3172	−1.44095	−0.720477	+	0.693479i	$0.756077\pi$
$504$	0	0
$505$	33.4172	1.48705
$506$	0	0
$507$	−1.07910	−0.0479245
$508$	0	0
$509$	−32.2212	−1.42818	−0.714089	−	0.700054i	$-0.753159\pi$
$509$	−32.2212	−1.42818	−0.714089	+	0.700054i	$0.753159\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.15415	−0.139259
$514$	0	0
$515$	−55.5579	−2.44817
$516$	0	0
$517$	−1.09448	−0.0481352
$518$	0	0
$519$	1.65096	0.0724693
$520$	0	0
$521$	−32.6548	−1.43063	−0.715316	−	0.698801i	$-0.753717\pi$
$521$	−32.6548	−1.43063	−0.715316	+	0.698801i	$0.753717\pi$
$522$	0	0
$523$	−23.9624	−1.04780	−0.523902	−	0.851778i	$-0.675524\pi$
$523$	−23.9624	−1.04780	−0.523902	+	0.851778i	$0.675524\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.4325	−0.498006
$528$	0	0
$529$	−18.4462	−0.802010
$530$	0	0
$531$	−37.9707	−1.64779
$532$	0	0
$533$	−0.749985	−0.0324855
$534$	0	0
$535$	43.2655	1.87053
$536$	0	0
$537$	1.74749	0.0754099
$538$	0	0
$539$	0	0
$540$	0	0
$541$	19.3057	0.830016	0.415008	−	0.909818i	$-0.363779\pi$
$541$	19.3057	0.830016	0.415008	+	0.909818i	$0.363779\pi$
$542$	0	0
$543$	−1.88258	−0.0807892
$544$	0	0
$545$	14.8463	0.635947
$546$	0	0
$547$	24.2803	1.03815	0.519076	−	0.854728i	$-0.326276\pi$
$547$	24.2803	1.03815	0.519076	+	0.854728i	$0.326276\pi$
$548$	0	0
$549$	−22.1279	−0.944396
$550$	0	0
$551$	−45.7476	−1.94891
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.51924	0.106936
$556$	0	0
$557$	−19.1509	−0.811448	−0.405724	−	0.913996i	$-0.632981\pi$
$557$	−19.1509	−0.811448	−0.405724	+	0.913996i	$0.632981\pi$
$558$	0	0
$559$	−2.20904	−0.0934326
$560$	0	0
$561$	−0.150532	−0.00635549
$562$	0	0
$563$	22.5643	0.950974	0.475487	−	0.879723i	$-0.342272\pi$
$563$	22.5643	0.950974	0.475487	+	0.879723i	$0.342272\pi$
$564$	0	0
$565$	−47.0903	−1.98110
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.1759	0.594283	0.297142	−	0.954833i	$-0.403967\pi$
$569$	14.1759	0.594283	0.297142	+	0.954833i	$0.403967\pi$
$570$	0	0
$571$	−0.657772	−0.0275269	−0.0137634	−	0.999905i	$-0.504381\pi$
$571$	−0.657772	−0.0275269	−0.0137634	+	0.999905i	$0.504381\pi$
$572$	0	0
$573$	0.180955	0.00755950
$574$	0	0
$575$	−20.4499	−0.852820
$576$	0	0
$577$	−18.8177	−0.783391	−0.391695	−	0.920095i	$-0.628111\pi$
$577$	−18.8177	−0.783391	−0.391695	+	0.920095i	$0.628111\pi$
$578$	0	0
$579$	1.52858	0.0635256
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.69402	0.152991
$584$	0	0
$585$	3.31270	0.136963
$586$	0	0
$587$	37.5628	1.55038	0.775192	−	0.631725i	$-0.217653\pi$
$587$	37.5628	1.55038	0.775192	+	0.631725i	$0.217653\pi$
$588$	0	0
$589$	40.7686	1.67984
$590$	0	0
$591$	0.00671135	0.000276068 0
$592$	0	0
$593$	20.5547	0.844079	0.422040	−	0.906577i	$-0.361314\pi$
$593$	20.5547	0.844079	0.422040	+	0.906577i	$0.361314\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.71768	−0.0703001
$598$	0	0
$599$	6.62343	0.270626	0.135313	−	0.990803i	$-0.456796\pi$
