LMFDB - Embedded newform 1380.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.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:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 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			1.1
Root			$4.73549$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -3.73549 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -3.73549 q^{7} +1.00000 q^{9} +4.84469 q^{11} -2.84469 q^{13} -1.00000 q^{15} +0.890804 q^{17} +6.84469 q^{19} -3.73549 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.890804 q^{29} +7.73549 q^{31} +4.84469 q^{33} +3.73549 q^{35} -1.95388 q^{37} -2.84469 q^{39} +12.3618 q^{41} -3.47098 q^{43} -1.00000 q^{45} -6.62629 q^{47} +6.95388 q^{49} +0.890804 q^{51} +12.3618 q^{53} -4.84469 q^{55} +6.84469 q^{57} +0.890804 q^{59} +8.62629 q^{61} -3.73549 q^{63} +2.84469 q^{65} +7.73549 q^{67} +1.00000 q^{69} -12.3618 q^{71} +16.5341 q^{73} +1.00000 q^{75} -18.0973 q^{77} +13.4710 q^{79} +1.00000 q^{81} -4.58018 q^{83} -0.890804 q^{85} -0.890804 q^{87} -15.1604 q^{89} +10.6263 q^{91} +7.73549 q^{93} -6.84469 q^{95} -17.2526 q^{97} +4.84469 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} - 3 q^{15} + 10 q^{19} + 2 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{39} + 8 q^{41} + 16 q^{43} - 3 q^{45} - 4 q^{47} + 13 q^{49} + 8 q^{53} - 4 q^{55} + 10 q^{57} + 10 q^{61} + 2 q^{63} - 2 q^{65} + 10 q^{67} + 3 q^{69} - 8 q^{71} + 18 q^{73} + 3 q^{75} - 12 q^{77} + 14 q^{79} + 3 q^{81} + 10 q^{83} + 2 q^{89} + 16 q^{91} + 10 q^{93} - 10 q^{95} - 20 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 - 3 * q^15 + 10 * q^19 + 2 * q^21 + 3 * q^23 + 3 * q^25 + 3 * q^27 + 10 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 + 2 * q^39 + 8 * q^41 + 16 * q^43 - 3 * q^45 - 4 * q^47 + 13 * q^49 + 8 * q^53 - 4 * q^55 + 10 * q^57 + 10 * q^61 + 2 * q^63 - 2 * q^65 + 10 * q^67 + 3 * q^69 - 8 * q^71 + 18 * q^73 + 3 * q^75 - 12 * q^77 + 14 * q^79 + 3 * q^81 + 10 * q^83 + 2 * q^89 + 16 * q^91 + 10 * q^93 - 10 * q^95 - 20 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.73549	−1.41188	−0.705941	−	0.708270i	$-0.749476\pi$
$7$	−3.73549	−1.41188	−0.705941	+	0.708270i	$0.749476\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.84469	1.46073	0.730364	−	0.683058i	$-0.239351\pi$
$11$	4.84469	1.46073	0.730364	+	0.683058i	$0.239351\pi$
$12$	0	0
$13$	−2.84469	−0.788974	−0.394487	−	0.918902i	$-0.629078\pi$
$13$	−2.84469	−0.788974	−0.394487	+	0.918902i	$0.629078\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.890804	0.216052	0.108026	−	0.994148i	$-0.465547\pi$
$17$	0.890804	0.216052	0.108026	+	0.994148i	$0.465547\pi$
$18$	0	0
$19$	6.84469	1.57028	0.785139	−	0.619319i	$-0.212591\pi$
$19$	6.84469	1.57028	0.785139	+	0.619319i	$0.212591\pi$
$20$	0	0
$21$	−3.73549	−0.815151
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.890804	−0.165418	−0.0827091	−	0.996574i	$-0.526357\pi$
$29$	−0.890804	−0.165418	−0.0827091	+	0.996574i	$0.526357\pi$
$30$	0	0
$31$	7.73549	1.38933	0.694667	−	0.719331i	$-0.255552\pi$
$31$	7.73549	1.38933	0.694667	+	0.719331i	$0.255552\pi$
$32$	0	0
$33$	4.84469	0.843352
$34$	0	0
$35$	3.73549	0.631413
$36$	0	0
$37$	−1.95388	−0.321216	−0.160608	−	0.987018i	$-0.551346\pi$
$37$	−1.95388	−0.321216	−0.160608	+	0.987018i	$0.551346\pi$
$38$	0	0
$39$	−2.84469	−0.455514
$40$	0	0
$41$	12.3618	1.93059	0.965293	−	0.261169i	$-0.0841080\pi$
$41$	12.3618	1.93059	0.965293	+	0.261169i	$0.0841080\pi$
$42$	0	0
$43$	−3.47098	−0.529319	−0.264660	−	0.964342i	$-0.585260\pi$
$43$	−3.47098	−0.529319	−0.264660	+	0.964342i	$0.585260\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−6.62629	−0.966544	−0.483272	−	0.875470i	$-0.660552\pi$
$47$	−6.62629	−0.966544	−0.483272	+	0.875470i	$0.660552\pi$
$48$	0	0
$49$	6.95388	0.993412
$50$	0	0
$51$	0.890804	0.124737
$52$	0	0
$53$	12.3618	1.69802	0.849011	−	0.528376i	$-0.177199\pi$
$53$	12.3618	1.69802	0.849011	+	0.528376i	$0.177199\pi$
$54$	0	0
$55$	−4.84469	−0.653257
$56$	0	0
$57$	6.84469	0.906601
$58$	0	0
$59$	0.890804	0.115973	0.0579864	−	0.998317i	$-0.481532\pi$
$59$	0.890804	0.115973	0.0579864	+	0.998317i	$0.481532\pi$
$60$	0	0
$61$	8.62629	1.10448	0.552242	−	0.833684i	$-0.313773\pi$
$61$	8.62629	1.10448	0.552242	+	0.833684i	$0.313773\pi$
$62$	0	0
$63$	−3.73549	−0.470627
$64$	0	0
$65$	2.84469	0.352840
$66$	0	0
$67$	7.73549	0.945040	0.472520	−	0.881320i	$-0.343344\pi$
$67$	7.73549	0.945040	0.472520	+	0.881320i	$0.343344\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−12.3618	−1.46707	−0.733537	−	0.679650i	$-0.762132\pi$
$71$	−12.3618	−1.46707	−0.733537	+	0.679650i	$0.762132\pi$
$72$	0	0
