LMFDB - Embedded newform 2704.2.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(1,2704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2704.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$21.5915487066$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2704.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} +1.73205 q^{5} +1.00000 q^{9} -3.46410 q^{15} +3.00000 q^{17} -3.46410 q^{19} -6.00000 q^{23} -2.00000 q^{25} +4.00000 q^{27} +3.00000 q^{29} +3.46410 q^{31} -8.66025 q^{37} -5.19615 q^{41} +8.00000 q^{43} +1.73205 q^{45} +3.46410 q^{47} -7.00000 q^{49} -6.00000 q^{51} -3.00000 q^{53} +6.92820 q^{57} -6.92820 q^{59} +1.00000 q^{61} -3.46410 q^{67} +12.0000 q^{69} +3.46410 q^{71} +1.73205 q^{73} +4.00000 q^{75} -4.00000 q^{79} -11.0000 q^{81} +13.8564 q^{83} +5.19615 q^{85} -6.00000 q^{87} +6.92820 q^{89} -6.92820 q^{93} -6.00000 q^{95} -6.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{9} + 6 q^{17} - 12 q^{23} - 4 q^{25} + 8 q^{27} + 6 q^{29} + 16 q^{43} - 14 q^{49} - 12 q^{51} - 6 q^{53} + 2 q^{61} + 24 q^{69} + 8 q^{75} - 8 q^{79} - 22 q^{81} - 12 q^{87} - 12 q^{95}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^9 + 6 * q^17 - 12 * q^23 - 4 * q^25 + 8 * q^27 + 6 * q^29 + 16 * q^43 - 14 * q^49 - 12 * q^51 - 6 * q^53 + 2 * q^61 + 24 * q^69 + 8 * q^75 - 8 * q^79 - 22 * q^81 - 12 * q^87 - 12 * q^95

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.46410	−0.894427
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	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
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.66025	−1.42374	−0.711868	−	0.702313i	$-0.752151\pi$
$37$	−8.66025	−1.42374	−0.711868	+	0.702313i	$0.752151\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.19615	−0.811503	−0.405751	−	0.913984i	$-0.632990\pi$
$41$	−5.19615	−0.811503	−0.405751	+	0.913984i	$0.632990\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	1.73205	0.258199
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.92820	0.917663
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	1.73205	0.202721	0.101361	−	0.994850i	$-0.467680\pi$
$73$	1.73205	0.202721	0.101361	+	0.994850i	$0.467680\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	13.8564	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$84$	0	0
$85$	5.19615	0.563602
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	6.92820	0.734388	0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820	0.734388	0.367194	+	0.930144i	$0.380318\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.92820	−0.718421
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−6.92820	−0.703452	−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	13.8564	1.32720	0.663602	−	0.748086i	$-0.269027\pi$
$109$	13.8564	1.32720	0.663602	+	0.748086i	$0.269027\pi$
$110$	0	0
$111$	17.3205	1.64399
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	−10.3923	−0.969087
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	10.3923	0.937043
$124$	0	0
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−16.0000	−1.40872
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.92820	0.596285
$136$	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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−6.92820	−0.583460
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.19615	0.431517
$146$	0	0
$147$	14.0000	1.15470
$148$	0	0
$149$	19.0526	1.56085	0.780423	−	0.625252i	$-0.215004\pi$
$149$	19.0526	1.56085	0.780423	+	0.625252i	$0.215004\pi$
$150$	0	0
$151$	−17.3205	−1.40952	−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205	−1.40952	−0.704761	+	0.709444i	$0.748946\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.7846	−1.62798	−0.813988	−	0.580881i	$-0.802708\pi$
$163$	−20.7846	−1.62798	−0.813988	+	0.580881i	$0.802708\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.46410	−0.264906
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.8564	1.04151
