LMFDB - Embedded newform 5440.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5440,2,Mod(1,5440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5440, 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("5440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5440 = 2^{6} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5440.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:	$43.4386186996$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	5440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -1.00000 q^{5} -2.73205 q^{7} +4.46410 q^{9} -4.73205 q^{11} +4.00000 q^{13} +2.73205 q^{15} -1.00000 q^{17} +1.46410 q^{19} +7.46410 q^{21} -8.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} +3.26795 q^{31} +12.9282 q^{33} +2.73205 q^{35} +0.535898 q^{37} -10.9282 q^{39} -3.46410 q^{41} +0.535898 q^{43} -4.46410 q^{45} +12.9282 q^{47} +0.464102 q^{49} +2.73205 q^{51} -6.00000 q^{53} +4.73205 q^{55} -4.00000 q^{57} -2.53590 q^{59} +4.92820 q^{61} -12.1962 q^{63} -4.00000 q^{65} +10.0000 q^{67} +22.3923 q^{69} +11.6603 q^{71} +6.39230 q^{73} -2.73205 q^{75} +12.9282 q^{77} +14.5885 q^{79} -2.46410 q^{81} -8.53590 q^{83} +1.00000 q^{85} -9.46410 q^{87} +4.39230 q^{89} -10.9282 q^{91} -8.92820 q^{93} -1.46410 q^{95} -4.92820 q^{97} -21.1244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 4 q^{19} + 8 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} + 10 q^{31} + 12 q^{33} + 2 q^{35} + 8 q^{37} - 8 q^{39} + 8 q^{43}+ \cdots - 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 4 * q^19 + 8 * q^21 - 6 * q^23 + 2 * q^25 - 8 * q^27 + 10 * q^31 + 12 * q^33 + 2 * q^35 + 8 * q^37 - 8 * q^39 + 8 * q^43 - 2 * q^45 + 12 * q^47 - 6 * q^49 + 2 * q^51 - 12 * q^53 + 6 * q^55 - 8 * q^57 - 12 * q^59 - 4 * q^61 - 14 * q^63 - 8 * q^65 + 20 * q^67 + 24 * q^69 + 6 * q^71 - 8 * q^73 - 2 * q^75 + 12 * q^77 - 2 * q^79 + 2 * q^81 - 24 * q^83 + 2 * q^85 - 12 * q^87 - 12 * q^89 - 8 * q^91 - 4 * q^93 + 4 * q^95 + 4 * q^97 - 18 * 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
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	2.73205	0.705412
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	7.46410	1.62880
$22$	0	0
$23$	−8.19615	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$23$	−8.19615	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	3.26795	0.586941	0.293471	−	0.955968i	$-0.405190\pi$
$31$	3.26795	0.586941	0.293471	+	0.955968i	$0.405190\pi$
$32$	0	0
$33$	12.9282	2.25051
$34$	0	0
$35$	2.73205	0.461801
$36$	0	0
$37$	0.535898	0.0881012	0.0440506	−	0.999029i	$-0.485974\pi$
$37$	0.535898	0.0881012	0.0440506	+	0.999029i	$0.485974\pi$
$38$	0	0
$39$	−10.9282	−1.74991
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	0.535898	0.0817237	0.0408619	−	0.999165i	$-0.486990\pi$
$43$	0.535898	0.0817237	0.0408619	+	0.999165i	$0.486990\pi$
$44$	0	0
$45$	−4.46410	−0.665469
$46$	0	0
$47$	12.9282	1.88577	0.942886	−	0.333115i	$-0.108100\pi$
$47$	12.9282	1.88577	0.942886	+	0.333115i	$0.108100\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	2.73205	0.382564
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	4.73205	0.638070
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	4.92820	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$61$	4.92820	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$62$	0	0
$63$	−12.1962	−1.53657
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	0	0
$69$	22.3923	2.69572
$70$	0	0
$71$	11.6603	1.38382	0.691909	−	0.721985i	$-0.256770\pi$
$71$	11.6603	1.38382	0.691909	+	0.721985i	$0.256770\pi$
$72$	0	0
$73$	6.39230	0.748163	0.374081	−	0.927396i	$-0.377958\pi$
$73$	6.39230	0.748163	0.374081	+	0.927396i	$0.377958\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	12.9282	1.47331
$78$	0	0
$79$	14.5885	1.64133	0.820665	−	0.571410i	$-0.193603\pi$
$79$	14.5885	1.64133	0.820665	+	0.571410i	$0.193603\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	−8.53590	−0.936937	−0.468468	−	0.883480i	$-0.655194\pi$
$83$	−8.53590	−0.936937	−0.468468	+	0.883480i	$0.655194\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−9.46410	−1.01466
$88$	0	0
$89$	4.39230	0.465583	0.232792	−	0.972527i	$-0.425214\pi$
$89$	4.39230	0.465583	0.232792	+	0.972527i	$0.425214\pi$
$90$	0	0
$91$	−10.9282	−1.14559
$92$	0	0
$93$	−8.92820	−0.925812
$94$	0	0
$95$	−1.46410	−0.150214
$96$	0	0