$73$	16.5341	1.93516	0.967582	−	0.252555i	$-0.0812709\pi$
$73$	16.5341	1.93516	0.967582	+	0.252555i	$0.0812709\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−18.0973	−2.06238
$78$	0	0
$79$	13.4710	1.51560	0.757802	−	0.652485i	$-0.226273\pi$
$79$	13.4710	1.51560	0.757802	+	0.652485i	$0.226273\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.58018	−0.502740	−0.251370	−	0.967891i	$-0.580881\pi$
$83$	−4.58018	−0.502740	−0.251370	+	0.967891i	$0.580881\pi$
$84$	0	0
$85$	−0.890804	−0.0966212
$86$	0	0
$87$	−0.890804	−0.0955042
$88$	0	0
$89$	−15.1604	−1.60699	−0.803497	−	0.595309i	$-0.797030\pi$
$89$	−15.1604	−1.60699	−0.803497	+	0.595309i	$0.797030\pi$
$90$	0	0
$91$	10.6263	1.11394
$92$	0	0
$93$	7.73549	0.802133
$94$	0	0
$95$	−6.84469	−0.702250
$96$	0	0
$97$	−17.2526	−1.75173	−0.875867	−	0.482552i	$-0.839710\pi$
$97$	−17.2526	−1.75173	−0.875867	+	0.482552i	$0.839710\pi$
$98$	0	0
$99$	4.84469	0.486909
$100$	0	0
$101$	−2.67241	−0.265915	−0.132957	−	0.991122i	$-0.542447\pi$
$101$	−2.67241	−0.265915	−0.132957	+	0.991122i	$0.542447\pi$
$102$	0	0
$103$	6.21839	0.612716	0.306358	−	0.951916i	$-0.400889\pi$
$103$	6.21839	0.612716	0.306358	+	0.951916i	$0.400889\pi$
$104$	0	0
$105$	3.73549	0.364546
$106$	0	0
$107$	−14.7986	−1.43063	−0.715316	−	0.698801i	$-0.753717\pi$
$107$	−14.7986	−1.43063	−0.715316	+	0.698801i	$0.753717\pi$
$108$	0	0
$109$	−12.5341	−1.20054	−0.600272	−	0.799796i	$-0.704941\pi$
$109$	−12.5341	−1.20054	−0.600272	+	0.799796i	$0.704941\pi$
$110$	0	0
$111$	−1.95388	−0.185454
$112$	0	0
$113$	12.3618	1.16290	0.581449	−	0.813583i	$-0.302486\pi$
$113$	12.3618	1.16290	0.581449	+	0.813583i	$0.302486\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−2.84469	−0.262991
$118$	0	0
$119$	−3.32759	−0.305040
$120$	0	0
$121$	12.4710	1.13373
$122$	0	0
$123$	12.3618	1.11462
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.7866	1.04590	0.522948	−	0.852365i	$-0.324832\pi$
$127$	11.7866	1.04590	0.522948	+	0.852365i	$0.324832\pi$
$128$	0	0
$129$	−3.47098	−0.305603
$130$	0	0
$131$	−17.4710	−1.52645	−0.763223	−	0.646135i	$-0.776384\pi$
$131$	−17.4710	−1.52645	−0.763223	+	0.646135i	$0.776384\pi$
$132$	0	0
$133$	−25.5683	−2.21705
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	15.1604	1.29524	0.647618	−	0.761965i	$-0.275765\pi$
$137$	15.1604	1.29524	0.647618	+	0.761965i	$0.275765\pi$
$138$	0	0
$139$	−5.51710	−0.467954	−0.233977	−	0.972242i	$-0.575174\pi$
$139$	−5.51710	−0.467954	−0.233977	+	0.972242i	$0.575174\pi$
$140$	0	0
$141$	−6.62629	−0.558035
$142$	0	0
$143$	−13.7816	−1.15248
$144$	0	0
$145$	0.890804	0.0739772
$146$	0	0
$147$	6.95388	0.573547
$148$	0	0
$149$	1.15531	0.0946470	0.0473235	−	0.998880i	$-0.484931\pi$
$149$	1.15531	0.0946470	0.0473235	+	0.998880i	$0.484931\pi$
$150$	0	0
$151$	−15.4710	−1.25901	−0.629505	−	0.776996i	$-0.716742\pi$
$151$	−15.4710	−1.25901	−0.629505	+	0.776996i	$0.716742\pi$
$152$	0	0
$153$	0.890804	0.0720172
$154$	0	0
$155$	−7.73549	−0.621329
$156$	0	0
$157$	18.6774	1.49062	0.745311	−	0.666717i	$-0.232301\pi$
$157$	18.6774	1.49062	0.745311	+	0.666717i	$0.232301\pi$
$158$	0	0
$159$	12.3618	0.980353
$160$	0	0
$161$	−3.73549	−0.294398
$162$	0	0
$163$	0.0922364	0.00722451	0.00361226	−	0.999993i	$-0.498850\pi$
$163$	0.0922364	0.00722451	0.00361226	+	0.999993i	$0.498850\pi$
$164$	0	0
$165$	−4.84469	−0.377158
$166$	0	0
$167$	−5.37371	−0.415830	−0.207915	−	0.978147i	$-0.566668\pi$
$167$	−5.37371	−0.415830	−0.207915	+	0.978147i	$0.566668\pi$
$168$	0	0
$169$	−4.90776	−0.377520
$170$	0	0
$171$	6.84469	0.523426
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−3.73549	−0.282376
$176$	0	0
$177$	0.890804	0.0669569
$178$	0	0
$179$	−9.56322	−0.714788	−0.357394	−	0.933954i	$-0.616335\pi$
$179$	−9.56322	−0.714788	−0.357394	+	0.933954i	$0.616335\pi$
$180$	0	0
$181$	−7.16035	−0.532225	−0.266112	−	0.963942i	$-0.585739\pi$
$181$	−7.16035	−0.532225	−0.266112	+	0.963942i	$0.585739\pi$
$182$	0	0
$183$	8.62629	0.637674
$184$	0	0
$185$	1.95388	0.143652
$186$	0	0
$187$	4.31566	0.315593
$188$	0	0
$189$	−3.73549	−0.271717
$190$	0	0
$191$	14.0050	1.01337	0.506684	−	0.862132i	$-0.330871\pi$
$191$	14.0050	1.01337	0.506684	+	0.862132i	$0.330871\pi$
$192$	0	0
$193$	26.7236	1.92360	0.961802	−	0.273745i	$-0.0882626\pi$
$193$	26.7236	1.92360	0.961802	+	0.273745i	$0.0882626\pi$
$194$	0	0
$195$	2.84469	0.203712
$196$	0	0
$197$	9.16035	0.652648	0.326324	−	0.945258i	$-0.394190\pi$
$197$	9.16035	0.652648	0.326324	+	0.945258i	$0.394190\pi$
$198$	0	0
$199$	−0.310629	−0.0220199	−0.0110099	−	0.999939i	$-0.503505\pi$
$199$	−0.310629	−0.0220199	−0.0110099	+	0.999939i	$0.503505\pi$
$200$	0	0
$201$	7.73549	0.545619