$599$	6.62343	0.270626	0.135313	+	0.990803i	$0.456796\pi$
$600$	0	0
$601$	−10.4996	−0.428288	−0.214144	−	0.976802i	$-0.568696\pi$
$601$	−10.4996	−0.428288	−0.214144	+	0.976802i	$0.568696\pi$
$602$	0	0
$603$	−30.3195	−1.23471
$604$	0	0
$605$	−38.0339	−1.54630
$606$	0	0
$607$	−24.9424	−1.01238	−0.506190	−	0.862422i	$-0.668947\pi$
$607$	−24.9424	−1.01238	−0.506190	+	0.862422i	$0.668947\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.311013	0.0125822
$612$	0	0
$613$	−34.6299	−1.39869	−0.699344	−	0.714786i	$-0.746524\pi$
$613$	−34.6299	−1.39869	−0.699344	+	0.714786i	$0.746524\pi$
$614$	0	0
$615$	0.825587	0.0332909
$616$	0	0
$617$	−18.0915	−0.728335	−0.364167	−	0.931334i	$-0.618646\pi$
$617$	−18.0915	−0.728335	−0.364167	+	0.931334i	$0.618646\pi$
$618$	0	0
$619$	31.0331	1.24733	0.623664	−	0.781693i	$-0.285643\pi$
$619$	31.0331	1.24733	0.623664	+	0.781693i	$0.285643\pi$
$620$	0	0
$621$	1.06848	0.0428765
$622$	0	0
$623$	0	0
$624$	0	0
$625$	18.9202	0.756806
$626$	0	0
$627$	0.536804	0.0214379
$628$	0	0
$629$	−13.9486	−0.556165
$630$	0	0
$631$	3.40426	0.135522	0.0677608	−	0.997702i	$-0.478415\pi$
$631$	3.40426	0.135522	0.0677608	+	0.997702i	$0.478415\pi$
$632$	0	0
$633$	0.740050	0.0294144
$634$	0	0
$635$	52.2671	2.07416
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.6406	−0.420935
$640$	0	0
$641$	−13.5416	−0.534860	−0.267430	−	0.963577i	$-0.586174\pi$
$641$	−13.5416	−0.534860	−0.267430	+	0.963577i	$0.586174\pi$
$642$	0	0
$643$	0.278984	0.0110021	0.00550104	−	0.999985i	$-0.498249\pi$
$643$	0.278984	0.0110021	0.00550104	+	0.999985i	$0.498249\pi$
$644$	0	0
$645$	2.43172	0.0957490
$646$	0	0
$647$	−14.6291	−0.575129	−0.287564	−	0.957761i	$-0.592845\pi$
$647$	−14.6291	−0.575129	−0.287564	+	0.957761i	$0.592845\pi$
$648$	0	0
$649$	12.9395	0.507920
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.9303	−1.21040	−0.605198	−	0.796075i	$-0.706906\pi$
$653$	−30.9303	−1.21040	−0.605198	+	0.796075i	$0.706906\pi$
$654$	0	0
$655$	−56.1768	−2.19501
$656$	0	0
$657$	24.6898	0.963241
$658$	0	0
$659$	−39.3965	−1.53467	−0.767335	−	0.641246i	$-0.778418\pi$
$659$	−39.3965	−1.53467	−0.767335	+	0.641246i	$0.778418\pi$
$660$	0	0
$661$	10.9496	0.425889	0.212945	−	0.977064i	$-0.431695\pi$
$661$	10.9496	0.425889	0.212945	+	0.977064i	$0.431695\pi$
$662$	0	0
$663$	0.0427760	0.00166128
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.4971	0.600050
$668$	0	0
$669$	1.50061	0.0580170
$670$	0	0
$671$	7.54066	0.291104
$672$	0	0
$673$	−2.89673	−0.111661	−0.0558303	−	0.998440i	$-0.517781\pi$
$673$	−2.89673	−0.111661	−0.0558303	+	0.998440i	$0.517781\pi$
$674$	0	0
$675$	−4.79827	−0.184686
$676$	0	0
$677$	5.32594	0.204692	0.102346	−	0.994749i	$-0.467365\pi$
$677$	5.32594	0.204692	0.102346	+	0.994749i	$0.467365\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.32780	0.0508812
$682$	0	0
$683$	38.9439	1.49015	0.745073	−	0.666983i	$-0.232414\pi$