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−15.0000	−1.10282
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	5.19615	0.374027	0.187014	−	0.982357i	$-0.440119\pi$
$193$	5.19615	0.374027	0.187014	+	0.982357i	$0.440119\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	6.92820	0.488678
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	−6.92820	−0.474713
$214$	0	0
$215$	13.8564	0.944999
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3.46410	−0.234082
$220$	0	0
$221$	0	0
$222$	0	0
$223$	10.3923	0.695920	0.347960	−	0.937509i	$-0.386874\pi$
$223$	10.3923	0.695920	0.347960	+	0.937509i	$0.386874\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	24.2487	1.60944	0.804722	−	0.593652i	$-0.202314\pi$
$227$	24.2487	1.60944	0.804722	+	0.593652i	$0.202314\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	6.00000	0.391397
$236$	0	0
$237$	8.00000	0.519656
$238$	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
$241$	−1.73205	−0.111571	−0.0557856	−	0.998443i	$-0.517766\pi$
$241$	−1.73205	−0.111571	−0.0557856	+	0.998443i	$0.517766\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	−12.1244	−0.774597
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−27.7128	−1.75623
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−10.3923	−0.650791
$256$	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
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−5.19615	−0.319197
$266$	0	0
$267$	−13.8564	−0.847998
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	20.7846	1.26258	0.631288	−	0.775549i	$-0.282527\pi$
$271$	20.7846	1.26258	0.631288	+	0.775549i	$0.282527\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.00000	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	0	0
$279$	3.46410	0.207390
$280$	0	0
$281$	−22.5167	−1.34323	−0.671616	−	0.740900i	$-0.734399\pi$
$281$	−22.5167	−1.34323	−0.671616	+	0.740900i	$0.734399\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	12.0000	0.710819
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	13.8564	0.812277
$292$	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
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	1.73205	0.0991769
$306$	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
$309$	20.0000	1.13776
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.19615	0.291845	0.145922	−	0.989296i	$-0.453385\pi$
$317$	5.19615	0.291845	0.145922	+	0.989296i	$0.453385\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−10.3923	−0.578243
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−27.7128	−1.53252
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−27.7128	−1.52323	−0.761617	−	0.648027i	$-0.775594\pi$
$331$	−27.7128	−1.52323	−0.761617	+	0.648027i	$0.775594\pi$
$332$	0	0
$333$	−8.66025	−0.474579
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	30.0000	1.62938
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	20.7846	1.11901
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	−13.8564	−0.741716	−0.370858	−	0.928689i	$-0.620936\pi$
$349$	−13.8564	−0.741716	−0.370858	+	0.928689i	$0.620936\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.9090	1.75157	0.875784	−	0.482704i	$-0.160345\pi$
$353$	32.9090	1.75157	0.875784	+	0.482704i	$0.160345\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.92820	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	22.0000	1.15470
$364$	0	0
$365$	3.00000	0.157027
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	0	0
$369$	−5.19615	−0.270501
$370$	0	0
$371$	0	0
$372$	0	0
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	24.2487	1.25220
$376$	0	0
$377$	0	0
$378$	0	0
$379$	24.2487	1.24557	0.622786	−	0.782392i	$-0.286001\pi$
$379$	24.2487	1.24557	0.622786	+	0.782392i	$0.286001\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	−20.7846	−1.06204	−0.531022	−	0.847358i	$-0.678192\pi$
$383$	−20.7846	−1.06204	−0.531022	+	0.847358i	$0.678192\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	36.0000	1.81596
$394$	0	0
$395$	−6.92820	−0.348596