$202$	0	0
$203$	3.32759	0.233551
$204$	0	0
$205$	−12.3618	−0.863384
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	33.1604	2.29375
$210$	0	0
$211$	5.95388	0.409882	0.204941	−	0.978774i	$-0.434300\pi$
$211$	5.95388	0.409882	0.204941	+	0.978774i	$0.434300\pi$
$212$	0	0
$213$	−12.3618	−0.847015
$214$	0	0
$215$	3.47098	0.236719
$216$	0	0
$217$	−28.8958	−1.96158
$218$	0	0
$219$	16.5341	1.11727
$220$	0	0
$221$	−2.53406	−0.170459
$222$	0	0
$223$	−11.9078	−0.797403	−0.398701	−	0.917081i	$-0.630539\pi$
$223$	−11.9078	−0.797403	−0.398701	+	0.917081i	$0.630539\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	7.37874	0.489744	0.244872	−	0.969555i	$-0.421254\pi$
$227$	7.37874	0.489744	0.244872	+	0.969555i	$0.421254\pi$
$228$	0	0
$229$	23.1604	1.53048	0.765240	−	0.643746i	$-0.222621\pi$
$229$	23.1604	1.53048	0.765240	+	0.643746i	$0.222621\pi$
$230$	0	0
$231$	−18.0973	−1.19071
$232$	0	0
$233$	−22.9420	−1.50298	−0.751489	−	0.659746i	$-0.770664\pi$
$233$	−22.9420	−1.50298	−0.751489	+	0.659746i	$0.770664\pi$
$234$	0	0
$235$	6.62629	0.432252
$236$	0	0
$237$	13.4710	0.875034
$238$	0	0
$239$	0.361783	0.0234018	0.0117009	−	0.999932i	$-0.496275\pi$
$239$	0.361783	0.0234018	0.0117009	+	0.999932i	$0.496275\pi$
$240$	0	0
$241$	28.5341	1.83804	0.919020	−	0.394211i	$-0.128982\pi$
$241$	28.5341	1.83804	0.919020	+	0.394211i	$0.128982\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.95388	−0.444267
$246$	0	0
$247$	−19.4710	−1.23891
$248$	0	0
$249$	−4.58018	−0.290257
$250$	0	0
$251$	−30.8497	−1.94722	−0.973609	−	0.228224i	$-0.926708\pi$
$251$	−30.8497	−1.94722	−0.973609	+	0.228224i	$0.926708\pi$
$252$	0	0
$253$	4.84469	0.304583
$254$	0	0
$255$	−0.890804	−0.0557843
$256$	0	0
$257$	−12.7524	−0.795476	−0.397738	−	0.917499i	$-0.630205\pi$
$257$	−12.7524	−0.795476	−0.397738	+	0.917499i	$0.630205\pi$
$258$	0	0
$259$	7.29870	0.453519
$260$	0	0
$261$	−0.890804	−0.0551394
$262$	0	0
$263$	6.36178	0.392284	0.196142	−	0.980575i	$-0.437159\pi$
$263$	6.36178	0.392284	0.196142	+	0.980575i	$0.437159\pi$
$264$	0	0
$265$	−12.3618	−0.759378
$266$	0	0
$267$	−15.1604	−0.927798
$268$	0	0
$269$	10.5802	0.645085	0.322542	−	0.946555i	$-0.395463\pi$
$269$	10.5802	0.645085	0.322542	+	0.946555i	$0.395463\pi$
$270$	0	0
$271$	−4.26451	−0.259051	−0.129525	−	0.991576i	$-0.541345\pi$
$271$	−4.26451	−0.259051	−0.129525	+	0.991576i	$0.541345\pi$
$272$	0	0
$273$	10.6263	0.643133
$274$	0	0
$275$	4.84469	0.292146
$276$	0	0
$277$	−29.3787	−1.76520	−0.882599	−	0.470127i	$-0.844208\pi$
$277$	−29.3787	−1.76520	−0.882599	+	0.470127i	$0.844208\pi$
$278$	0	0
$279$	7.73549	0.463112
$280$	0	0
$281$	−18.7524	−1.11868	−0.559339	−	0.828939i	$-0.688945\pi$
$281$	−18.7524	−1.11868	−0.559339	+	0.828939i	$0.688945\pi$
$282$	0	0
$283$	−12.8958	−0.766578	−0.383289	−	0.923628i	$-0.625209\pi$
$283$	−12.8958	−0.766578	−0.383289	+	0.923628i	$0.625209\pi$
$284$	0	0
$285$	−6.84469	−0.405444
$286$	0	0
$287$	−46.1773	−2.72576
$288$	0	0
$289$	−16.2065	−0.953322
$290$	0	0
$291$	−17.2526	−1.01136
$292$	0	0
$293$	8.79857	0.514018	0.257009	−	0.966409i	$-0.417263\pi$
$293$	8.79857	0.514018	0.257009	+	0.966409i	$0.417263\pi$
$294$	0	0
$295$	−0.890804	−0.0518646
$296$	0	0
$297$	4.84469	0.281117
$298$	0	0
$299$	−2.84469	−0.164512
$300$	0	0
$301$	12.9658	0.747337
$302$	0	0
$303$	−2.67241	−0.153526
$304$	0	0
$305$	−8.62629	−0.493940
$306$	0	0
$307$	10.0050	0.571018	0.285509	−	0.958376i	$-0.407837\pi$
$307$	10.0050	0.571018	0.285509	+	0.958376i	$0.407837\pi$
$308$	0	0
$309$	6.21839	0.353752
$310$	0	0
$311$	−18.5290	−1.05068	−0.525342	−	0.850891i	$-0.676063\pi$
$311$	−18.5290	−1.05068	−0.525342	+	0.850891i	$0.676063\pi$
$312$	0	0
$313$	2.39067	0.135128	0.0675642	−	0.997715i	$-0.478477\pi$
$313$	2.39067	0.135128	0.0675642	+	0.997715i	$0.478477\pi$
$314$	0	0
$315$	3.73549	0.210471
$316$	0	0
$317$	−16.3157	−0.916379	−0.458190	−	0.888855i	$-0.651502\pi$
$317$	−16.3157	−0.916379	−0.458190	+	0.888855i	$0.651502\pi$
$318$	0	0
$319$	−4.31566	−0.241631
$320$	0	0
$321$	−14.7986	−0.825975
$322$	0	0
$323$	6.09727	0.339261
$324$	0	0
$325$	−2.84469	−0.157795
$326$	0	0
$327$	−12.5341	−0.693135
$328$	0	0
$329$	24.7524	1.36465
$330$	0	0
$331$	−24.8958	−1.36840	−0.684200	−	0.729295i	$-0.739848\pi$
$331$	−24.8958	−1.36840	−0.684200	+	0.729295i	$0.739848\pi$
$332$	0	0
$333$	−1.95388	−0.107072
$334$	0	0
$335$	−7.73549	−0.422635
$336$	0	0
$337$	−27.4710	−1.49644	−0.748220	−	0.663451i	$-0.769091\pi$
$337$	−27.4710	−1.49644	−0.748220	+	0.663451i	$0.769091\pi$
$338$	0	0
$339$	12.3618	0.671400
$340$	0	0
$341$	37.4760	2.02944
$342$	0	0
$343$	0.172274	0.00930193
