LMFDB - Embedded newform 676.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,2,Mod(1,676)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(676, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("676.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	676.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:	$5.39788717664$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	676.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.73205 q^{7} -2.00000 q^{9} -5.19615 q^{11} -3.00000 q^{17} +5.19615 q^{19} -1.73205 q^{21} -3.00000 q^{23} -5.00000 q^{25} +5.00000 q^{27} -9.00000 q^{29} +3.46410 q^{31} +5.19615 q^{33} -5.19615 q^{37} +5.19615 q^{41} -5.00000 q^{43} -10.3923 q^{47} -4.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} -5.19615 q^{57} -5.19615 q^{59} -5.00000 q^{61} -3.46410 q^{63} -1.73205 q^{67} +3.00000 q^{69} +5.19615 q^{71} +6.92820 q^{73} +5.00000 q^{75} -9.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} +10.3923 q^{83} +9.00000 q^{87} +15.5885 q^{89} -3.46410 q^{93} -12.1244 q^{97} +10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{9} - 6 q^{17} - 6 q^{23} - 10 q^{25} + 10 q^{27} - 18 q^{29} - 10 q^{43} - 8 q^{49} + 6 q^{51} - 12 q^{53} - 10 q^{61} + 6 q^{69} + 10 q^{75} - 18 q^{77} + 8 q^{79} + 2 q^{81} + 18 q^{87}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^9 - 6 * q^17 - 6 * q^23 - 10 * q^25 + 10 * q^27 - 18 * q^29 - 10 * q^43 - 8 * q^49 + 6 * q^51 - 12 * q^53 - 10 * q^61 + 6 * q^69 + 10 * q^75 - 18 * q^77 + 8 * q^79 + 2 * q^81 + 18 * q^87

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−5.19615	−1.56670	−0.783349	−	0.621582i	$-0.786490\pi$
$11$	−5.19615	−1.56670	−0.783349	+	0.621582i	$0.786490\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	5.19615	1.19208	0.596040	−	0.802955i	$-0.296740\pi$
$19$	5.19615	1.19208	0.596040	+	0.802955i	$0.296740\pi$
$20$	0	0
$21$	−1.73205	−0.377964
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	5.19615	0.904534
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.19615	−0.854242	−0.427121	−	0.904194i	$-0.640472\pi$
$37$	−5.19615	−0.854242	−0.427121	+	0.904194i	$0.640472\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.19615	0.811503	0.405751	−	0.913984i	$-0.367010\pi$
$41$	5.19615	0.811503	0.405751	+	0.913984i	$0.367010\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	3.00000	0.420084
$52$	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
$55$	0	0
$56$	0	0
$57$	−5.19615	−0.688247
$58$	0	0
$59$	−5.19615	−0.676481	−0.338241	−	0.941060i	$-0.609832\pi$
$59$	−5.19615	−0.676481	−0.338241	+	0.941060i	$0.609832\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	−3.46410	−0.436436
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.73205	−0.211604	−0.105802	−	0.994387i	$-0.533741\pi$
$67$	−1.73205	−0.211604	−0.105802	+	0.994387i	$0.533741\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	5.19615	0.616670	0.308335	−	0.951278i	$-0.400228\pi$
$71$	5.19615	0.616670	0.308335	+	0.951278i	$0.400228\pi$
$72$	0	0
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	0	0
$77$	−9.00000	−1.02565
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.3923	1.14070	0.570352	−	0.821401i	$-0.306807\pi$
$83$	10.3923	1.14070	0.570352	+	0.821401i	$0.306807\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.00000	0.964901
$88$	0	0
$89$	15.5885	1.65237	0.826187	−	0.563397i	$-0.190506\pi$
$89$	15.5885	1.65237	0.826187	+	0.563397i	$0.190506\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.46410	−0.359211
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.1244	−1.23104	−0.615521	−	0.788121i	$-0.711054\pi$
$97$	−12.1244	−1.23104	−0.615521	+	0.788121i	$0.711054\pi$