$683$	38.9439	1.49015	0.745073	+	0.666983i	$0.232414\pi$
$684$	0	0
$685$	8.00715	0.305937
$686$	0	0
$687$	2.27213	0.0866871
$688$	0	0
$689$	−1.04971	−0.0399908
$690$	0	0
$691$	22.4591	0.854385	0.427192	−	0.904161i	$-0.359503\pi$
$691$	22.4591	0.854385	0.427192	+	0.904161i	$0.359503\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	68.1080	2.58348
$696$	0	0
$697$	−4.57112	−0.173143
$698$	0	0
$699$	0.832741	0.0314972
$700$	0	0
$701$	−3.51755	−0.132856	−0.0664280	−	0.997791i	$-0.521160\pi$
$701$	−3.51755	−0.132856	−0.0664280	+	0.997791i	$0.521160\pi$
$702$	0	0
$703$	49.7410	1.87602
$704$	0	0
$705$	−0.342364	−0.0128942
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12.2052	−0.458374	−0.229187	−	0.973382i	$-0.573607\pi$
$709$	−12.2052	−0.458374	−0.229187	+	0.973382i	$0.573607\pi$
$710$	0	0
$711$	−10.2998	−0.386272
$712$	0	0
$713$	−13.8105	−0.517205
$714$	0	0
$715$	−1.12889	−0.0422181
$716$	0	0
$717$	−0.896431	−0.0334778
$718$	0	0
$719$	24.5971	0.917315	0.458658	−	0.888613i	$-0.348330\pi$
$719$	24.5971	0.917315	0.458658	+	0.888613i	$0.348330\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−0.992007	−0.0368931
$724$	0	0
$725$	−69.5938	−2.58465
$726$	0	0
$727$	−18.4633	−0.684765	−0.342383	−	0.939561i	$-0.611234\pi$
$727$	−18.4633	−0.684765	−0.342383	+	0.939561i	$0.611234\pi$
$728$	0	0
$729$	−26.6238	−0.986067
$730$	0	0
$731$	−13.4640	−0.497984
$732$	0	0
$733$	5.44110	0.200972	0.100486	−	0.994938i	$-0.467960\pi$
$733$	5.44110	0.200972	0.100486	+	0.994938i	$0.467960\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.3322	0.380591
$738$	0	0
$739$	−31.9092	−1.17380	−0.586900	−	0.809660i	$-0.699652\pi$
$739$	−31.9092	−1.17380	−0.586900	+	0.809660i	$0.699652\pi$
$740$	0	0
$741$	−0.152541	−0.00560372
$742$	0	0
$743$	44.8535	1.64552	0.822758	−	0.568392i	$-0.192434\pi$
$743$	44.8535	1.64552	0.822758	+	0.568392i	$0.192434\pi$
$744$	0	0
$745$	−31.0305	−1.13687
$746$	0	0
$747$	20.6794	0.756619
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−36.2989	−1.32456	−0.662282	−	0.749254i	$-0.730412\pi$
$751$	−36.2989	−1.32456	−0.662282	+	0.749254i	$0.730412\pi$
$752$	0	0
$753$	0.0444470	0.00161974
$754$	0	0
$755$	10.2809	0.374160
$756$	0	0
$757$	38.2370	1.38975	0.694874	−	0.719132i	$-0.255460\pi$
$757$	38.2370	1.38975	0.694874	+	0.719132i	$0.255460\pi$
$758$	0	0
$759$	−0.181844	−0.00660050
$760$	0	0
$761$	40.8511	1.48085	0.740426	−	0.672138i	$-0.234624\pi$
$761$	40.8511	1.48085	0.740426	+	0.672138i	$0.234624\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	20.1907	0.729998
$766$	0	0
$767$	−3.67694	−0.132767
$768$	0	0
$769$	7.48085	0.269766	0.134883	−	0.990862i	$-0.456934\pi$
$769$	7.48085	0.269766	0.134883	+	0.990862i	$0.456934\pi$
$770$	0	0
$771$	0.338555	0.0121927
$772$	0	0
$773$	−13.7872	−0.495891	−0.247946	−	0.968774i	$-0.579755\pi$
$773$	−13.7872	−0.495891	−0.247946	+	0.968774i	$0.579755\pi$