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−1.25259	−0.0672424	−0.0336212	−	0.999435i	$-0.510704\pi$
$347$	−1.25259	−0.0672424	−0.0336212	+	0.999435i	$0.510704\pi$
$348$	0	0
$349$	9.64325	0.516192	0.258096	−	0.966119i	$-0.416905\pi$
$349$	9.64325	0.516192	0.258096	+	0.966119i	$0.416905\pi$
$350$	0	0
$351$	−2.84469	−0.151838
$352$	0	0
$353$	−12.7524	−0.678744	−0.339372	−	0.940652i	$-0.610215\pi$
$353$	−12.7524	−0.678744	−0.339372	+	0.940652i	$0.610215\pi$
$354$	0	0
$355$	12.3618	0.656095
$356$	0	0
$357$	−3.32759	−0.176115
$358$	0	0
$359$	−6.62629	−0.349722	−0.174861	−	0.984593i	$-0.555948\pi$
$359$	−6.62629	−0.349722	−0.174861	+	0.984593i	$0.555948\pi$
$360$	0	0
$361$	27.8497	1.46577
$362$	0	0
$363$	12.4710	0.654557
$364$	0	0
$365$	−16.5341	−0.865432
$366$	0	0
$367$	−9.60933	−0.501603	−0.250802	−	0.968039i	$-0.580694\pi$
$367$	−9.60933	−0.501603	−0.250802	+	0.968039i	$0.580694\pi$
$368$	0	0
$369$	12.3618	0.643529
$370$	0	0
$371$	−46.1773	−2.39741
$372$	0	0
$373$	29.8839	1.54733	0.773665	−	0.633595i	$-0.218421\pi$
$373$	29.8839	1.54733	0.773665	+	0.633595i	$0.218421\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	2.53406	0.130511
$378$	0	0
$379$	2.52902	0.129907	0.0649535	−	0.997888i	$-0.479310\pi$
$379$	2.52902	0.129907	0.0649535	+	0.997888i	$0.479310\pi$
$380$	0	0
$381$	11.7866	0.603848
$382$	0	0
$383$	25.2115	1.28825	0.644124	−	0.764921i	$-0.277222\pi$
$383$	25.2115	1.28825	0.644124	+	0.764921i	$0.277222\pi$
$384$	0	0
$385$	18.0973	0.922322
$386$	0	0
$387$	−3.47098	−0.176440
$388$	0	0
$389$	35.0681	1.77802	0.889012	−	0.457884i	$-0.151392\pi$
$389$	35.0681	1.77802	0.889012	+	0.457884i	$0.151392\pi$
$390$	0	0
$391$	0.890804	0.0450499
$392$	0	0
$393$	−17.4710	−0.881294
$394$	0	0
$395$	−13.4710	−0.677799
$396$	0	0
$397$	8.84972	0.444155	0.222077	−	0.975029i	$-0.428716\pi$
$397$	8.84972	0.444155	0.222077	+	0.975029i	$0.428716\pi$
$398$	0	0
$399$	−25.5683	−1.28001
$400$	0	0
$401$	−23.0681	−1.15197	−0.575983	−	0.817461i	$-0.695381\pi$
$401$	−23.0681	−1.15197	−0.575983	+	0.817461i	$0.695381\pi$
$402$	0	0
$403$	−22.0050	−1.09615
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−9.46594	−0.469209
$408$	0	0
$409$	5.29870	0.262004	0.131002	−	0.991382i	$-0.458181\pi$
$409$	5.29870	0.262004	0.131002	+	0.991382i	$0.458181\pi$
$410$	0	0
$411$	15.1604	0.747805
$412$	0	0
$413$	−3.32759	−0.163740
$414$	0	0
$415$	4.58018	0.224832
$416$	0	0
$417$	−5.51710	−0.270173
$418$	0	0
$419$	2.53406	0.123797	0.0618984	−	0.998082i	$-0.480285\pi$
$419$	2.53406	0.123797	0.0618984	+	0.998082i	$0.480285\pi$
$420$	0	0
$421$	22.4079	1.09209	0.546047	−	0.837754i	$-0.316132\pi$
$421$	22.4079	1.09209	0.546047	+	0.837754i	$0.316132\pi$
$422$	0	0
$423$	−6.62629	−0.322181
$424$	0	0
$425$	0.890804	0.0432103
$426$	0	0
$427$	−32.2234	−1.55940
$428$	0	0
$429$	−13.7816	−0.665382
$430$	0	0
$431$	31.5733	1.52083	0.760416	−	0.649436i	$-0.224995\pi$
$431$	31.5733	1.52083	0.760416	+	0.649436i	$0.224995\pi$
$432$	0	0
$433$	7.20647	0.346321	0.173160	−	0.984894i	$-0.444602\pi$
$433$	7.20647	0.346321	0.173160	+	0.984894i	$0.444602\pi$
$434$	0	0
$435$	0.890804	0.0427108
$436$	0	0
$437$	6.84469	0.327426
$438$	0	0
$439$	−4.72357	−0.225443	−0.112722	−	0.993627i	$-0.535957\pi$
$439$	−4.72357	−0.225443	−0.112722	+	0.993627i	$0.535957\pi$
$440$	0	0
$441$	6.95388	0.331137
$442$	0	0
$443$	−7.87888	−0.374337	−0.187168	−	0.982328i	$-0.559931\pi$
$443$	−7.87888	−0.374337	−0.187168	+	0.982328i	$0.559931\pi$
$444$	0	0
$445$	15.1604	0.718670
$446$	0	0
$447$	1.15531	0.0546445
$448$	0	0
$449$	−4.70633	−0.222105	−0.111053	−	0.993815i	$-0.535422\pi$
$449$	−4.70633	−0.222105	−0.111053	+	0.993815i	$0.535422\pi$
$450$	0	0
$451$	59.8890	2.82006
$452$	0	0
$453$	−15.4710	−0.726890
$454$	0	0
$455$	−10.6263	−0.498168
$456$	0	0
$457$	−6.04612	−0.282825	−0.141413	−	0.989951i	$-0.545164\pi$
$457$	−6.04612	−0.282825	−0.141413	+	0.989951i	$0.545164\pi$
$458$	0	0
$459$	0.890804	0.0415792
$460$	0	0
$461$	−19.2526	−0.896682	−0.448341	−	0.893863i	$-0.647985\pi$
$461$	−19.2526	−0.896682	−0.448341	+	0.893863i	$0.647985\pi$
$462$	0	0
$463$	−26.4418	−1.22886	−0.614428	−	0.788973i	$-0.710613\pi$
$463$	−26.4418	−1.22886	−0.614428	+	0.788973i	$0.710613\pi$
$464$	0	0
$465$	−7.73549	−0.358725
$466$	0	0
$467$	9.20143	0.425792	0.212896	−	0.977075i	$-0.431711\pi$
$467$	9.20143	0.425792	0.212896	+	0.977075i	$0.431711\pi$
$468$	0	0
$469$	−28.8958	−1.33429
$470$	0	0
$471$	18.6774	0.860611
$472$	0	0
$473$	−16.8158	−0.773191
$474$	0	0
$475$	6.84469	0.314056
$476$	0	0
$477$	12.3618	0.566007
$478$	0	0
$479$	−12.7524	−0.582674	−0.291337	−	0.956620i	$-0.594100\pi$
$479$	−12.7524	−0.582674	−0.291337	+	0.956620i	$0.594100\pi$