$98$	0	0
$99$	10.3923	1.04447
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	0	0
$109$	−13.8564	−1.32720	−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564	−1.32720	−0.663602	+	0.748086i	$0.730973\pi$
$110$	0	0
$111$	5.19615	0.493197
$112$	0	0
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.19615	−0.476331
$120$	0	0
$121$	16.0000	1.45455
$122$	0	0
$123$	−5.19615	−0.468521
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	0	0
$129$	5.00000	0.440225
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	9.00000	0.780399
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.5885	−1.33181	−0.665906	−	0.746036i	$-0.731955\pi$
$137$	−15.5885	−1.33181	−0.665906	+	0.746036i	$0.731955\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	10.3923	0.875190
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.00000	0.329914
$148$	0	0
$149$	−5.19615	−0.425685	−0.212843	−	0.977086i	$-0.568272\pi$
$149$	−5.19615	−0.425685	−0.212843	+	0.977086i	$0.568272\pi$
$150$	0	0
$151$	10.3923	0.845714	0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923	0.845714	0.422857	+	0.906196i	$0.361027\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−5.19615	−0.409514
$162$	0	0
$163$	−12.1244	−0.949653	−0.474826	−	0.880079i	$-0.657489\pi$
$163$	−12.1244	−0.949653	−0.474826	+	0.880079i	$0.657489\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.5885	1.20627	0.603136	−	0.797639i	$-0.293918\pi$
$167$	15.5885	1.20627	0.603136	+	0.797639i	$0.293918\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−10.3923	−0.794719
$172$	0	0
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	0	0
$175$	−8.66025	−0.654654
$176$	0	0
$177$	5.19615	0.390567
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.5885	1.13994
$188$	0	0
$189$	8.66025	0.629941
$190$	0	0
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	0	0
$193$	8.66025	0.623379	0.311689	−	0.950184i	$-0.399105\pi$
$193$	8.66025	0.623379	0.311689	+	0.950184i	$0.399105\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.9808	1.85105	0.925526	−	0.378683i	$-0.123623\pi$
$197$	25.9808	1.85105	0.925526	+	0.378683i	$0.123623\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	0	0
$201$	1.73205	0.122169
$202$	0	0
$203$	−15.5885	−1.09410
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−27.0000	−1.86763
$210$	0	0
$211$	7.00000	0.481900	0.240950	−	0.970538i	$-0.422541\pi$
$211$	7.00000	0.481900	0.240950	+	0.970538i	$0.422541\pi$
$212$	0	0
$213$	−5.19615	−0.356034
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	−6.92820	−0.468165
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−15.5885	−1.04388	−0.521940	−	0.852982i	$-0.674792\pi$
$223$	−15.5885	−1.04388	−0.521940	+	0.852982i	$0.674792\pi$
$224$	0	0
$225$	10.0000	0.666667
$226$	0	0
$227$	5.19615	0.344881	0.172440	−	0.985020i	$-0.444835\pi$
$227$	5.19615	0.344881	0.172440	+	0.985020i	$0.444835\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	9.00000	0.592157
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−10.3923	−0.672222	−0.336111	−	0.941822i	$-0.609112\pi$
$239$	−10.3923	−0.672222	−0.336111	+	0.941822i	$0.609112\pi$
$240$	0	0
$241$	5.19615	0.334714	0.167357	−	0.985896i	$-0.446477\pi$
$241$	5.19615	0.334714	0.167357	+	0.985896i	$0.446477\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.3923	−0.658586
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	15.5885	0.980038
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	18.0000	1.11417
$262$	0	0
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−15.5885	−0.953998
$268$	0	0
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	−12.1244	−0.736502	−0.368251	−	0.929726i	$-0.620043\pi$