$396$	0	0
$397$	−13.8564	−0.695433	−0.347717	−	0.937600i	$-0.613043\pi$
$397$	−13.8564	−0.695433	−0.347717	+	0.937600i	$0.613043\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.73205	0.0864945	0.0432472	−	0.999064i	$-0.486230\pi$
$401$	1.73205	0.0864945	0.0432472	+	0.999064i	$0.486230\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−19.0526	−0.946729
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−15.5885	−0.770800	−0.385400	−	0.922750i	$-0.625936\pi$
$409$	−15.5885	−0.770800	−0.385400	+	0.922750i	$0.625936\pi$
$410$	0	0
$411$	31.1769	1.53784
$412$	0	0
$413$	0	0
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−15.5885	−0.759735	−0.379867	−	0.925041i	$-0.624030\pi$
$421$	−15.5885	−0.759735	−0.379867	+	0.925041i	$0.624030\pi$
$422$	0	0
$423$	3.46410	0.168430
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.92820	0.333720	0.166860	−	0.985981i	$-0.446637\pi$
$431$	6.92820	0.333720	0.166860	+	0.985981i	$0.446637\pi$
$432$	0	0
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	0	0
$435$	−10.3923	−0.498273
$436$	0	0
$437$	20.7846	0.994263
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	−38.1051	−1.80231
$448$	0	0
$449$	−6.92820	−0.326962	−0.163481	−	0.986546i	$-0.552272\pi$
$449$	−6.92820	−0.326962	−0.163481	+	0.986546i	$0.552272\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	34.6410	1.62758
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.73205	0.0810219	0.0405110	−	0.999179i	$-0.487101\pi$
$457$	1.73205	0.0810219	0.0405110	+	0.999179i	$0.487101\pi$
$458$	0	0
$459$	12.0000	0.560112
$460$	0	0
$461$	−22.5167	−1.04871	−0.524353	−	0.851501i	$-0.675693\pi$
$461$	−22.5167	−1.04871	−0.524353	+	0.851501i	$0.675693\pi$
$462$	0	0
$463$	−13.8564	−0.643962	−0.321981	−	0.946746i	$-0.604349\pi$
$463$	−13.8564	−0.643962	−0.321981	+	0.946746i	$0.604349\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	26.0000	1.19802
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.92820	0.317888
$476$	0	0
$477$	−3.00000	−0.137361
$478$	0	0
$479$	24.2487	1.10795	0.553976	−	0.832533i	$-0.313110\pi$
$479$	24.2487	1.10795	0.553976	+	0.832533i	$0.313110\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	−6.92820	−0.313947	−0.156973	−	0.987603i	$-0.550174\pi$
$487$	−6.92820	−0.313947	−0.156973	+	0.987603i	$0.550174\pi$
$488$	0	0
$489$	41.5692	1.87983
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	31.1769	1.39567	0.697835	−	0.716258i	$-0.254147\pi$
$499$	31.1769	1.39567	0.697835	+	0.716258i	$0.254147\pi$
$500$	0	0
$501$	27.7128	1.23812
$502$	0	0
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	5.19615	0.231226
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.0526	0.844490	0.422245	−	0.906482i	$-0.361242\pi$
$509$	19.0526	0.844490	0.422245	+	0.906482i	$0.361242\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−13.8564	−0.611775
$514$	0	0
$515$	−17.3205	−0.763233
$516$	0	0
$517$	0	0
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.3923	0.452696
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−6.92820	−0.300658
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−10.3923	−0.449299
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	29.4449	1.26593	0.632967	−	0.774179i	$-0.281837\pi$
$541$	29.4449	1.26593	0.632967	+	0.774179i	$0.281837\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	24.0000	1.02805
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	0	0
$549$	1.00000	0.0426790
$550$	0	0
$551$	−10.3923	−0.442727
$552$	0	0
$553$	0	0
$554$	0	0
$555$	30.0000	1.27343
$556$	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
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−25.9808	−1.09302
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	36.0000	1.50392
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	19.0526	0.793168	0.396584	−	0.917998i	$-0.370195\pi$
$577$	19.0526	0.793168	0.396584	+	0.917998i	$0.370195\pi$
$578$	0	0
$579$	−10.3923	−0.431889