$480$	0	0
$481$	5.55818	0.253431
$482$	0	0
$483$	−3.73549	−0.169971
$484$	0	0
$485$	17.2526	0.783400
$486$	0	0
$487$	−30.5341	−1.38363	−0.691815	−	0.722075i	$-0.743189\pi$
$487$	−30.5341	−1.38363	−0.691815	+	0.722075i	$0.743189\pi$
$488$	0	0
$489$	0.0922364	0.00417107
$490$	0	0
$491$	−2.67241	−0.120604	−0.0603021	−	0.998180i	$-0.519206\pi$
$491$	−2.67241	−0.120604	−0.0603021	+	0.998180i	$0.519206\pi$
$492$	0	0
$493$	−0.793532	−0.0357389
$494$	0	0
$495$	−4.84469	−0.217752
$496$	0	0
$497$	46.1773	2.07134
$498$	0	0
$499$	−20.8036	−0.931297	−0.465649	−	0.884970i	$-0.654179\pi$
$499$	−20.8036	−0.931297	−0.465649	+	0.884970i	$0.654179\pi$
$500$	0	0
$501$	−5.37371	−0.240080
$502$	0	0
$503$	3.32759	0.148370	0.0741849	−	0.997245i	$-0.476364\pi$
$503$	3.32759	0.148370	0.0741849	+	0.997245i	$0.476364\pi$
$504$	0	0
$505$	2.67241	0.118921
$506$	0	0
$507$	−4.90776	−0.217961
$508$	0	0
$509$	16.2184	0.718868	0.359434	−	0.933171i	$-0.382970\pi$
$509$	16.2184	0.718868	0.359434	+	0.933171i	$0.382970\pi$
$510$	0	0
$511$	−61.7628	−2.73223
$512$	0	0
$513$	6.84469	0.302200
$514$	0	0
$515$	−6.21839	−0.274015
$516$	0	0
$517$	−32.1023	−1.41186
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.18923	0.139723	0.0698614	−	0.997557i	$-0.477744\pi$
$521$	3.18923	0.139723	0.0698614	+	0.997557i	$0.477744\pi$
$522$	0	0
$523$	23.8155	1.04138	0.520690	−	0.853746i	$-0.325675\pi$
$523$	23.8155	1.04138	0.520690	+	0.853746i	$0.325675\pi$
$524$	0	0
$525$	−3.73549	−0.163030
$526$	0	0
$527$	6.89080	0.300168
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.890804	0.0386576
$532$	0	0
$533$	−35.1654	−1.52318
$534$	0	0
$535$	14.7986	0.639798
$536$	0	0
$537$	−9.56322	−0.412683
$538$	0	0
$539$	33.6894	1.45110
$540$	0	0
$541$	15.7816	0.678504	0.339252	−	0.940695i	$-0.389826\pi$
$541$	15.7816	0.678504	0.339252	+	0.940695i	$0.389826\pi$
$542$	0	0
$543$	−7.16035	−0.307280
$544$	0	0
$545$	12.5341	0.536900
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	8.62629	0.368161
$550$	0	0
$551$	−6.09727	−0.259753
$552$	0	0
$553$	−50.3207	−2.13985
$554$	0	0
$555$	1.95388	0.0829377
$556$	0	0
$557$	31.7405	1.34489	0.672445	−	0.740147i	$-0.265244\pi$
$557$	31.7405	1.34489	0.672445	+	0.740147i	$0.265244\pi$
$558$	0	0
$559$	9.87384	0.417619
$560$	0	0
$561$	4.31566	0.182208
$562$	0	0
$563$	21.9250	0.924029	0.462014	−	0.886872i	$-0.347127\pi$
$563$	21.9250	0.924029	0.462014	+	0.886872i	$0.347127\pi$
$564$	0	0
$565$	−12.3618	−0.520064
$566$	0	0
$567$	−3.73549	−0.156876
$568$	0	0
$569$	−14.6313	−0.613377	−0.306689	−	0.951810i	$-0.599221\pi$
$569$	−14.6313	−0.613377	−0.306689	+	0.951810i	$0.599221\pi$
$570$	0	0
$571$	1.24755	0.0522084	0.0261042	−	0.999659i	$-0.491690\pi$
$571$	1.24755	0.0522084	0.0261042	+	0.999659i	$0.491690\pi$
$572$	0	0
$573$	14.0050	0.585069
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	18.0922	0.753190	0.376595	−	0.926378i	$-0.377095\pi$
$577$	18.0922	0.753190	0.376595	+	0.926378i	$0.377095\pi$
$578$	0	0
$579$	26.7236	1.11059
$580$	0	0
$581$	17.1092	0.709809
$582$	0	0
$583$	59.8890	2.48035
$584$	0	0
$585$	2.84469	0.117613
$586$	0	0
$587$	12.5290	0.517128	0.258564	−	0.965994i	$-0.416751\pi$
$587$	12.5290	0.517128	0.258564	+	0.965994i	$0.416751\pi$
$588$	0	0
$589$	52.9470	2.18164
$590$	0	0
$591$	9.16035	0.376806
$592$	0	0
$593$	−24.9470	−1.02445	−0.512225	−	0.858851i	$-0.671179\pi$
$593$	−24.9470	−1.02445	−0.512225	+	0.858851i	$0.671179\pi$
$594$	0	0
$595$	3.32759	0.136418
$596$	0	0
$597$	−0.310629	−0.0127132
$598$	0	0
$599$	−34.5391	−1.41123	−0.705615	−	0.708596i	$-0.749329\pi$
$599$	−34.5391	−1.41123	−0.705615	+	0.708596i	$0.749329\pi$
$600$	0	0
$601$	37.9300	1.54720	0.773599	−	0.633675i	$-0.218454\pi$
$601$	37.9300	1.54720	0.773599	+	0.633675i	$0.218454\pi$
$602$	0	0
$603$	7.73549	0.315013
$604$	0	0
$605$	−12.4710	−0.507017
$606$	0	0
$607$	26.0973	1.05926	0.529628	−	0.848230i	$-0.322332\pi$
$607$	26.0973	1.05926	0.529628	+	0.848230i	$0.322332\pi$
$608$	0	0
$609$	3.32759	0.134841
$610$	0	0
$611$	18.8497	0.762578
$612$	0	0
$613$	2.84972	0.115099	0.0575496	−	0.998343i	$-0.481671\pi$
$613$	2.84972	0.115099	0.0575496	+	0.998343i	$0.481671\pi$
$614$	0	0
$615$	−12.3618	−0.498475
$616$	0	0
$617$	−9.52213	−0.383347	−0.191673	−	0.981459i	$-0.561391\pi$
$617$	−9.52213	−0.383347	−0.191673	+	0.981459i	$0.561391\pi$
$618$	0	0
$619$	−23.9762	−0.963683	−0.481841	−	0.876258i	$-0.660032\pi$
$619$	−23.9762	−0.963683	−0.481841	+	0.876258i	$0.660032\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	56.6313	2.26889
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	33.1604	1.32430
$628$	0	0
$629$	−1.74053	−0.0693993
$630$	0	0