$271$	−12.1244	−0.736502	−0.368251	+	0.929726i	$0.620043\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.9808	1.56670
$276$	0	0
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	0	0
$279$	−6.92820	−0.414781
$280$	0	0
$281$	−20.7846	−1.23991	−0.619953	−	0.784639i	$-0.712848\pi$
$281$	−20.7846	−1.23991	−0.619953	+	0.784639i	$0.712848\pi$
$282$	0	0
$283$	−7.00000	−0.416107	−0.208053	−	0.978117i	$-0.566713\pi$
$283$	−7.00000	−0.416107	−0.208053	+	0.978117i	$0.566713\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	12.1244	0.710742
$292$	0	0
$293$	5.19615	0.303562	0.151781	−	0.988414i	$-0.451499\pi$
$293$	5.19615	0.303562	0.151781	+	0.988414i	$0.451499\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−25.9808	−1.50756
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−8.66025	−0.499169
$302$	0	0
$303$	3.00000	0.172345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.3205	0.988534	0.494267	−	0.869310i	$-0.335437\pi$
$307$	17.3205	0.988534	0.494267	+	0.869310i	$0.335437\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.7846	1.16738	0.583690	−	0.811977i	$-0.301608\pi$
$317$	20.7846	1.16738	0.583690	+	0.811977i	$0.301608\pi$
$318$	0	0
$319$	46.7654	2.61836
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	−15.5885	−0.867365
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.8564	0.766261
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	12.1244	0.666415	0.333207	−	0.942854i	$-0.391869\pi$
$331$	12.1244	0.666415	0.333207	+	0.942854i	$0.391869\pi$
$332$	0	0
$333$	10.3923	0.569495
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	−3.00000	−0.162938
$340$	0	0
$341$	−18.0000	−0.974755
$342$	0	0
$343$	−19.0526	−1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	−19.0526	−1.01986	−0.509930	−	0.860216i	$-0.670329\pi$
$349$	−19.0526	−1.01986	−0.509930	+	0.860216i	$0.670329\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.5885	−0.829690	−0.414845	−	0.909892i	$-0.636164\pi$
$353$	−15.5885	−0.829690	−0.414845	+	0.909892i	$0.636164\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.19615	0.275010
$358$	0	0
$359$	−31.1769	−1.64545	−0.822727	−	0.568436i	$-0.807549\pi$
$359$	−31.1769	−1.64545	−0.822727	+	0.568436i	$0.807549\pi$
$360$	0	0
$361$	8.00000	0.421053
$362$	0	0
$363$	−16.0000	−0.839782
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.0000	−0.678594	−0.339297	−	0.940679i	$-0.610189\pi$
$367$	−13.0000	−0.678594	−0.339297	+	0.940679i	$0.610189\pi$
$368$	0	0
$369$	−10.3923	−0.541002
$370$	0	0
$371$	−10.3923	−0.539542
$372$	0	0
$373$	7.00000	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$373$	7.00000	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−8.66025	−0.444847	−0.222424	−	0.974950i	$-0.571397\pi$
$379$	−8.66025	−0.444847	−0.222424	+	0.974950i	$0.571397\pi$
$380$	0	0
$381$	−17.0000	−0.870936
$382$	0	0
$383$	25.9808	1.32755	0.663777	−	0.747930i	$-0.268952\pi$
$383$	25.9808	1.32755	0.663777	+	0.747930i	$0.268952\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.73205	−0.0869291	−0.0434646	−	0.999055i	$-0.513840\pi$
$397$	−1.73205	−0.0869291	−0.0434646	+	0.999055i	$0.513840\pi$
$398$	0	0
$399$	−9.00000	−0.450564
$400$	0	0
$401$	−5.19615	−0.259483	−0.129742	−	0.991548i	$-0.541415\pi$
$401$	−5.19615	−0.259483	−0.129742	+	0.991548i	$0.541415\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	27.0000	1.33834
$408$	0	0
$409$	8.66025	0.428222	0.214111	−	0.976809i	$-0.431315\pi$
$409$	8.66025	0.428222	0.214111	+	0.976809i	$0.431315\pi$
$410$	0	0
$411$	15.5885	0.768922
$412$	0	0
$413$	−9.00000	−0.442861
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.00000	0.342791