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	0	0
$99$	−21.1244	−2.12308
$100$	0	0
$101$	9.46410	0.941713	0.470857	−	0.882210i	$-0.343945\pi$
$101$	9.46410	0.941713	0.470857	+	0.882210i	$0.343945\pi$
$102$	0	0
$103$	8.92820	0.879722	0.439861	−	0.898066i	$-0.355028\pi$
$103$	8.92820	0.879722	0.439861	+	0.898066i	$0.355028\pi$
$104$	0	0
$105$	−7.46410	−0.728422
$106$	0	0
$107$	−17.6603	−1.70728	−0.853641	−	0.520862i	$-0.825610\pi$
$107$	−17.6603	−1.70728	−0.853641	+	0.520862i	$0.825610\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−1.46410	−0.138966
$112$	0	0
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	8.19615	0.764295
$116$	0	0
$117$	17.8564	1.65083
$118$	0	0
$119$	2.73205	0.250447
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	9.46410	0.853349
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.3923	−1.27711	−0.638555	−	0.769576i	$-0.720468\pi$
$127$	−14.3923	−1.27711	−0.638555	+	0.769576i	$0.720468\pi$
$128$	0	0
$129$	−1.46410	−0.128907
$130$	0	0
$131$	2.19615	0.191879	0.0959394	−	0.995387i	$-0.469415\pi$
$131$	2.19615	0.191879	0.0959394	+	0.995387i	$0.469415\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	3.66025	0.310459	0.155229	−	0.987878i	$-0.450388\pi$
$139$	3.66025	0.310459	0.155229	+	0.987878i	$0.450388\pi$
$140$	0	0
$141$	−35.3205	−2.97452
$142$	0	0
$143$	−18.9282	−1.58286
$144$	0	0
$145$	−3.46410	−0.287678
$146$	0	0
$147$	−1.26795	−0.104579
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−1.46410	−0.119147	−0.0595734	−	0.998224i	$-0.518974\pi$
$151$	−1.46410	−0.119147	−0.0595734	+	0.998224i	$0.518974\pi$
$152$	0	0
$153$	−4.46410	−0.360901
$154$	0	0
$155$	−3.26795	−0.262488
$156$	0	0
$157$	−8.92820	−0.712548	−0.356274	−	0.934381i	$-0.615953\pi$
$157$	−8.92820	−0.712548	−0.356274	+	0.934381i	$0.615953\pi$
$158$	0	0
$159$	16.3923	1.29999
$160$	0	0
$161$	22.3923	1.76476
$162$	0	0
$163$	0.196152	0.0153638	0.00768192	−	0.999970i	$-0.497555\pi$
$163$	0.196152	0.0153638	0.00768192	+	0.999970i	$0.497555\pi$
$164$	0	0
$165$	−12.9282	−1.00646
$166$	0	0
$167$	12.5885	0.974124	0.487062	−	0.873367i	$-0.338069\pi$
$167$	12.5885	0.974124	0.487062	+	0.873367i	$0.338069\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	6.53590	0.499813
$172$	0	0
$173$	−3.46410	−0.263371	−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410	−0.263371	−0.131685	+	0.991292i	$0.542039\pi$
$174$	0	0
$175$	−2.73205	−0.206524
$176$	0	0
$177$	6.92820	0.520756
$178$	0	0
$179$	11.3205	0.846135	0.423067	−	0.906098i	$-0.360953\pi$
$179$	11.3205	0.846135	0.423067	+	0.906098i	$0.360953\pi$
$180$	0	0
$181$	2.39230	0.177819	0.0889093	−	0.996040i	$-0.471662\pi$
$181$	2.39230	0.177819	0.0889093	+	0.996040i	$0.471662\pi$
$182$	0	0
$183$	−13.4641	−0.995295
$184$	0	0
$185$	−0.535898	−0.0394000
$186$	0	0
$187$	4.73205	0.346042
$188$	0	0
$189$	10.9282	0.794910
$190$	0	0
$191$	1.85641	0.134325	0.0671624	−	0.997742i	$-0.478605\pi$
$191$	1.85641	0.134325	0.0671624	+	0.997742i	$0.478605\pi$
$192$	0	0
$193$	16.5359	1.19028	0.595140	−	0.803622i	$-0.297097\pi$
$193$	16.5359	1.19028	0.595140	+	0.803622i	$0.297097\pi$
$194$	0	0
$195$	10.9282	0.782585
$196$	0	0
$197$	−17.3205	−1.23404	−0.617018	−	0.786949i	$-0.711659\pi$
$197$	−17.3205	−1.23404	−0.617018	+	0.786949i	$0.711659\pi$
$198$	0	0
$199$	10.1962	0.722786	0.361393	−	0.932414i	$-0.382301\pi$
$199$	10.1962	0.722786	0.361393	+	0.932414i	$0.382301\pi$
$200$	0	0
$201$	−27.3205	−1.92704
$202$	0	0
$203$	−9.46410	−0.664250
$204$	0	0
$205$	3.46410	0.241943
$206$	0	0
$207$	−36.5885	−2.54307
$208$	0	0
$209$	−6.92820	−0.479234
$210$	0	0
$211$	−10.1962	−0.701932	−0.350966	−	0.936388i	$-0.614147\pi$
$211$	−10.1962	−0.701932	−0.350966	+	0.936388i	$0.614147\pi$
$212$	0	0
$213$	−31.8564	−2.18277
$214$	0	0
$215$	−0.535898	−0.0365480
$216$	0	0
$217$	−8.92820	−0.606086
$218$	0	0
$219$	−17.4641	−1.18011
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−26.3923	−1.76736	−0.883680	−	0.468092i	$-0.844942\pi$
$223$	−26.3923	−1.76736	−0.883680	+	0.468092i	$0.844942\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	−22.7321	−1.50878	−0.754390	−	0.656427i	$-0.772067\pi$
$227$	−22.7321	−1.50878	−0.754390	+	0.656427i	$0.772067\pi$
$228$	0	0
$229$	8.39230	0.554579	0.277290	−	0.960786i	$-0.410564\pi$
$229$	8.39230	0.554579	0.277290	+	0.960786i	$0.410564\pi$
$230$	0	0
$231$	−35.3205	−2.32392