$631$	−17.8789	−0.711747	−0.355873	−	0.934534i	$-0.615817\pi$
$631$	−17.8789	−0.711747	−0.355873	+	0.934534i	$0.615817\pi$
$632$	0	0
$633$	5.95388	0.236646
$634$	0	0
$635$	−11.7866	−0.467739
$636$	0	0
$637$	−19.7816	−0.783776
$638$	0	0
$639$	−12.3618	−0.489025
$640$	0	0
$641$	2.93692	0.116001	0.0580007	−	0.998317i	$-0.481527\pi$
$641$	2.93692	0.116001	0.0580007	+	0.998317i	$0.481527\pi$
$642$	0	0
$643$	−0.895840	−0.0353285	−0.0176642	−	0.999844i	$-0.505623\pi$
$643$	−0.895840	−0.0353285	−0.0176642	+	0.999844i	$0.505623\pi$
$644$	0	0
$645$	3.47098	0.136670
$646$	0	0
$647$	−21.4659	−0.843913	−0.421957	−	0.906616i	$-0.638657\pi$
$647$	−21.4659	−0.843913	−0.421957	+	0.906616i	$0.638657\pi$
$648$	0	0
$649$	4.31566	0.169405
$650$	0	0
$651$	−28.8958	−1.13252
$652$	0	0
$653$	−4.84469	−0.189587	−0.0947936	−	0.995497i	$-0.530219\pi$
$653$	−4.84469	−0.189587	−0.0947936	+	0.995497i	$0.530219\pi$
$654$	0	0
$655$	17.4710	0.682648
$656$	0	0
$657$	16.5341	0.645055
$658$	0	0
$659$	−18.0973	−0.704970	−0.352485	−	0.935818i	$-0.614663\pi$
$659$	−18.0973	−0.704970	−0.352485	+	0.935818i	$0.614663\pi$
$660$	0	0
$661$	0.747413	0.0290710	0.0145355	−	0.999894i	$-0.495373\pi$
$661$	0.747413	0.0290710	0.0145355	+	0.999894i	$0.495373\pi$
$662$	0	0
$663$	−2.53406	−0.0984146
$664$	0	0
$665$	25.5683	0.991494
$666$	0	0
$667$	−0.890804	−0.0344921
$668$	0	0
$669$	−11.9078	−0.460381
$670$	0	0
$671$	41.7917	1.61335
$672$	0	0
$673$	−19.1892	−0.739691	−0.369845	−	0.929093i	$-0.620589\pi$
$673$	−19.1892	−0.739691	−0.369845	+	0.929093i	$0.620589\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−4.17731	−0.160547	−0.0802735	−	0.996773i	$-0.525579\pi$
$677$	−4.17731	−0.160547	−0.0802735	+	0.996773i	$0.525579\pi$
$678$	0	0
$679$	64.4469	2.47324
$680$	0	0
$681$	7.37874	0.282754
$682$	0	0
$683$	−26.0050	−0.995055	−0.497528	−	0.867448i	$-0.665759\pi$
$683$	−26.0050	−0.995055	−0.497528	+	0.867448i	$0.665759\pi$
$684$	0	0
$685$	−15.1604	−0.579247
$686$	0	0
$687$	23.1604	0.883622
$688$	0	0
$689$	−35.1654	−1.33969
$690$	0	0
$691$	26.8497	1.02141	0.510706	−	0.859756i	$-0.329384\pi$
$691$	26.8497	1.02141	0.510706	+	0.859756i	$0.329384\pi$
$692$	0	0
$693$	−18.0973	−0.687459
$694$	0	0
$695$	5.51710	0.209275
$696$	0	0
$697$	11.0119	0.417106
$698$	0	0
$699$	−22.9420	−0.867745
$700$	0	0
$701$	11.9027	0.449560	0.224780	−	0.974410i	$-0.427834\pi$
$701$	11.9027	0.449560	0.224780	+	0.974410i	$0.427834\pi$
$702$	0	0
$703$	−13.3737	−0.504399
$704$	0	0
$705$	6.62629	0.249561
$706$	0	0
$707$	9.98277	0.375441
$708$	0	0
$709$	−39.3737	−1.47871	−0.739355	−	0.673315i	$-0.764870\pi$
$709$	−39.3737	−1.47871	−0.739355	+	0.673315i	$0.764870\pi$
$710$	0	0
$711$	13.4710	0.505201
$712$	0	0
$713$	7.73549	0.289696
$714$	0	0
$715$	13.7816	0.515403
$716$	0	0
$717$	0.361783	0.0135110
$718$	0	0
$719$	23.6382	0.881557	0.440778	−	0.897616i	$-0.354702\pi$
$719$	23.6382	0.881557	0.440778	+	0.897616i	$0.354702\pi$
$720$	0	0
$721$	−23.2287	−0.865083
$722$	0	0
$723$	28.5341	1.06119
$724$	0	0
$725$	−0.890804	−0.0330836
$726$	0	0
$727$	12.5513	0.465502	0.232751	−	0.972536i	$-0.425227\pi$
$727$	12.5513	0.465502	0.232751	+	0.972536i	$0.425227\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.09196	−0.114360
$732$	0	0
$733$	39.1142	1.44472	0.722359	−	0.691519i	$-0.243058\pi$
$733$	39.1142	1.44472	0.722359	+	0.691519i	$0.243058\pi$
$734$	0	0
$735$	−6.95388	−0.256498
$736$	0	0
$737$	37.4760	1.38045
$738$	0	0
$739$	35.7456	1.31492	0.657461	−	0.753489i	$-0.271630\pi$
$739$	35.7456	1.31492	0.657461	+	0.753489i	$0.271630\pi$
$740$	0	0
$741$	−19.4710	−0.715284
$742$	0	0
$743$	20.4368	0.749753	0.374876	−	0.927075i	$-0.377685\pi$
$743$	20.4368	0.749753	0.374876	+	0.927075i	$0.377685\pi$
$744$	0	0
$745$	−1.15531	−0.0423274
$746$	0	0
$747$	−4.58018	−0.167580
$748$	0	0
$749$	55.2799	2.01988
$750$	0	0
$751$	−0.534057	−0.0194880	−0.00974401	−	0.999953i	$-0.503102\pi$
$751$	−0.534057	−0.0194880	−0.00974401	+	0.999953i	$0.503102\pi$
$752$	0	0
$753$	−30.8497	−1.12423
$754$	0	0
$755$	15.4710	0.563047
$756$	0	0
$757$	−29.5171	−1.07282	−0.536409	−	0.843958i	$-0.680219\pi$
$757$	−29.5171	−1.07282	−0.536409	+	0.843958i	$0.680219\pi$
$758$	0	0
$759$	4.84469	0.175851
$760$	0	0
$761$	38.1434	1.38270	0.691348	−	0.722522i	$-0.257017\pi$
$761$	38.1434	1.38270	0.691348	+	0.722522i	$0.257017\pi$
$762$	0	0
$763$	46.8208	1.69503
$764$	0	0
$765$	−0.890804	−0.0322071
$766$	0	0
$767$	−2.53406	−0.0914995
$768$	0	0
$769$	20.8786	0.752902	0.376451	−	0.926437i	$-0.377144\pi$
$769$	20.8786	0.752902	0.376451	+	0.926437i	$0.377144\pi$
$770$	0	0
$771$	−12.7524	−0.459268
$772$	0	0
$773$	8.31063	0.298913	0.149456	−	0.988768i	$-0.452248\pi$