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	13.8564	0.675320	0.337660	−	0.941268i	$-0.390365\pi$
$421$	13.8564	0.675320	0.337660	+	0.941268i	$0.390365\pi$
$422$	0	0
$423$	20.7846	1.01058
$424$	0	0
$425$	15.0000	0.727607
$426$	0	0
$427$	−8.66025	−0.419099
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.19615	0.250290	0.125145	−	0.992138i	$-0.460060\pi$
$431$	5.19615	0.250290	0.125145	+	0.992138i	$0.460060\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.5885	−0.745697
$438$	0	0
$439$	−23.0000	−1.09773	−0.548865	−	0.835911i	$-0.684940\pi$
$439$	−23.0000	−1.09773	−0.548865	+	0.835911i	$0.684940\pi$
$440$	0	0
$441$	8.00000	0.380952
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.19615	0.245770
$448$	0	0
$449$	5.19615	0.245222	0.122611	−	0.992455i	$-0.460873\pi$
$449$	5.19615	0.245222	0.122611	+	0.992455i	$0.460873\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	0	0
$453$	−10.3923	−0.488273
$454$	0	0
$455$	0	0
$456$	0	0
$457$	39.8372	1.86350	0.931752	−	0.363095i	$-0.118280\pi$
$457$	39.8372	1.86350	0.931752	+	0.363095i	$0.118280\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	−25.9808	−1.21004	−0.605022	−	0.796208i	$-0.706836\pi$
$461$	−25.9808	−1.21004	−0.605022	+	0.796208i	$0.706836\pi$
$462$	0	0
$463$	−17.3205	−0.804952	−0.402476	−	0.915430i	$-0.631850\pi$
$463$	−17.3205	−0.804952	−0.402476	+	0.915430i	$0.631850\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−3.00000	−0.138527
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	25.9808	1.19460
$474$	0	0
$475$	−25.9808	−1.19208
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	36.3731	1.66193	0.830964	−	0.556326i	$-0.187790\pi$
$479$	36.3731	1.66193	0.830964	+	0.556326i	$0.187790\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	5.19615	0.236433
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.3731	−1.64822	−0.824110	−	0.566429i	$-0.808325\pi$
$487$	−36.3731	−1.64822	−0.824110	+	0.566429i	$0.808325\pi$
$488$	0	0
$489$	12.1244	0.548282
$490$	0	0
$491$	−27.0000	−1.21849	−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000	−1.21849	−0.609246	+	0.792981i	$0.708528\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.00000	0.403705
$498$	0	0
$499$	−31.1769	−1.39567	−0.697835	−	0.716258i	$-0.745853\pi$
$499$	−31.1769	−1.39567	−0.697835	+	0.716258i	$0.745853\pi$
$500$	0	0
$501$	−15.5885	−0.696441
$502$	0	0
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.3731	−1.61221	−0.806104	−	0.591774i	$-0.798428\pi$
$509$	−36.3731	−1.61221	−0.806104	+	0.591774i	$0.798428\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	0	0
$513$	25.9808	1.14708
$514$	0	0
$515$	0	0
$516$	0	0
$517$	54.0000	2.37492
$518$	0	0
$519$	3.00000	0.131685
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	0	0
$525$	8.66025	0.377964
$526$	0	0
$527$	−10.3923	−0.452696
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	10.3923	0.450988
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.0000	0.647298
$538$	0	0
$539$	20.7846	0.895257
$540$	0	0
$541$	−6.92820	−0.297867	−0.148933	−	0.988847i	$-0.547584\pi$
$541$	−6.92820	−0.297867	−0.148933	+	0.988847i	$0.547584\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−46.7654	−1.99227
$552$	0	0
$553$	6.92820	0.294617
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.5885	0.660504	0.330252	−	0.943893i	$-0.392866\pi$
$557$	15.5885	0.660504	0.330252	+	0.943893i	$0.392866\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−15.5885	−0.658145
$562$	0	0
$563$	15.0000	0.632175	0.316087	−	0.948730i	$-0.397631\pi$
$563$	15.0000	0.632175	0.316087	+	0.948730i	$0.397631\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.73205	0.0727393
$568$	0	0
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−21.0000	−0.877288