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−12.9282	−0.843343
$236$	0	0
$237$	−39.8564	−2.58895
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	−5.60770	−0.361223	−0.180612	−	0.983554i	$-0.557808\pi$
$241$	−5.60770	−0.361223	−0.180612	+	0.983554i	$0.557808\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	−0.464102	−0.0296504
$246$	0	0
$247$	5.85641	0.372634
$248$	0	0
$249$	23.3205	1.47788
$250$	0	0
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	0	0
$253$	38.7846	2.43837
$254$	0	0
$255$	−2.73205	−0.171088
$256$	0	0
$257$	−6.92820	−0.432169	−0.216085	−	0.976375i	$-0.569329\pi$
$257$	−6.92820	−0.432169	−0.216085	+	0.976375i	$0.569329\pi$
$258$	0	0
$259$	−1.46410	−0.0909748
$260$	0	0
$261$	15.4641	0.957204
$262$	0	0
$263$	−1.60770	−0.0991347	−0.0495674	−	0.998771i	$-0.515784\pi$
$263$	−1.60770	−0.0991347	−0.0495674	+	0.998771i	$0.515784\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−12.0000	−0.734388
$268$	0	0
$269$	−0.928203	−0.0565935	−0.0282968	−	0.999600i	$-0.509008\pi$
$269$	−0.928203	−0.0565935	−0.0282968	+	0.999600i	$0.509008\pi$
$270$	0	0
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	0	0
$273$	29.8564	1.80699
$274$	0	0
$275$	−4.73205	−0.285353
$276$	0	0
$277$	−20.9282	−1.25745	−0.628727	−	0.777626i	$-0.716424\pi$
$277$	−20.9282	−1.25745	−0.628727	+	0.777626i	$0.716424\pi$
$278$	0	0
$279$	14.5885	0.873388
$280$	0	0
$281$	−12.9282	−0.771232	−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282	−0.771232	−0.385616	+	0.922659i	$0.626011\pi$
$282$	0	0
$283$	5.26795	0.313147	0.156574	−	0.987666i	$-0.449955\pi$
$283$	5.26795	0.313147	0.156574	+	0.987666i	$0.449955\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	9.46410	0.558648
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	13.4641	0.789280
$292$	0	0
$293$	0.928203	0.0542262	0.0271131	−	0.999632i	$-0.491369\pi$
$293$	0.928203	0.0542262	0.0271131	+	0.999632i	$0.491369\pi$
$294$	0	0
$295$	2.53590	0.147646
$296$	0	0
$297$	18.9282	1.09833
$298$	0	0
$299$	−32.7846	−1.89598
$300$	0	0
$301$	−1.46410	−0.0843894
$302$	0	0
$303$	−25.8564	−1.48541
$304$	0	0
$305$	−4.92820	−0.282188
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	−24.3923	−1.38763
$310$	0	0
$311$	−16.0526	−0.910257	−0.455129	−	0.890426i	$-0.650407\pi$
$311$	−16.0526	−0.910257	−0.455129	+	0.890426i	$0.650407\pi$
$312$	0	0
$313$	−26.3923	−1.49178	−0.745891	−	0.666068i	$-0.767976\pi$
$313$	−26.3923	−1.49178	−0.745891	+	0.666068i	$0.767976\pi$
$314$	0	0
$315$	12.1962	0.687175
$316$	0	0
$317$	24.9282	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$317$	24.9282	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$318$	0	0
$319$	−16.3923	−0.917793
$320$	0	0
$321$	48.2487	2.69298
$322$	0	0
$323$	−1.46410	−0.0814648
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−27.3205	−1.51083
$328$	0	0
$329$	−35.3205	−1.94728
$330$	0	0
$331$	6.53590	0.359245	0.179623	−	0.983736i	$-0.442512\pi$
$331$	6.53590	0.359245	0.179623	+	0.983736i	$0.442512\pi$
$332$	0	0
$333$	2.39230	0.131097
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−6.78461	−0.369581	−0.184791	−	0.982778i	$-0.559161\pi$
$337$	−6.78461	−0.369581	−0.184791	+	0.982778i	$0.559161\pi$
$338$	0	0
$339$	47.3205	2.57010
$340$	0	0
$341$	−15.4641	−0.837428
$342$	0	0
$343$	17.8564	0.964155
$344$	0	0
$345$	−22.3923	−1.20556
$346$	0	0
$347$	−3.80385	−0.204201	−0.102101	−	0.994774i	$-0.532556\pi$
$347$	−3.80385	−0.204201	−0.102101	+	0.994774i	$0.532556\pi$
$348$	0	0
$349$	−10.7846	−0.577287	−0.288643	−	0.957437i	$-0.593204\pi$
$349$	−10.7846	−0.577287	−0.288643	+	0.957437i	$0.593204\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	26.7846	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$353$	26.7846	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$354$	0	0
$355$	−11.6603	−0.618862
$356$	0	0
$357$	−7.46410	−0.395042
$358$	0	0
$359$	−21.4641	−1.13283	−0.566416	−	0.824119i	$-0.691670\pi$
$359$	−21.4641	−1.13283	−0.566416	+	0.824119i	$0.691670\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	0	0
$363$	−31.1244	−1.63361
$364$	0	0
$365$	−6.39230	−0.334589
$366$	0	0
$367$	−7.80385	−0.407358	−0.203679	−	0.979038i	$-0.565290\pi$
$367$	−7.80385	−0.407358	−0.203679	+	0.979038i	$0.565290\pi$
$368$	0	0
$369$	−15.4641	−0.805029
$370$	0	0
$371$	16.3923	0.851046
$372$	0	0
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	0	0