$773$	8.31063	0.298913	0.149456	+	0.988768i	$0.452248\pi$
$774$	0	0
$775$	7.73549	0.277867
$776$	0	0
$777$	7.29870	0.261840
$778$	0	0
$779$	84.6125	3.03156
$780$	0	0
$781$	−59.8890	−2.14300
$782$	0	0
$783$	−0.890804	−0.0318347
$784$	0	0
$785$	−18.6774	−0.666627
$786$	0	0
$787$	−1.95388	−0.0696484	−0.0348242	−	0.999393i	$-0.511087\pi$
$787$	−1.95388	−0.0696484	−0.0348242	+	0.999393i	$0.511087\pi$
$788$	0	0
$789$	6.36178	0.226485
$790$	0	0
$791$	−46.1773	−1.64188
$792$	0	0
$793$	−24.5391	−0.871409
$794$	0	0
$795$	−12.3618	−0.438427
$796$	0	0
$797$	−16.1773	−0.573030	−0.286515	−	0.958076i	$-0.592497\pi$
$797$	−16.1773	−0.573030	−0.286515	+	0.958076i	$0.592497\pi$
$798$	0	0
$799$	−5.90273	−0.208823
$800$	0	0
$801$	−15.1604	−0.535665
$802$	0	0
$803$	80.1023	2.82675
$804$	0	0
$805$	3.73549	0.131659
$806$	0	0
$807$	10.5802	0.372440
$808$	0	0
$809$	−26.8670	−0.944592	−0.472296	−	0.881440i	$-0.656575\pi$
$809$	−26.8670	−0.944592	−0.472296	+	0.881440i	$0.656575\pi$
$810$	0	0
$811$	2.13835	0.0750878	0.0375439	−	0.999295i	$-0.488047\pi$
$811$	2.13835	0.0750878	0.0375439	+	0.999295i	$0.488047\pi$
$812$	0	0
$813$	−4.26451	−0.149563
$814$	0	0
$815$	−0.0922364	−0.00323090
$816$	0	0
$817$	−23.7578	−0.831179
$818$	0	0
$819$	10.6263	0.371313
$820$	0	0
$821$	15.1604	0.529100	0.264550	−	0.964372i	$-0.414777\pi$
$821$	15.1604	0.529100	0.264550	+	0.964372i	$0.414777\pi$
$822$	0	0
$823$	−23.1264	−0.806137	−0.403068	−	0.915170i	$-0.632056\pi$
$823$	−23.1264	−0.806137	−0.403068	+	0.915170i	$0.632056\pi$
$824$	0	0
$825$	4.84469	0.168670
$826$	0	0
$827$	47.4299	1.64930	0.824650	−	0.565644i	$-0.191372\pi$
$827$	47.4299	1.64930	0.824650	+	0.565644i	$0.191372\pi$
$828$	0	0
$829$	−30.1723	−1.04793	−0.523963	−	0.851741i	$-0.675547\pi$
$829$	−30.1723	−1.04793	−0.523963	+	0.851741i	$0.675547\pi$
$830$	0	0
$831$	−29.3787	−1.01914
$832$	0	0
$833$	6.19454	0.214628
$834$	0	0
$835$	5.37371	0.185965
$836$	0	0
$837$	7.73549	0.267378
$838$	0	0
$839$	18.8497	0.650765	0.325382	−	0.945583i	$-0.394507\pi$
$839$	18.8497	0.650765	0.325382	+	0.945583i	$0.394507\pi$
$840$	0	0
$841$	−28.2065	−0.972637
$842$	0	0
$843$	−18.7524	−0.645869
$844$	0	0
$845$	4.90776	0.168832
$846$	0	0
$847$	−46.5852	−1.60069
$848$	0	0
$849$	−12.8958	−0.442584
$850$	0	0
$851$	−1.95388	−0.0669782
$852$	0	0
$853$	29.7578	1.01889	0.509443	−	0.860504i	$-0.329851\pi$
$853$	29.7578	1.01889	0.509443	+	0.860504i	$0.329851\pi$
$854$	0	0
$855$	−6.84469	−0.234083
$856$	0	0
$857$	−15.2865	−0.522177	−0.261089	−	0.965315i	$-0.584081\pi$
$857$	−15.2865	−0.522177	−0.261089	+	0.965315i	$0.584081\pi$
$858$	0	0
$859$	33.5171	1.14359	0.571794	−	0.820397i	$-0.306248\pi$
$859$	33.5171	1.14359	0.571794	+	0.820397i	$0.306248\pi$
$860$	0	0
$861$	−46.1773	−1.57372
$862$	0	0
$863$	−31.0443	−1.05676	−0.528380	−	0.849008i	$-0.677200\pi$
$863$	−31.0443	−1.05676	−0.528380	+	0.849008i	$0.677200\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.2065	−0.550401
$868$	0	0
$869$	65.2627	2.21388
$870$	0	0
$871$	−22.0050	−0.745612
$872$	0	0
$873$	−17.2526	−0.583912
$874$	0	0
$875$	3.73549	0.126283
$876$	0	0
$877$	10.6313	0.358994	0.179497	−	0.983758i	$-0.442553\pi$
$877$	10.6313	0.358994	0.179497	+	0.983758i	$0.442553\pi$
$878$	0	0
$879$	8.79857	0.296768
$880$	0	0
$881$	20.5341	0.691810	0.345905	−	0.938270i	$-0.387572\pi$
$881$	20.5341	0.691810	0.345905	+	0.938270i	$0.387572\pi$
$882$	0	0
$883$	36.0390	1.21281	0.606404	−	0.795157i	$-0.292612\pi$
$883$	36.0390	1.21281	0.606404	+	0.795157i	$0.292612\pi$
$884$	0	0
$885$	−0.890804	−0.0299440
$886$	0	0
$887$	−24.9759	−0.838608	−0.419304	−	0.907846i	$-0.637726\pi$
$887$	−24.9759	−0.838608	−0.419304	+	0.907846i	$0.637726\pi$
$888$	0	0
$889$	−44.0289	−1.47668
$890$	0	0
$891$	4.84469	0.162303
$892$	0	0
$893$	−45.3549	−1.51774
$894$	0	0
$895$	9.56322	0.319663
$896$	0	0
$897$	−2.84469	−0.0949813
$898$	0	0
$899$	−6.89080	−0.229821
$900$	0	0
$901$	11.0119	0.366860
$902$	0	0
$903$	12.9658	0.431475
$904$	0	0
$905$	7.16035	0.238018
$906$	0	0
$907$	41.6194	1.38195	0.690975	−	0.722879i	$-0.257182\pi$
$907$	41.6194	1.38195	0.690975	+	0.722879i	$0.257182\pi$
$908$	0	0
$909$	−2.67241	−0.0886383
$910$	0	0
$911$	3.81553	0.126414	0.0632070	−	0.998000i	$-0.479867\pi$
$911$	3.81553	0.126414	0.0632070	+	0.998000i	$0.479867\pi$
$912$	0	0
$913$	−22.1895	−0.734366
$914$	0	0
$915$	−8.62629	−0.285176
$916$	0	0
$917$	65.2627	2.15516
$918$	0	0
$919$	−39.7917	−1.31261	−0.656303	−	0.754497i	$-0.727881\pi$
$919$	−39.7917	−1.31261	−0.656303	+	0.754497i	$0.727881\pi$
$920$	0	0
$921$	10.0050	0.329677
$922$	0	0
$923$	35.1654	1.15748
$924$	0	0
$925$	−1.95388	−0.0642432
$926$	0	0