$774$	0	0
$775$	62.0194	2.22780
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.3008	0.584035
$780$	0	0
$781$	3.62606	0.129750
$782$	0	0
$783$	3.63617	0.129946
$784$	0	0
$785$	23.5463	0.840405
$786$	0	0
$787$	−3.44749	−0.122890	−0.0614449	−	0.998110i	$-0.519571\pi$
$787$	−3.44749	−0.122890	−0.0614449	+	0.998110i	$0.519571\pi$
$788$	0	0
$789$	−1.74607	−0.0621618
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.14279	−0.0760926
$794$	0	0
$795$	1.15553	0.0409823
$796$	0	0
$797$	−21.8968	−0.775623	−0.387811	−	0.921739i	$-0.626769\pi$
$797$	−21.8968	−0.775623	−0.387811	+	0.921739i	$0.626769\pi$
$798$	0	0
$799$	1.89560	0.0670616
$800$	0	0
$801$	53.4175	1.88742
$802$	0	0
$803$	−8.41369	−0.296913
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.17361	0.0413129
$808$	0	0
$809$	26.8869	0.945294	0.472647	−	0.881252i	$-0.343299\pi$
$809$	26.8869	0.945294	0.472647	+	0.881252i	$0.343299\pi$
$810$	0	0
$811$	−34.9560	−1.22747	−0.613736	−	0.789512i	$-0.710334\pi$
$811$	−34.9560	−1.22747	−0.613736	+	0.789512i	$0.710334\pi$
$812$	0	0
$813$	1.12663	0.0395126
$814$	0	0
$815$	−85.2587	−2.98648
$816$	0	0
$817$	48.0131	1.67976
$818$	0	0
$819$	0	0
$820$	0	0
$821$	51.2931	1.79014	0.895070	−	0.445926i	$-0.147125\pi$
$821$	51.2931	1.79014	0.895070	+	0.445926i	$0.147125\pi$
$822$	0	0
$823$	3.71495	0.129495	0.0647475	−	0.997902i	$-0.479376\pi$
$823$	3.71495	0.129495	0.0647475	+	0.997902i	$0.479376\pi$
$824$	0	0
$825$	0.816616	0.0284309
$826$	0	0
$827$	1.47542	0.0513053	0.0256526	−	0.999671i	$-0.491834\pi$
$827$	1.47542	0.0513053	0.0256526	+	0.999671i	$0.491834\pi$
$828$	0	0
$829$	47.8382	1.66149	0.830745	−	0.556653i	$-0.187915\pi$
$829$	47.8382	1.66149	0.830745	+	0.556653i	$0.187915\pi$
$830$	0	0
$831$	−0.572973	−0.0198762
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−93.1966	−3.22520
$836$	0	0
$837$	−3.24042	−0.112005
$838$	0	0
$839$	−32.8485	−1.13406	−0.567028	−	0.823699i	$-0.691907\pi$
$839$	−32.8485	−1.13406	−0.567028	+	0.823699i	$0.691907\pi$
$840$	0	0
$841$	23.7387	0.818576
$842$	0	0
$843$	−1.07912	−0.0371668
$844$	0	0
$845$	−49.3234	−1.69678
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.81590	−0.0623216
$850$	0	0
$851$	−16.8499	−0.577606
$852$	0	0
$853$	1.86457	0.0638418	0.0319209	−	0.999490i	$-0.489838\pi$
$853$	1.86457	0.0638418	0.0319209	+	0.999490i	$0.489838\pi$
$854$	0	0
$855$	−72.0009	−2.46238
$856$	0	0
$857$	41.9242	1.43210	0.716052	−	0.698047i	$-0.245947\pi$
$857$	41.9242	1.43210	0.716052	+	0.698047i	$0.245947\pi$
$858$	0	0
$859$	−34.8918	−1.19049	−0.595246	−	0.803544i	$-0.702945\pi$
$859$	−34.8918	−1.19049	−0.595246	+	0.803544i	$0.702945\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.9567	−0.951658	−0.475829	−	0.879538i	$-0.657852\pi$
$863$	−27.9567	−0.951658	−0.475829	+	0.879538i	$0.657852\pi$
$864$	0	0
$865$	75.4621	2.56579
$866$	0	0
$867$	−1.15959	−0.0393818
$868$	0	0