$377$	13.8564	0.713641
$378$	0	0
$379$	−17.8038	−0.914522	−0.457261	−	0.889332i	$-0.651170\pi$
$379$	−17.8038	−0.914522	−0.457261	+	0.889332i	$0.651170\pi$
$380$	0	0
$381$	39.3205	2.01445
$382$	0	0
$383$	15.4641	0.790179	0.395089	−	0.918643i	$-0.370714\pi$
$383$	15.4641	0.790179	0.395089	+	0.918643i	$0.370714\pi$
$384$	0	0
$385$	−12.9282	−0.658882
$386$	0	0
$387$	2.39230	0.121608
$388$	0	0
$389$	4.39230	0.222699	0.111349	−	0.993781i	$-0.464483\pi$
$389$	4.39230	0.222699	0.111349	+	0.993781i	$0.464483\pi$
$390$	0	0
$391$	8.19615	0.414497
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	−14.5885	−0.734025
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	10.9282	0.547094
$400$	0	0
$401$	12.9282	0.645604	0.322802	−	0.946467i	$-0.395375\pi$
$401$	12.9282	0.645604	0.322802	+	0.946467i	$0.395375\pi$
$402$	0	0
$403$	13.0718	0.651153
$404$	0	0
$405$	2.46410	0.122442
$406$	0	0
$407$	−2.53590	−0.125700
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.92820	0.340915
$414$	0	0
$415$	8.53590	0.419011
$416$	0	0
$417$	−10.0000	−0.489702
$418$	0	0
$419$	−38.1962	−1.86600	−0.933002	−	0.359871i	$-0.882821\pi$
$419$	−38.1962	−1.86600	−0.933002	+	0.359871i	$0.882821\pi$
$420$	0	0
$421$	−5.46410	−0.266304	−0.133152	−	0.991096i	$-0.542510\pi$
$421$	−5.46410	−0.266304	−0.133152	+	0.991096i	$0.542510\pi$
$422$	0	0
$423$	57.7128	2.80609
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−13.4641	−0.651574
$428$	0	0
$429$	51.7128	2.49672
$430$	0	0
$431$	9.80385	0.472235	0.236117	−	0.971725i	$-0.424125\pi$
$431$	9.80385	0.472235	0.236117	+	0.971725i	$0.424125\pi$
$432$	0	0
$433$	−17.8564	−0.858124	−0.429062	−	0.903275i	$-0.641156\pi$
$433$	−17.8564	−0.858124	−0.429062	+	0.903275i	$0.641156\pi$
$434$	0	0
$435$	9.46410	0.453769
$436$	0	0
$437$	−12.0000	−0.574038
$438$	0	0
$439$	−20.0526	−0.957056	−0.478528	−	0.878072i	$-0.658830\pi$
$439$	−20.0526	−0.957056	−0.478528	+	0.878072i	$0.658830\pi$
$440$	0	0
$441$	2.07180	0.0986570
$442$	0	0
$443$	12.9282	0.614237	0.307119	−	0.951671i	$-0.400635\pi$
$443$	12.9282	0.614237	0.307119	+	0.951671i	$0.400635\pi$
$444$	0	0
$445$	−4.39230	−0.208215
$446$	0	0
$447$	−16.3923	−0.775329
$448$	0	0
$449$	−34.3923	−1.62307	−0.811537	−	0.584302i	$-0.801369\pi$
$449$	−34.3923	−1.62307	−0.811537	+	0.584302i	$0.801369\pi$
$450$	0	0
$451$	16.3923	0.771883
$452$	0	0
$453$	4.00000	0.187936
$454$	0	0
$455$	10.9282	0.512322
$456$	0	0
$457$	−36.7846	−1.72071	−0.860356	−	0.509694i	$-0.829759\pi$
$457$	−36.7846	−1.72071	−0.860356	+	0.509694i	$0.829759\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	24.9282	1.16102	0.580511	−	0.814252i	$-0.302853\pi$
$461$	24.9282	1.16102	0.580511	+	0.814252i	$0.302853\pi$
$462$	0	0
$463$	−23.8564	−1.10870	−0.554351	−	0.832283i	$-0.687033\pi$
$463$	−23.8564	−1.10870	−0.554351	+	0.832283i	$0.687033\pi$
$464$	0	0
$465$	8.92820	0.414036
$466$	0	0
$467$	−1.60770	−0.0743953	−0.0371976	−	0.999308i	$-0.511843\pi$
$467$	−1.60770	−0.0743953	−0.0371976	+	0.999308i	$0.511843\pi$
$468$	0	0
$469$	−27.3205	−1.26154
$470$	0	0
$471$	24.3923	1.12394
$472$	0	0
$473$	−2.53590	−0.116601
$474$	0	0
$475$	1.46410	0.0671776
$476$	0	0
$477$	−26.7846	−1.22638
$478$	0	0
$479$	−11.6603	−0.532771	−0.266385	−	0.963867i	$-0.585829\pi$
$479$	−11.6603	−0.532771	−0.266385	+	0.963867i	$0.585829\pi$
$480$	0	0
$481$	2.14359	0.0977395
$482$	0	0
$483$	−61.1769	−2.78365
$484$	0	0
$485$	4.92820	0.223778
$486$	0	0
$487$	24.9808	1.13199	0.565993	−	0.824410i	$-0.308493\pi$
$487$	24.9808	1.13199	0.565993	+	0.824410i	$0.308493\pi$
$488$	0	0
$489$	−0.535898	−0.0242342
$490$	0	0
$491$	19.6077	0.884883	0.442441	−	0.896797i	$-0.354112\pi$
$491$	19.6077	0.884883	0.442441	+	0.896797i	$0.354112\pi$
$492$	0	0
$493$	−3.46410	−0.156015
$494$	0	0
$495$	21.1244	0.949469
$496$	0	0
$497$	−31.8564	−1.42896
$498$	0	0
$499$	15.6603	0.701049	0.350525	−	0.936554i	$-0.386003\pi$
$499$	15.6603	0.701049	0.350525	+	0.936554i	$0.386003\pi$
$500$	0	0
$501$	−34.3923	−1.53653
$502$	0	0
$503$	15.1244	0.674362	0.337181	−	0.941440i	$-0.390527\pi$
$503$	15.1244	0.674362	0.337181	+	0.941440i	$0.390527\pi$
$504$	0	0
$505$	−9.46410	−0.421147
$506$	0	0
$507$	−8.19615	−0.364004
$508$	0	0
$509$	−19.8564	−0.880120	−0.440060	−	0.897968i	$-0.645043\pi$
$509$	−19.8564	−0.880120	−0.440060	+	0.897968i	$0.645043\pi$
$510$	0	0
$511$	−17.4641	−0.772566