$574$	0	0
$575$	15.0000	0.625543
$576$	0	0
$577$	20.7846	0.865275	0.432637	−	0.901568i	$-0.357583\pi$
$577$	20.7846	0.865275	0.432637	+	0.901568i	$0.357583\pi$
$578$	0	0
$579$	−8.66025	−0.359908
$580$	0	0
$581$	18.0000	0.746766
$582$	0	0
$583$	31.1769	1.29122
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.9808	1.07234	0.536170	−	0.844110i	$-0.319870\pi$
$587$	25.9808	1.07234	0.536170	+	0.844110i	$0.319870\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	0	0
$591$	−25.9808	−1.06871
$592$	0	0
$593$	20.7846	0.853522	0.426761	−	0.904365i	$-0.359655\pi$
$593$	20.7846	0.853522	0.426761	+	0.904365i	$0.359655\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.00000	0.0409273
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	3.46410	0.141069
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.0000	1.66414	0.832069	−	0.554672i	$-0.187156\pi$
$607$	41.0000	1.66414	0.832069	+	0.554672i	$0.187156\pi$
$608$	0	0
$609$	15.5885	0.631676
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−22.5167	−0.909439	−0.454720	−	0.890635i	$-0.650261\pi$
$613$	−22.5167	−0.909439	−0.454720	+	0.890635i	$0.650261\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.9808	−1.04595	−0.522973	−	0.852349i	$-0.675177\pi$
$617$	−25.9808	−1.04595	−0.522973	+	0.852349i	$0.675177\pi$
$618$	0	0
$619$	45.0333	1.81004	0.905021	−	0.425367i	$-0.139855\pi$
$619$	45.0333	1.81004	0.905021	+	0.425367i	$0.139855\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	0	0
$623$	27.0000	1.08173
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	27.0000	1.07828
$628$	0	0
$629$	15.5885	0.621552
$630$	0	0
$631$	8.66025	0.344759	0.172380	−	0.985031i	$-0.444854\pi$
$631$	8.66025	0.344759	0.172380	+	0.985031i	$0.444854\pi$
$632$	0	0
$633$	−7.00000	−0.278225
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.3923	−0.411113
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	0	0
$643$	−43.3013	−1.70764	−0.853818	−	0.520572i	$-0.825719\pi$
$643$	−43.3013	−1.70764	−0.853818	+	0.520572i	$0.825719\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	0	0
$649$	27.0000	1.05984
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.8564	−0.540590
$658$	0	0
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	0	0
$661$	22.5167	0.875797	0.437898	−	0.899025i	$-0.355723\pi$
$661$	22.5167	0.875797	0.437898	+	0.899025i	$0.355723\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27.0000	1.04544
$668$	0	0
$669$	15.5885	0.602685
$670$	0	0
$671$	25.9808	1.00298
$672$	0	0
$673$	−23.0000	−0.886585	−0.443292	−	0.896377i	$-0.646190\pi$
$673$	−23.0000	−0.886585	−0.443292	+	0.896377i	$0.646190\pi$
$674$	0	0
$675$	−25.0000	−0.962250
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	−21.0000	−0.805906
$680$	0	0
$681$	−5.19615	−0.199117
$682$	0	0
$683$	−5.19615	−0.198825	−0.0994126	−	0.995046i	$-0.531696\pi$
$683$	−5.19615	−0.198825	−0.0994126	+	0.995046i	$0.531696\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−8.66025	−0.329452	−0.164726	−	0.986339i	$-0.552674\pi$
$691$	−8.66025	−0.329452	−0.164726	+	0.986339i	$0.552674\pi$
$692$	0	0
$693$	18.0000	0.683763
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.5885	−0.590455
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−27.0000	−1.01832
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.19615	−0.195421
$708$	0	0
$709$	−29.4449	−1.10583	−0.552913	−	0.833239i	$-0.686484\pi$
$709$	−29.4449	−1.10583	−0.552913	+	0.833239i	$0.686484\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−10.3923	−0.389195
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.3923	0.388108
$718$	0	0
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	0	0