$869$	3.50992	0.119066
$870$	0	0
$871$	−2.93603	−0.0994838
$872$	0	0
$873$	39.4313	1.33455
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.24257	−0.109494	−0.0547469	−	0.998500i	$-0.517435\pi$
$877$	−3.24257	−0.109494	−0.0547469	+	0.998500i	$0.517435\pi$
$878$	0	0
$879$	−0.859602	−0.0289936
$880$	0	0
$881$	19.9706	0.672825	0.336412	−	0.941715i	$-0.390786\pi$
$881$	19.9706	0.672825	0.336412	+	0.941715i	$0.390786\pi$
$882$	0	0
$883$	27.4770	0.924674	0.462337	−	0.886704i	$-0.347011\pi$
$883$	27.4770	0.924674	0.462337	+	0.886704i	$0.347011\pi$
$884$	0	0
$885$	4.04759	0.136058
$886$	0	0
$887$	−11.7632	−0.394971	−0.197485	−	0.980306i	$-0.563278\pi$
$887$	−11.7632	−0.394971	−0.197485	+	0.980306i	$0.563278\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.11552	0.305381
$892$	0	0
$893$	−6.75979	−0.226208
$894$	0	0
$895$	79.8742	2.66990
$896$	0	0
$897$	0.0516734	0.00172533
$898$	0	0
$899$	−46.9988	−1.56750
$900$	0	0
$901$	−6.39792	−0.213146
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−86.0487	−2.86036
$906$	0	0
$907$	−8.92374	−0.296308	−0.148154	−	0.988964i	$-0.547333\pi$
$907$	−8.92374	−0.296308	−0.148154	+	0.988964i	$0.547333\pi$
$908$	0	0
$909$	−26.1912	−0.868706
$910$	0	0
$911$	−56.5205	−1.87261	−0.936304	−	0.351191i	$-0.885777\pi$
$911$	−56.5205	−1.87261	−0.936304	+	0.351191i	$0.885777\pi$
$912$	0	0
$913$	−7.04704	−0.233223
$914$	0	0
$915$	2.35879	0.0779791
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.47184	−0.147512	−0.0737562	−	0.997276i	$-0.523499\pi$
$919$	−4.47184	−0.147512	−0.0737562	+	0.997276i	$0.523499\pi$
$920$	0	0
$921$	−1.34541	−0.0443329
$922$	0	0
$923$	−1.03040	−0.0339159
$924$	0	0
$925$	75.6688	2.48797
$926$	0	0
$927$	43.5442	1.43018
$928$	0	0
$929$	55.2167	1.81160	0.905801	−	0.423703i	$-0.139270\pi$
$929$	55.2167	1.81160	0.905801	+	0.423703i	$0.139270\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0.808024	0.0264535
$934$	0	0
$935$	−6.88052	−0.225017
$936$	0	0
$937$	−35.3159	−1.15372	−0.576860	−	0.816843i	$-0.695722\pi$
$937$	−35.3159	−1.15372	−0.576860	+	0.816843i	$0.695722\pi$
$938$	0	0
$939$	−0.0342442	−0.00111752
$940$	0	0
$941$	−22.0941	−0.720248	−0.360124	−	0.932904i	$-0.617266\pi$
$941$	−22.0941	−0.720248	−0.360124	+	0.932904i	$0.617266\pi$
$942$	0	0
$943$	−5.52192	−0.179818
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0177	1.17042	0.585208	−	0.810883i	$-0.301013\pi$
$947$	36.0177	1.17042	0.585208	+	0.810883i	$0.301013\pi$
$948$	0	0
$949$	2.39087	0.0776109
$950$	0	0
$951$	−1.51471	−0.0491177
$952$	0	0
$953$	7.41930	0.240335	0.120167	−	0.992754i	$-0.461657\pi$
$953$	7.41930	0.240335	0.120167	+	0.992754i	$0.461657\pi$
$954$	0	0
$955$	8.27107	0.267645
$956$	0	0
$957$	−0.618838	−0.0200042
$958$	0	0
$959$	0	0
$960$	0	0
$961$	10.8836	0.351085
$962$	0	0
$963$	−33.9099	−1.09273
$964$	0	0
$965$	69.8682	2.24914
$966$	0	0
$967$	−39.2927	−1.26357	−0.631785	−	0.775144i	$-0.717677\pi$