$512$	0	0
$513$	−5.85641	−0.258567
$514$	0	0
$515$	−8.92820	−0.393424
$516$	0	0
$517$	−61.1769	−2.69056
$518$	0	0
$519$	9.46410	0.415428
$520$	0	0
$521$	4.14359	0.181534	0.0907671	−	0.995872i	$-0.471068\pi$
$521$	4.14359	0.181534	0.0907671	+	0.995872i	$0.471068\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	0	0
$525$	7.46410	0.325760
$526$	0	0
$527$	−3.26795	−0.142354
$528$	0	0
$529$	44.1769	1.92074
$530$	0	0
$531$	−11.3205	−0.491268
$532$	0	0
$533$	−13.8564	−0.600188
$534$	0	0
$535$	17.6603	0.763519
$536$	0	0
$537$	−30.9282	−1.33465
$538$	0	0
$539$	−2.19615	−0.0945950
$540$	0	0
$541$	−39.1769	−1.68435	−0.842174	−	0.539207i	$-0.818724\pi$
$541$	−39.1769	−1.68435	−0.842174	+	0.539207i	$0.818724\pi$
$542$	0	0
$543$	−6.53590	−0.280482
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	39.9090	1.70638	0.853192	−	0.521597i	$-0.174663\pi$
$547$	39.9090	1.70638	0.853192	+	0.521597i	$0.174663\pi$
$548$	0	0
$549$	22.0000	0.938937
$550$	0	0
$551$	5.07180	0.216066
$552$	0	0
$553$	−39.8564	−1.69487
$554$	0	0
$555$	1.46410	0.0621477
$556$	0	0
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	0	0
$559$	2.14359	0.0906643
$560$	0	0
$561$	−12.9282	−0.545829
$562$	0	0
$563$	−27.4641	−1.15747	−0.578737	−	0.815514i	$-0.696454\pi$
$563$	−27.4641	−1.15747	−0.578737	+	0.815514i	$0.696454\pi$
$564$	0	0
$565$	17.3205	0.728679
$566$	0	0
$567$	6.73205	0.282720
$568$	0	0
$569$	40.6410	1.70376	0.851880	−	0.523737i	$-0.175463\pi$
$569$	40.6410	1.70376	0.851880	+	0.523737i	$0.175463\pi$
$570$	0	0
$571$	36.4449	1.52517	0.762585	−	0.646888i	$-0.223930\pi$
$571$	36.4449	1.52517	0.762585	+	0.646888i	$0.223930\pi$
$572$	0	0
$573$	−5.07180	−0.211877
$574$	0	0
$575$	−8.19615	−0.341803
$576$	0	0
$577$	−38.6410	−1.60865	−0.804323	−	0.594192i	$-0.797472\pi$
$577$	−38.6410	−1.60865	−0.804323	+	0.594192i	$0.797472\pi$
$578$	0	0
$579$	−45.1769	−1.87749
$580$	0	0
$581$	23.3205	0.967498
$582$	0	0
$583$	28.3923	1.17589
$584$	0	0
$585$	−17.8564	−0.738272
$586$	0	0
$587$	−46.3923	−1.91482	−0.957408	−	0.288740i	$-0.906764\pi$
$587$	−46.3923	−1.91482	−0.957408	+	0.288740i	$0.906764\pi$
$588$	0	0
$589$	4.78461	0.197146
$590$	0	0
$591$	47.3205	1.94651
$592$	0	0
$593$	19.8564	0.815405	0.407702	−	0.913115i	$-0.366330\pi$
$593$	19.8564	0.815405	0.407702	+	0.913115i	$0.366330\pi$
$594$	0	0
$595$	−2.73205	−0.112003
$596$	0	0
$597$	−27.8564	−1.14009
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	8.24871	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$601$	8.24871	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$602$	0	0
$603$	44.6410	1.81792
$604$	0	0
$605$	−11.3923	−0.463163
$606$	0	0
$607$	−21.6603	−0.879163	−0.439581	−	0.898203i	$-0.644873\pi$
$607$	−21.6603	−0.879163	−0.439581	+	0.898203i	$0.644873\pi$
$608$	0	0
$609$	25.8564	1.04775
$610$	0	0
$611$	51.7128	2.09208
$612$	0	0
$613$	−15.8564	−0.640434	−0.320217	−	0.947344i	$-0.603756\pi$
$613$	−15.8564	−0.640434	−0.320217	+	0.947344i	$0.603756\pi$
$614$	0	0
$615$	−9.46410	−0.381629
$616$	0	0
$617$	−27.4641	−1.10566	−0.552832	−	0.833293i	$-0.686453\pi$
$617$	−27.4641	−1.10566	−0.552832	+	0.833293i	$0.686453\pi$
$618$	0	0
$619$	−38.5885	−1.55100	−0.775501	−	0.631347i	$-0.782502\pi$
$619$	−38.5885	−1.55100	−0.775501	+	0.631347i	$0.782502\pi$
$620$	0	0
$621$	32.7846	1.31560
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	18.9282	0.755920
$628$	0	0
$629$	−0.535898	−0.0213677
$630$	0	0
$631$	−32.3923	−1.28952	−0.644759	−	0.764386i	$-0.723042\pi$
$631$	−32.3923	−1.28952	−0.644759	+	0.764386i	$0.723042\pi$
$632$	0	0
$633$	27.8564	1.10719
$634$	0	0
$635$	14.3923	0.571141
$636$	0	0
$637$	1.85641	0.0735535
$638$	0	0
$639$	52.0526	2.05917
$640$	0	0
$641$	31.1769	1.23141	0.615707	−	0.787975i	$-0.288870\pi$
$641$	31.1769	1.23141	0.615707	+	0.787975i	$0.288870\pi$
$642$	0	0
$643$	24.1962	0.954203	0.477102	−	0.878848i	$-0.341687\pi$
$643$	24.1962	0.954203	0.477102	+	0.878848i	$0.341687\pi$
$644$	0	0
$645$	1.46410	0.0576489
$646$	0	0
$647$	−38.7846	−1.52478	−0.762390	−	0.647118i	$-0.775974\pi$
$647$	−38.7846	−1.52478	−0.762390	+	0.647118i	$0.775974\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	24.3923	0.956010
$652$	0	0
$653$	25.6077	1.00211	0.501053	−	0.865416i	$-0.332946\pi$
$653$	25.6077	1.00211	0.501053	+	0.865416i	$0.332946\pi$
$654$	0	0
$655$	−2.19615	−0.0858108
$656$	0	0