$721$	27.7128	1.03208
$722$	0	0
$723$	−5.19615	−0.193247
$724$	0	0
$725$	45.0000	1.67126
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	15.0000	0.554795
$732$	0	0
$733$	−34.6410	−1.27950	−0.639748	−	0.768585i	$-0.720961\pi$
$733$	−34.6410	−1.27950	−0.639748	+	0.768585i	$0.720961\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.00000	0.331519
$738$	0	0
$739$	15.5885	0.573431	0.286715	−	0.958016i	$-0.407437\pi$
$739$	15.5885	0.573431	0.286715	+	0.958016i	$0.407437\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.19615	0.190628	0.0953142	−	0.995447i	$-0.469614\pi$
$743$	5.19615	0.190628	0.0953142	+	0.995447i	$0.469614\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−20.7846	−0.760469
$748$	0	0
$749$	25.9808	0.949316
$750$	0	0
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	0	0
$753$	21.0000	0.765283
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.0000	1.56286	0.781431	−	0.623992i	$-0.214490\pi$
$757$	43.0000	1.56286	0.781431	+	0.623992i	$0.214490\pi$
$758$	0	0
$759$	−15.5885	−0.565825
$760$	0	0
$761$	5.19615	0.188360	0.0941802	−	0.995555i	$-0.469977\pi$
$761$	5.19615	0.188360	0.0941802	+	0.995555i	$0.469977\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−36.3731	−1.31165	−0.655823	−	0.754915i	$-0.727678\pi$
$769$	−36.3731	−1.31165	−0.655823	+	0.754915i	$0.727678\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	0	0
$773$	−25.9808	−0.934463	−0.467232	−	0.884135i	$-0.654749\pi$
$773$	−25.9808	−0.934463	−0.467232	+	0.884135i	$0.654749\pi$
$774$	0	0
$775$	−17.3205	−0.622171
$776$	0	0
$777$	9.00000	0.322873
$778$	0	0
$779$	27.0000	0.967375
$780$	0	0
$781$	−27.0000	−0.966136
$782$	0	0
$783$	−45.0000	−1.60817
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.73205	0.0617409	0.0308705	−	0.999523i	$-0.490172\pi$
$787$	1.73205	0.0617409	0.0308705	+	0.999523i	$0.490172\pi$
$788$	0	0
$789$	21.0000	0.747620
$790$	0	0
$791$	5.19615	0.184754
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.0000	−0.956389	−0.478195	−	0.878254i	$-0.658709\pi$
$797$	−27.0000	−0.956389	−0.478195	+	0.878254i	$0.658709\pi$
$798$	0	0
$799$	31.1769	1.10296
$800$	0	0
$801$	−31.1769	−1.10158
$802$	0	0
$803$	−36.0000	−1.27041
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.00000	0.316815
$808$	0	0
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	45.0333	1.58133	0.790667	−	0.612247i	$-0.209734\pi$
$811$	45.0333	1.58133	0.790667	+	0.612247i	$0.209734\pi$
$812$	0	0
$813$	12.1244	0.425220
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−25.9808	−0.908952
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.9808	0.906735	0.453367	−	0.891324i	$-0.350223\pi$
$821$	25.9808	0.906735	0.453367	+	0.891324i	$0.350223\pi$
$822$	0	0
$823$	−41.0000	−1.42917	−0.714585	−	0.699549i	$-0.753384\pi$
$823$	−41.0000	−1.42917	−0.714585	+	0.699549i	$0.753384\pi$
$824$	0	0
$825$	−25.9808	−0.904534
$826$	0	0
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	−55.0000	−1.91023	−0.955114	−	0.296237i	$-0.904268\pi$
$829$	−55.0000	−1.91023	−0.955114	+	0.296237i	$0.904268\pi$
$830$	0	0
$831$	−1.00000	−0.0346896
$832$	0	0
$833$	12.0000	0.415775
$834$	0	0
$835$	0	0
$836$	0	0
$837$	17.3205	0.598684
$838$	0	0
$839$	36.3731	1.25574	0.627869	−	0.778319i	$-0.283927\pi$
$839$	36.3731	1.25574	0.627869	+	0.778319i	$0.283927\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	20.7846	0.715860
$844$	0	0
$845$	0	0
$846$	0	0
$847$	27.7128	0.952224
$848$	0	0
$849$	7.00000	0.240239
$850$	0	0
$851$	15.5885	0.534365
$852$	0	0
$853$	13.8564	0.474434	0.237217	−	0.971457i	$-0.423765\pi$
$853$	13.8564	0.474434	0.237217	+	0.971457i	$0.423765\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	0	0