$967$	−39.2927	−1.26357	−0.631785	+	0.775144i	$0.717677\pi$
$968$	0	0
$969$	−0.929726	−0.0298671
$970$	0	0
$971$	−17.4114	−0.558759	−0.279380	−	0.960181i	$-0.590129\pi$
$971$	−17.4114	−0.558759	−0.279380	+	0.960181i	$0.590129\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.232053	−0.00743165
$976$	0	0
$977$	24.2665	0.776355	0.388178	−	0.921585i	$-0.373105\pi$
$977$	24.2665	0.776355	0.388178	+	0.921585i	$0.373105\pi$
$978$	0	0
$979$	−18.2034	−0.581784
$980$	0	0
$981$	−11.6360	−0.371509
$982$	0	0
$983$	−45.1386	−1.43970	−0.719849	−	0.694131i	$-0.755789\pi$
$983$	−45.1386	−1.43970	−0.719849	+	0.694131i	$0.755789\pi$
$984$	0	0
$985$	0.306761	0.00977423
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.2645	−0.517182
$990$	0	0
$991$	−5.71502	−0.181544	−0.0907718	−	0.995872i	$-0.528933\pi$
$991$	−5.71502	−0.181544	−0.0907718	+	0.995872i	$0.528933\pi$
$992$	0	0
$993$	2.06640	0.0655753
$994$	0	0
$995$	−78.5117	−2.48899
$996$	0	0
$997$	53.2192	1.68547	0.842735	−	0.538328i	$-0.180944\pi$
$997$	53.2192	1.68547	0.842735	+	0.538328i	$0.180944\pi$
$998$	0	0
$999$	−3.95358	−0.125086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2744.2.a.h.1.7	yes	12
4.3	odd	2		5488.2.a.w.1.6		12
7.6	odd	2	inner	2744.2.a.h.1.6	✓	12
28.27	even	2		5488.2.a.w.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2744.2.a.h.1.6	✓	12	7.6	odd	2	inner
2744.2.a.h.1.7	yes	12	1.1	even	1	trivial
5488.2.a.w.1.6		12	4.3	odd	2
5488.2.a.w.1.7		12	28.27	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:	\( N \)	\(=\)	\( 2744 = 2^{3} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2744.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0835475 q^{3} +3.81878 q^{5} -2.99302 q^{9} +O(q^{10})\)	\(q+0.0835475 q^{3} +3.81878 q^{5} -2.99302 q^{9} +1.01995 q^{11} -0.289833 q^{13} +0.319050 q^{15} -1.76652 q^{17} +6.29946 q^{19} -2.13396 q^{23} +9.58309 q^{25} -0.500702 q^{27} -7.26214 q^{29} +6.47176 q^{31} +0.0852143 q^{33} +7.89607 q^{37} -0.0242149 q^{39} +2.58764 q^{41} +7.62177 q^{43} -11.4297 q^{45} -1.07307 q^{47} -0.147588 q^{51} +3.62177 q^{53} +3.89496 q^{55} +0.526304 q^{57} +12.6864 q^{59} +7.39317 q^{61} -1.10681 q^{65} +10.1301 q^{67} -0.178287 q^{69} +3.55513 q^{71} -8.24913 q^{73} +0.800643 q^{75} +3.44126 q^{79} +8.93723 q^{81} -6.90920 q^{83} -6.74594 q^{85} -0.606734 q^{87} -17.8474 q^{89} +0.540699 q^{93} +24.0563 q^{95} -13.1744 q^{97} -3.05273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 22 q^{9}+O(q^{10}) \) 12 * q + 22 * q^9	\( 12 q + 22 q^{9} + 18 q^{11} + 28 q^{15} + 20 q^{23} + 14 q^{25} - 18 q^{29} - 18 q^{37} + 36 q^{39} + 10 q^{43} + 48 q^{51} - 38 q^{53} + 12 q^{57} + 8 q^{65} + 42 q^{67} + 56 q^{71} + 56 q^{79} + 52 q^{81} + 8 q^{85} - 48 q^{93} + 84 q^{95} + 90 q^{99}+O(q^{100}) \) 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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