$657$	28.5359	1.11329
$658$	0	0
$659$	32.7846	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$659$	32.7846	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$660$	0	0
$661$	8.14359	0.316749	0.158375	−	0.987379i	$-0.449375\pi$
$661$	8.14359	0.316749	0.158375	+	0.987379i	$0.449375\pi$
$662$	0	0
$663$	10.9282	0.424416
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	−28.3923	−1.09935
$668$	0	0
$669$	72.1051	2.78774
$670$	0	0
$671$	−23.3205	−0.900278
$672$	0	0
$673$	23.4641	0.904475	0.452237	−	0.891898i	$-0.350626\pi$
$673$	23.4641	0.904475	0.452237	+	0.891898i	$0.350626\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−2.78461	−0.107021	−0.0535106	−	0.998567i	$-0.517041\pi$
$677$	−2.78461	−0.107021	−0.0535106	+	0.998567i	$0.517041\pi$
$678$	0	0
$679$	13.4641	0.516705
$680$	0	0
$681$	62.1051	2.37987
$682$	0	0
$683$	30.8372	1.17995	0.589976	−	0.807421i	$-0.299137\pi$
$683$	30.8372	1.17995	0.589976	+	0.807421i	$0.299137\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.9282	−0.874766
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	38.9808	1.48290	0.741449	−	0.671009i	$-0.234139\pi$
$691$	38.9808	1.48290	0.741449	+	0.671009i	$0.234139\pi$
$692$	0	0
$693$	57.7128	2.19233
$694$	0	0
$695$	−3.66025	−0.138841
$696$	0	0
$697$	3.46410	0.131212
$698$	0	0
$699$	−16.3923	−0.620014
$700$	0	0
$701$	−11.3205	−0.427570	−0.213785	−	0.976881i	$-0.568579\pi$
$701$	−11.3205	−0.427570	−0.213785	+	0.976881i	$0.568579\pi$
$702$	0	0
$703$	0.784610	0.0295921
$704$	0	0
$705$	35.3205	1.33025
$706$	0	0
$707$	−25.8564	−0.972430
$708$	0	0
$709$	−4.53590	−0.170349	−0.0851746	−	0.996366i	$-0.527145\pi$
$709$	−4.53590	−0.170349	−0.0851746	+	0.996366i	$0.527145\pi$
$710$	0	0
$711$	65.1244	2.44235
$712$	0	0
$713$	−26.7846	−1.00309
$714$	0	0
$715$	18.9282	0.707875
$716$	0	0
$717$	56.7846	2.12066
$718$	0	0
$719$	−5.41154	−0.201816	−0.100908	−	0.994896i	$-0.532175\pi$
$719$	−5.41154	−0.201816	−0.100908	+	0.994896i	$0.532175\pi$
$720$	0	0
$721$	−24.3923	−0.908417
$722$	0	0
$723$	15.3205	0.569776
$724$	0	0
$725$	3.46410	0.128654
$726$	0	0
$727$	0.143594	0.00532559	0.00266279	−	0.999996i	$-0.499152\pi$
$727$	0.143594	0.00532559	0.00266279	+	0.999996i	$0.499152\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−0.535898	−0.0198209
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	1.26795	0.0467690
$736$	0	0
$737$	−47.3205	−1.74307
$738$	0	0
$739$	11.6077	0.426996	0.213498	−	0.976944i	$-0.431514\pi$
$739$	11.6077	0.426996	0.213498	+	0.976944i	$0.431514\pi$
$740$	0	0
$741$	−16.0000	−0.587775
$742$	0	0
$743$	−22.7321	−0.833958	−0.416979	−	0.908916i	$-0.636911\pi$
$743$	−22.7321	−0.833958	−0.416979	+	0.908916i	$0.636911\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−38.1051	−1.39419
$748$	0	0
$749$	48.2487	1.76297
$750$	0	0
$751$	43.6603	1.59319	0.796593	−	0.604516i	$-0.206634\pi$
$751$	43.6603	1.59319	0.796593	+	0.604516i	$0.206634\pi$
$752$	0	0
$753$	18.9282	0.689782
$754$	0	0
$755$	1.46410	0.0532841
$756$	0	0
$757$	−30.6410	−1.11367	−0.556833	−	0.830624i	$-0.687984\pi$
$757$	−30.6410	−1.11367	−0.556833	+	0.830624i	$0.687984\pi$
$758$	0	0
$759$	−105.962	−3.84616
$760$	0	0
$761$	−28.3923	−1.02922	−0.514610	−	0.857424i	$-0.672063\pi$
$761$	−28.3923	−1.02922	−0.514610	+	0.857424i	$0.672063\pi$
$762$	0	0
$763$	−27.3205	−0.989069
$764$	0	0
$765$	4.46410	0.161400
$766$	0	0
$767$	−10.1436	−0.366264
$768$	0	0
$769$	36.3923	1.31234	0.656170	−	0.754613i	$-0.272175\pi$
$769$	36.3923	1.31234	0.656170	+	0.754613i	$0.272175\pi$
$770$	0	0
$771$	18.9282	0.681683
$772$	0	0
$773$	46.6410	1.67756	0.838780	−	0.544470i	$-0.183269\pi$
$773$	46.6410	1.67756	0.838780	+	0.544470i	$0.183269\pi$
$774$	0	0
$775$	3.26795	0.117388
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−5.07180	−0.181716
$780$	0	0
$781$	−55.1769	−1.97439
$782$	0	0
$783$	−13.8564	−0.495188
$784$	0	0
$785$	8.92820	0.318661
$786$	0	0
$787$	−43.9090	−1.56519	−0.782593	−	0.622534i	$-0.786103\pi$
$787$	−43.9090	−1.56519	−0.782593	+	0.622534i	$0.786103\pi$
$788$	0	0
$789$	4.39230	0.156370
$790$	0	0
$791$	47.3205	1.68252
$792$	0	0
$793$	19.7128	0.700023
$794$	0	0
$795$	−16.3923	−0.581375
$796$	0	0
$797$	24.9282	0.883002	0.441501	−	0.897261i	$-0.354446\pi$
$797$	24.9282	0.883002	0.441501	+	0.897261i	$0.354446\pi$
$798$	0	0
$799$	−12.9282	−0.457367
$800$	0	0
$801$	19.6077	0.692804
$802$	0	0