$863$	−10.3923	−0.353758	−0.176879	−	0.984233i	$-0.556600\pi$
$863$	−10.3923	−0.353758	−0.176879	+	0.984233i	$0.556600\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−20.7846	−0.705070
$870$	0	0
$871$	0	0
$872$	0	0
$873$	24.2487	0.820695
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.3731	1.22823	0.614116	−	0.789216i	$-0.289513\pi$
$877$	36.3731	1.22823	0.614116	+	0.789216i	$0.289513\pi$
$878$	0	0
$879$	−5.19615	−0.175262
$880$	0	0
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.0000	0.503651	0.251825	−	0.967773i	$-0.418969\pi$
$887$	15.0000	0.503651	0.251825	+	0.967773i	$0.418969\pi$
$888$	0	0
$889$	29.4449	0.987549
$890$	0	0
$891$	−5.19615	−0.174078
$892$	0	0
$893$	−54.0000	−1.80704
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−31.1769	−1.03981
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	8.66025	0.288195
$904$	0	0
$905$	0	0
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−54.0000	−1.78714
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.7846	−0.686368
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	−17.3205	−0.570730
$922$	0	0
$923$	0	0
$924$	0	0
$925$	25.9808	0.854242
$926$	0	0
$927$	−32.0000	−1.05102
$928$	0	0
$929$	36.3731	1.19336	0.596681	−	0.802479i	$-0.296486\pi$
$929$	36.3731	1.19336	0.596681	+	0.802479i	$0.296486\pi$
$930$	0	0
$931$	−20.7846	−0.681188
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−20.7846	−0.677559	−0.338779	−	0.940866i	$-0.610014\pi$
$941$	−20.7846	−0.677559	−0.338779	+	0.940866i	$0.610014\pi$
$942$	0	0
$943$	−15.5885	−0.507630
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.7654	−1.51967	−0.759835	−	0.650116i	$-0.774720\pi$
$947$	−46.7654	−1.51967	−0.759835	+	0.650116i	$0.774720\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−20.7846	−0.673987
$952$	0	0
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−46.7654	−1.51171
$958$	0	0
$959$	−27.0000	−0.871875
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	−30.0000	−0.966736
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−10.3923	−0.334194	−0.167097	−	0.985940i	$-0.553439\pi$
$967$	−10.3923	−0.334194	−0.167097	+	0.985940i	$0.553439\pi$
$968$	0	0
$969$	15.5885	0.500773
$970$	0	0
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	0	0
$973$	−12.1244	−0.388689
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.5885	−0.498719	−0.249359	−	0.968411i	$-0.580220\pi$
$977$	−15.5885	−0.498719	−0.249359	+	0.968411i	$0.580220\pi$
$978$	0	0
$979$	−81.0000	−2.58877
$980$	0	0
$981$	27.7128	0.884802
$982$	0	0
$983$	10.3923	0.331463	0.165732	−	0.986171i	$-0.447001\pi$
$983$	10.3923	0.331463	0.165732	+	0.986171i	$0.447001\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	18.0000	0.572946
$988$	0	0
$989$	15.0000	0.476972
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	0	0
$993$	−12.1244	−0.384755
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	0	0
$999$	−25.9808	−0.821995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.2.a.f.1.2		2
3.2	odd	2		6084.2.a.t.1.2		2
4.3	odd	2		2704.2.a.u.1.1		2
13.2	odd	12		676.2.h.a.485.1		2
13.3	even	3		676.2.e.e.529.1		4
13.4	even	6		676.2.e.e.653.2		4
13.5	odd	4		676.2.d.b.337.2		2
13.6	odd	12		52.2.h.a.49.1	yes	2
13.7	odd	12		676.2.h.a.361.1		2
13.8	odd	4		676.2.d.b.337.1		2
13.9	even	3		676.2.e.e.653.1		4
13.10	even	6		676.2.e.e.529.2		4
13.11	odd	12		52.2.h.a.17.1	✓	2
13.12	even	2	inner	676.2.a.f.1.1		2
39.5	even	4		6084.2.b.d.4393.2		2
39.8	even	4		6084.2.b.d.4393.1		2
39.11	even	12		468.2.t.a.433.1		2
39.32	even	12		468.2.t.a.361.1		2
39.38	odd	2		6084.2.a.t.1.1		2
52.11	even	12		208.2.w.a.17.1		2
52.19	even	12		208.2.w.a.49.1		2
52.31	even	4		2704.2.f.h.337.1		2