$927$	6.21839	0.204239
$928$	0	0
$929$	12.6141	0.413855	0.206928	−	0.978356i	$-0.433654\pi$
$929$	12.6141	0.413855	0.206928	+	0.978356i	$0.433654\pi$
$930$	0	0
$931$	47.5971	1.55993
$932$	0	0
$933$	−18.5290	−0.606613
$934$	0	0
$935$	−4.31566	−0.141137
$936$	0	0
$937$	−22.1262	−0.722830	−0.361415	−	0.932405i	$-0.617706\pi$
$937$	−22.1262	−0.722830	−0.361415	+	0.932405i	$0.617706\pi$
$938$	0	0
$939$	2.39067	0.0780164
$940$	0	0
$941$	−11.0392	−0.359869	−0.179934	−	0.983679i	$-0.557589\pi$
$941$	−11.0392	−0.359869	−0.179934	+	0.983679i	$0.557589\pi$
$942$	0	0
$943$	12.3618	0.402555
$944$	0	0
$945$	3.73549	0.121515
$946$	0	0
$947$	9.68937	0.314862	0.157431	−	0.987530i	$-0.449679\pi$
$947$	9.68937	0.314862	0.157431	+	0.987530i	$0.449679\pi$
$948$	0	0
$949$	−47.0342	−1.52679
$950$	0	0
$951$	−16.3157	−0.529072
$952$	0	0
$953$	−14.1845	−0.459480	−0.229740	−	0.973252i	$-0.573788\pi$
$953$	−14.1845	−0.459480	−0.229740	+	0.973252i	$0.573788\pi$
$954$	0	0
$955$	−14.0050	−0.453192
$956$	0	0
$957$	−4.31566	−0.139506
$958$	0	0
$959$	−56.6313	−1.82872
$960$	0	0
$961$	28.8378	0.930252
$962$	0	0
$963$	−14.7986	−0.476877
$964$	0	0
$965$	−26.7236	−0.860262
$966$	0	0
$967$	12.8447	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$967$	12.8447	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$968$	0	0
$969$	6.09727	0.195873
$970$	0	0
$971$	−24.1945	−0.776440	−0.388220	−	0.921567i	$-0.626910\pi$
$971$	−24.1945	−0.776440	−0.388220	+	0.921567i	$0.626910\pi$
$972$	0	0
$973$	20.6091	0.660696
$974$	0	0
$975$	−2.84469	−0.0911029
$976$	0	0
$977$	18.4879	0.591482	0.295741	−	0.955268i	$-0.404434\pi$
$977$	18.4879	0.591482	0.295741	+	0.955268i	$0.404434\pi$
$978$	0	0
$979$	−73.4471	−2.34738
$980$	0	0
$981$	−12.5341	−0.400182
$982$	0	0
$983$	−28.8325	−0.919614	−0.459807	−	0.888019i	$-0.652081\pi$
$983$	−28.8325	−0.919614	−0.459807	+	0.888019i	$0.652081\pi$
$984$	0	0
$985$	−9.16035	−0.291873
$986$	0	0
$987$	24.7524	0.787879
$988$	0	0
$989$	−3.47098	−0.110371
$990$	0	0
$991$	40.8958	1.29910	0.649550	−	0.760319i	$-0.274957\pi$
$991$	40.8958	1.29910	0.649550	+	0.760319i	$0.274957\pi$
$992$	0	0
$993$	−24.8958	−0.790046
$994$	0	0
$995$	0.310629	0.00984759
$996$	0	0
$997$	−4.65518	−0.147431	−0.0737155	−	0.997279i	$-0.523486\pi$
$997$	−4.65518	−0.147431	−0.0737155	+	0.997279i	$0.523486\pi$
$998$	0	0
$999$	−1.95388	−0.0618181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.a.j.1.1	✓	3
3.2	odd	2		4140.2.a.s.1.1		3
4.3	odd	2		5520.2.a.bv.1.3		3
5.2	odd	4		6900.2.f.r.6349.1		6
5.3	odd	4		6900.2.f.r.6349.6		6
5.4	even	2		6900.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.j.1.1	✓	3	1.1	even	1	trivial
4140.2.a.s.1.1		3	3.2	odd	2
5520.2.a.bv.1.3		3	4.3	odd	2
6900.2.a.x.1.3		3	5.4	even	2
6900.2.f.r.6349.1		6	5.2	odd	4
6900.2.f.r.6349.6		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.73549 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.73549 q^{7} +1.00000 q^{9} +4.84469 q^{11} -2.84469 q^{13} -1.00000 q^{15} +0.890804 q^{17} +6.84469 q^{19} -3.73549 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.890804 q^{29} +7.73549 q^{31} +4.84469 q^{33} +3.73549 q^{35} -1.95388 q^{37} -2.84469 q^{39} +12.3618 q^{41} -3.47098 q^{43} -1.00000 q^{45} -6.62629 q^{47} +6.95388 q^{49} +0.890804 q^{51} +12.3618 q^{53} -4.84469 q^{55} +6.84469 q^{57} +0.890804 q^{59} +8.62629 q^{61} -3.73549 q^{63} +2.84469 q^{65} +7.73549 q^{67} +1.00000 q^{69} -12.3618 q^{71} +16.5341 q^{73} +1.00000 q^{75} -18.0973 q^{77} +13.4710 q^{79} +1.00000 q^{81} -4.58018 q^{83} -0.890804 q^{85} -0.890804 q^{87} -15.1604 q^{89} +10.6263 q^{91} +7.73549 q^{93} -6.84469 q^{95} -17.2526 q^{97} +4.84469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} - 3 q^{15} + 10 q^{19} + 2 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{39} + 8 q^{41} + 16 q^{43} - 3 q^{45} - 4 q^{47} + 13 q^{49} + 8 q^{53} - 4 q^{55} + 10 q^{57} + 10 q^{61} + 2 q^{63} - 2 q^{65} + 10 q^{67} + 3 q^{69} - 8 q^{71} + 18 q^{73} + 3 q^{75} - 12 q^{77} + 14 q^{79} + 3 q^{81} + 10 q^{83} + 2 q^{89} + 16 q^{91} + 10 q^{93} - 10 q^{95} - 20 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 - 3 * q^15 + 10 * q^19 + 2 * q^21 + 3 * q^23 + 3 * q^25 + 3 * q^27 + 10 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 + 2 * q^39 + 8 * q^41 + 16 * q^43 - 3 * q^45 - 4 * q^47 + 13 * q^49 + 8 * q^53 - 4 * q^55 + 10 * q^57 + 10 * q^61 + 2 * q^63 - 2 * q^65 + 10 * q^67 + 3 * q^69 - 8 * q^71 + 18 * q^73 + 3 * q^75 - 12 * q^77 + 14 * q^79 + 3 * q^81 + 10 * q^83 + 2 * q^89 + 16 * q^91 + 10 * q^93 - 10 * q^95 - 20 * q^97 + 4 * q^99

Introduction

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