$803$	−30.2487	−1.06745
$804$	0	0
$805$	−22.3923	−0.789225
$806$	0	0
$807$	2.53590	0.0892679
$808$	0	0
$809$	−55.8564	−1.96381	−0.981903	−	0.189383i	$-0.939351\pi$
$809$	−55.8564	−1.96381	−0.981903	+	0.189383i	$0.939351\pi$
$810$	0	0
$811$	−44.8372	−1.57445	−0.787223	−	0.616668i	$-0.788482\pi$
$811$	−44.8372	−1.57445	−0.787223	+	0.616668i	$0.788482\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	−0.196152	−0.00687092
$816$	0	0
$817$	0.784610	0.0274500
$818$	0	0
$819$	−48.7846	−1.70467
$820$	0	0
$821$	24.9282	0.870000	0.435000	−	0.900430i	$-0.356748\pi$
$821$	24.9282	0.870000	0.435000	+	0.900430i	$0.356748\pi$
$822$	0	0
$823$	−8.98076	−0.313050	−0.156525	−	0.987674i	$-0.550029\pi$
$823$	−8.98076	−0.313050	−0.156525	+	0.987674i	$0.550029\pi$
$824$	0	0
$825$	12.9282	0.450102
$826$	0	0
$827$	15.8038	0.549554	0.274777	−	0.961508i	$-0.411396\pi$
$827$	15.8038	0.549554	0.274777	+	0.961508i	$0.411396\pi$
$828$	0	0
$829$	−17.7128	−0.615191	−0.307596	−	0.951517i	$-0.599524\pi$
$829$	−17.7128	−0.615191	−0.307596	+	0.951517i	$0.599524\pi$
$830$	0	0
$831$	57.1769	1.98345
$832$	0	0
$833$	−0.464102	−0.0160802
$834$	0	0
$835$	−12.5885	−0.435642
$836$	0	0
$837$	−13.0718	−0.451827
$838$	0	0
$839$	19.2679	0.665203	0.332602	−	0.943067i	$-0.392074\pi$
$839$	19.2679	0.665203	0.332602	+	0.943067i	$0.392074\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	35.3205	1.21650
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−31.1244	−1.06945
$848$	0	0
$849$	−14.3923	−0.493943
$850$	0	0
$851$	−4.39230	−0.150566
$852$	0	0
$853$	23.1769	0.793562	0.396781	−	0.917913i	$-0.370127\pi$
$853$	23.1769	0.793562	0.396781	+	0.917913i	$0.370127\pi$
$854$	0	0
$855$	−6.53590	−0.223523
$856$	0	0
$857$	31.1769	1.06498	0.532492	−	0.846435i	$-0.321256\pi$
$857$	31.1769	1.06498	0.532492	+	0.846435i	$0.321256\pi$
$858$	0	0
$859$	25.4641	0.868824	0.434412	−	0.900714i	$-0.356956\pi$
$859$	25.4641	0.868824	0.434412	+	0.900714i	$0.356956\pi$
$860$	0	0
$861$	−25.8564	−0.881184
$862$	0	0
$863$	23.0718	0.785373	0.392687	−	0.919672i	$-0.371546\pi$
$863$	23.0718	0.785373	0.392687	+	0.919672i	$0.371546\pi$
$864$	0	0
$865$	3.46410	0.117783
$866$	0	0
$867$	−2.73205	−0.0927853
$868$	0	0
$869$	−69.0333	−2.34180
$870$	0	0
$871$	40.0000	1.35535
$872$	0	0
$873$	−22.0000	−0.744587
$874$	0	0
$875$	2.73205	0.0923602
$876$	0	0
$877$	1.21539	0.0410408	0.0205204	−	0.999789i	$-0.493468\pi$
$877$	1.21539	0.0410408	0.0205204	+	0.999789i	$0.493468\pi$
$878$	0	0
$879$	−2.53590	−0.0855337
$880$	0	0
$881$	−41.3205	−1.39212	−0.696062	−	0.717982i	$-0.745066\pi$
$881$	−41.3205	−1.39212	−0.696062	+	0.717982i	$0.745066\pi$
$882$	0	0
$883$	10.0000	0.336527	0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000	0.336527	0.168263	+	0.985742i	$0.446184\pi$
$884$	0	0
$885$	−6.92820	−0.232889
$886$	0	0
$887$	3.12436	0.104906	0.0524528	−	0.998623i	$-0.483296\pi$
$887$	3.12436	0.104906	0.0524528	+	0.998623i	$0.483296\pi$
$888$	0	0
$889$	39.3205	1.31877
$890$	0	0
$891$	11.6603	0.390633
$892$	0	0
$893$	18.9282	0.633408
$894$	0	0
$895$	−11.3205	−0.378403
$896$	0	0
$897$	89.5692	2.99063
$898$	0	0
$899$	11.3205	0.377560
$900$	0	0
$901$	6.00000	0.199889
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−2.39230	−0.0795229
$906$	0	0
$907$	−30.7321	−1.02044	−0.510220	−	0.860044i	$-0.670436\pi$
$907$	−30.7321	−1.02044	−0.510220	+	0.860044i	$0.670436\pi$
$908$	0	0
$909$	42.2487	1.40130
$910$	0	0
$911$	36.3397	1.20399	0.601995	−	0.798500i	$-0.294373\pi$
$911$	36.3397	1.20399	0.601995	+	0.798500i	$0.294373\pi$
$912$	0	0
$913$	40.3923	1.33679
$914$	0	0
$915$	13.4641	0.445109
$916$	0	0
$917$	−6.00000	−0.198137
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−27.3205	−0.900241
$922$	0	0
$923$	46.6410	1.53521
$924$	0	0
$925$	0.535898	0.0176202
$926$	0	0
$927$	39.8564	1.30906
$928$	0	0
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	0	0
$931$	0.679492	0.0222694
$932$	0	0
$933$	43.8564	1.43579
$934$	0	0
$935$	−4.73205	−0.154755
$936$	0	0
$937$	−32.6410	−1.06634	−0.533168	−	0.846010i	$-0.678999\pi$
$937$	−32.6410	−1.06634	−0.533168	+	0.846010i	$0.678999\pi$
$938$	0	0
$939$	72.1051	2.35306
$940$	0	0
$941$	−48.2487	−1.57286	−0.786432	−	0.617677i	$-0.788074\pi$
$941$	−48.2487	−1.57286	−0.786432	+	0.617677i	$0.788074\pi$
$942$	0	0
$943$	28.3923	0.924581
$944$	0	0
$945$	−10.9282	−0.355494
$946$	0	0