52.47	even	4		2704.2.f.h.337.2		2
52.51	odd	2		2704.2.a.u.1.2		2
65.19	odd	12		1300.2.y.a.101.1		2
65.24	odd	12		1300.2.y.a.901.1		2
65.32	even	12		1300.2.ba.a.49.2		4
65.37	even	12		1300.2.ba.a.849.1		4
65.58	even	12		1300.2.ba.a.49.1		4
65.63	even	12		1300.2.ba.a.849.2		4
91.6	even	12		2548.2.u.a.1765.1		2
91.11	odd	12		2548.2.bq.a.1941.1		2
91.19	even	12		2548.2.bq.b.361.1		2
91.24	even	12		2548.2.bq.b.1941.1		2
91.32	odd	12		2548.2.bb.b.569.1		2
91.37	odd	12		2548.2.bb.b.1733.1		2
91.45	even	12		2548.2.bb.a.569.1		2
91.58	odd	12		2548.2.bq.a.361.1		2
91.76	even	12		2548.2.u.a.589.1		2
91.89	even	12		2548.2.bb.a.1733.1		2
104.11	even	12		832.2.w.c.641.1		2
104.19	even	12		832.2.w.c.257.1		2
104.37	odd	12		832.2.w.b.641.1		2
104.45	odd	12		832.2.w.b.257.1		2
156.11	odd	12		1872.2.by.e.433.1		2
156.71	odd	12		1872.2.by.e.1297.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.h.a.17.1	✓	2	13.11	odd	12
52.2.h.a.49.1	yes	2	13.6	odd	12
208.2.w.a.17.1		2	52.11	even	12
208.2.w.a.49.1		2	52.19	even	12
468.2.t.a.361.1		2	39.32	even	12
468.2.t.a.433.1		2	39.11	even	12
676.2.a.f.1.1		2	13.12	even	2	inner
676.2.a.f.1.2		2	1.1	even	1	trivial
676.2.d.b.337.1		2	13.8	odd	4
676.2.d.b.337.2		2	13.5	odd	4
676.2.e.e.529.1		4	13.3	even	3
676.2.e.e.529.2		4	13.10	even	6
676.2.e.e.653.1		4	13.9	even	3
676.2.e.e.653.2		4	13.4	even	6
676.2.h.a.361.1		2	13.7	odd	12
676.2.h.a.485.1		2	13.2	odd	12
832.2.w.b.257.1		2	104.45	odd	12
832.2.w.b.641.1		2	104.37	odd	12
832.2.w.c.257.1		2	104.19	even	12
832.2.w.c.641.1		2	104.11	even	12
1300.2.y.a.101.1		2	65.19	odd	12
1300.2.y.a.901.1		2	65.24	odd	12
1300.2.ba.a.49.1		4	65.58	even	12
1300.2.ba.a.49.2		4	65.32	even	12
1300.2.ba.a.849.1		4	65.37	even	12
1300.2.ba.a.849.2		4	65.63	even	12
1872.2.by.e.433.1		2	156.11	odd	12
1872.2.by.e.1297.1		2	156.71	odd	12
2548.2.u.a.589.1		2	91.76	even	12
2548.2.u.a.1765.1		2	91.6	even	12
2548.2.bb.a.569.1		2	91.45	even	12
2548.2.bb.a.1733.1		2	91.89	even	12
2548.2.bb.b.569.1		2	91.32	odd	12
2548.2.bb.b.1733.1		2	91.37	odd	12
2548.2.bq.a.361.1		2	91.58	odd	12
2548.2.bq.a.1941.1		2	91.11	odd	12
2548.2.bq.b.361.1		2	91.19	even	12
2548.2.bq.b.1941.1		2	91.24	even	12
2704.2.a.u.1.1		2	4.3	odd	2
2704.2.a.u.1.2		2	52.51	odd	2
2704.2.f.h.337.1		2	52.31	even	4
2704.2.f.h.337.2		2	52.47	even	4
6084.2.a.t.1.1		2	39.38	odd	2
6084.2.a.t.1.2		2	3.2	odd	2
6084.2.b.d.4393.1		2	39.8	even	4
6084.2.b.d.4393.2		2	39.5	even	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:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.73205 q^{7} -2.00000 q^{9} -5.19615 q^{11} -3.00000 q^{17} +5.19615 q^{19} -1.73205 q^{21} -3.00000 q^{23} -5.00000 q^{25} +5.00000 q^{27} -9.00000 q^{29} +3.46410 q^{31} +5.19615 q^{33} -5.19615 q^{37} +5.19615 q^{41} -5.00000 q^{43} -10.3923 q^{47} -4.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} -5.19615 q^{57} -5.19615 q^{59} -5.00000 q^{61} -3.46410 q^{63} -1.73205 q^{67} +3.00000 q^{69} +5.19615 q^{71} +6.92820 q^{73} +5.00000 q^{75} -9.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} +10.3923 q^{83} +9.00000 q^{87} +15.5885 q^{89} -3.46410 q^{93} -12.1244 q^{97} +10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{9} - 6 q^{17} - 6 q^{23} - 10 q^{25} + 10 q^{27} - 18 q^{29} - 10 q^{43} - 8 q^{49} + 6 q^{51} - 12 q^{53} - 10 q^{61} + 6 q^{69} + 10 q^{75} - 18 q^{77} + 8 q^{79} + 2 q^{81} + 18 q^{87}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^9 - 6 * q^17 - 6 * q^23 - 10 * q^25 + 10 * q^27 - 18 * q^29 - 10 * q^43 - 8 * q^49 + 6 * q^51 - 12 * q^53 - 10 * q^61 + 6 * q^69 + 10 * q^75 - 18 * q^77 + 8 * q^79 + 2 * q^81 + 18 * q^87

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