$947$	−8.19615	−0.266339	−0.133170	−	0.991093i	$-0.542515\pi$
$947$	−8.19615	−0.266339	−0.133170	+	0.991093i	$0.542515\pi$
$948$	0	0
$949$	25.5692	0.830012
$950$	0	0
$951$	−68.1051	−2.20846
$952$	0	0
$953$	−8.78461	−0.284561	−0.142281	−	0.989826i	$-0.545444\pi$
$953$	−8.78461	−0.284561	−0.142281	+	0.989826i	$0.545444\pi$
$954$	0	0
$955$	−1.85641	−0.0600719
$956$	0	0
$957$	44.7846	1.44768
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.3205	−0.655500
$962$	0	0
$963$	−78.8372	−2.54049
$964$	0	0
$965$	−16.5359	−0.532309
$966$	0	0
$967$	39.1769	1.25984	0.629922	−	0.776658i	$-0.283087\pi$
$967$	39.1769	1.25984	0.629922	+	0.776658i	$0.283087\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−54.2487	−1.74092	−0.870462	−	0.492236i	$-0.836180\pi$
$971$	−54.2487	−1.74092	−0.870462	+	0.492236i	$0.836180\pi$
$972$	0	0
$973$	−10.0000	−0.320585
$974$	0	0
$975$	−10.9282	−0.349983
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	−20.7846	−0.664279
$980$	0	0
$981$	44.6410	1.42528
$982$	0	0
$983$	58.7321	1.87326	0.936631	−	0.350318i	$-0.113927\pi$
$983$	58.7321	1.87326	0.936631	+	0.350318i	$0.113927\pi$
$984$	0	0
$985$	17.3205	0.551877
$986$	0	0
$987$	96.4974	3.07155
$988$	0	0
$989$	−4.39230	−0.139667
$990$	0	0
$991$	−2.98076	−0.0946870	−0.0473435	−	0.998879i	$-0.515076\pi$
$991$	−2.98076	−0.0946870	−0.0473435	+	0.998879i	$0.515076\pi$
$992$	0	0
$993$	−17.8564	−0.566656
$994$	0	0
$995$	−10.1962	−0.323240
$996$	0	0
$997$	2.39230	0.0757651	0.0378825	−	0.999282i	$-0.487939\pi$
$997$	2.39230	0.0757651	0.0378825	+	0.999282i	$0.487939\pi$
$998$	0	0
$999$	−2.14359	−0.0678203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5440.2.a.bb.1.1		2
4.3	odd	2		5440.2.a.bl.1.2		2
8.3	odd	2		1360.2.a.k.1.1		2
8.5	even	2		85.2.a.c.1.1	✓	2
24.5	odd	2		765.2.a.g.1.2		2
40.13	odd	4		425.2.b.d.324.4		4
40.19	odd	2		6800.2.a.bg.1.2		2
40.29	even	2		425.2.a.e.1.2		2
40.37	odd	4		425.2.b.d.324.1		4
56.13	odd	2		4165.2.a.t.1.1		2
120.29	odd	2		3825.2.a.v.1.1		2
136.13	even	4		1445.2.d.e.866.4		4
136.21	even	4		1445.2.d.e.866.3		4
136.101	even	2		1445.2.a.g.1.1		2
680.509	even	2		7225.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.1	✓	2	8.5	even	2
425.2.a.e.1.2		2	40.29	even	2
425.2.b.d.324.1		4	40.37	odd	4
425.2.b.d.324.4		4	40.13	odd	4
765.2.a.g.1.2		2	24.5	odd	2
1360.2.a.k.1.1		2	8.3	odd	2
1445.2.a.g.1.1		2	136.101	even	2
1445.2.d.e.866.3		4	136.21	even	4
1445.2.d.e.866.4		4	136.13	even	4
3825.2.a.v.1.1		2	120.29	odd	2
4165.2.a.t.1.1		2	56.13	odd	2
5440.2.a.bb.1.1		2	1.1	even	1	trivial
5440.2.a.bl.1.2		2	4.3	odd	2
6800.2.a.bg.1.2		2	40.19	odd	2
7225.2.a.l.1.2		2	680.509	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:	\( N \)	\(=\)	\( 5440 = 2^{6} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5440.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -1.00000 q^{5} -2.73205 q^{7} +4.46410 q^{9} -4.73205 q^{11} +4.00000 q^{13} +2.73205 q^{15} -1.00000 q^{17} +1.46410 q^{19} +7.46410 q^{21} -8.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} +3.26795 q^{31} +12.9282 q^{33} +2.73205 q^{35} +0.535898 q^{37} -10.9282 q^{39} -3.46410 q^{41} +0.535898 q^{43} -4.46410 q^{45} +12.9282 q^{47} +0.464102 q^{49} +2.73205 q^{51} -6.00000 q^{53} +4.73205 q^{55} -4.00000 q^{57} -2.53590 q^{59} +4.92820 q^{61} -12.1962 q^{63} -4.00000 q^{65} +10.0000 q^{67} +22.3923 q^{69} +11.6603 q^{71} +6.39230 q^{73} -2.73205 q^{75} +12.9282 q^{77} +14.5885 q^{79} -2.46410 q^{81} -8.53590 q^{83} +1.00000 q^{85} -9.46410 q^{87} +4.39230 q^{89} -10.9282 q^{91} -8.92820 q^{93} -1.46410 q^{95} -4.92820 q^{97} -21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 4 q^{19} + 8 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} + 10 q^{31} + 12 q^{33} + 2 q^{35} + 8 q^{37} - 8 q^{39} + 8 q^{43}+ \cdots - 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 4 * q^19 + 8 * q^21 - 6 * q^23 + 2 * q^25 - 8 * q^27 + 10 * q^31 + 12 * q^33 + 2 * q^35 + 8 * q^37 - 8 * q^39 + 8 * q^43 - 2 * q^45 + 12 * q^47 - 6 * q^49 + 2 * q^51 - 12 * q^53 + 6 * q^55 - 8 * q^57 - 12 * q^59 - 4 * q^61 - 14 * q^63 - 8 * q^65 + 20 * q^67 + 24 * q^69 + 6 * q^71 - 8 * q^73 - 2 * q^75 + 12 * q^77 - 2 * q^79 + 2 * q^81 - 24 * q^83 + 2 * q^85 - 12 * q^87 - 12 * q^89 - 8 * q^91 - 4 * q^93 + 4 * q^95 + 4 * q^97 - 18 * q^99

Introduction

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