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.7846	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	−27.7128	−1.13995
$592$	0	0
$593$	−25.9808	−1.06690	−0.533451	−	0.845831i	$-0.679105\pi$
$593$	−25.9808	−1.06690	−0.533451	+	0.845831i	$0.679105\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.00000	0.163709
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	25.0000	1.01977	0.509886	−	0.860242i	$-0.329688\pi$
$601$	25.0000	1.01977	0.509886	+	0.860242i	$0.329688\pi$
$602$	0	0
$603$	−3.46410	−0.141069
$604$	0	0
$605$	−19.0526	−0.774597
$606$	0	0
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−12.1244	−0.489698	−0.244849	−	0.969561i	$-0.578738\pi$
$613$	−12.1244	−0.489698	−0.244849	+	0.969561i	$0.578738\pi$
$614$	0	0
$615$	18.0000	0.725830
$616$	0	0
$617$	22.5167	0.906487	0.453243	−	0.891387i	$-0.350267\pi$
$617$	22.5167	0.906487	0.453243	+	0.891387i	$0.350267\pi$
$618$	0	0
$619$	20.7846	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$619$	20.7846	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25.9808	−1.03592
$630$	0	0
$631$	48.4974	1.93065	0.965326	−	0.261048i	$-0.0840679\pi$
$631$	48.4974	1.93065	0.965326	+	0.261048i	$0.0840679\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	−3.46410	−0.137469
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.46410	0.137038
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	13.8564	0.546443	0.273222	−	0.961951i	$-0.411911\pi$
$643$	13.8564	0.546443	0.273222	+	0.961951i	$0.411911\pi$
$644$	0	0
$645$	−27.7128	−1.09119
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	−31.1769	−1.21818
$656$	0	0
$657$	1.73205	0.0675737
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	46.7654	1.81896	0.909481	−	0.415745i	$-0.136479\pi$
$661$	46.7654	1.81896	0.909481	+	0.415745i	$0.136479\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.0000	−0.696963
$668$	0	0
$669$	−20.7846	−0.803579
$670$	0	0
$671$	0	0
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	−8.00000	−0.307920
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−48.4974	−1.85843
$682$	0	0
$683$	−24.2487	−0.927851	−0.463926	−	0.885874i	$-0.653559\pi$
$683$	−24.2487	−0.927851	−0.463926	+	0.885874i	$0.653559\pi$
$684$	0	0
$685$	−27.0000	−1.03162
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−13.8564	−0.527123	−0.263561	−	0.964643i	$-0.584897\pi$
$691$	−13.8564	−0.527123	−0.263561	+	0.964643i	$0.584897\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.92820	0.262802
$696$	0	0
$697$	−15.5885	−0.590455
$698$	0	0
$699$	12.0000	0.453882
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	30.0000	1.13147
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.19615	−0.195146	−0.0975728	−	0.995228i	$-0.531108\pi$
$709$	−5.19615	−0.195146	−0.0975728	+	0.995228i	$0.531108\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−20.7846	−0.778390
$714$	0	0
$715$	0	0
$716$	0	0
$717$	41.5692	1.55243
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.46410	0.128831
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	−12.1244	−0.447823	−0.223912	−	0.974609i	$-0.571883\pi$
$733$	−12.1244	−0.447823	−0.223912	+	0.974609i	$0.571883\pi$
$734$	0	0
$735$	24.2487	0.894427
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−20.7846	−0.764574	−0.382287	−	0.924044i	$-0.624863\pi$
$739$	−20.7846	−0.764574	−0.382287	+	0.924044i	$0.624863\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.6410	1.27086	0.635428	−	0.772160i	$-0.280824\pi$
$743$	34.6410	1.27086	0.635428	+	0.772160i	$0.280824\pi$
$744$	0	0
$745$	33.0000	1.20903
$746$	0	0
$747$	13.8564	0.506979
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	36.0000	1.31191
$754$	0	0
$755$	−30.0000	−1.09181
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.6410	1.25574	0.627868	−	0.778320i	$-0.283928\pi$
$761$	34.6410	1.25574	0.627868	+	0.778320i	$0.283928\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.19615	0.187867
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.92820	0.249837	0.124919	−	0.992167i	$-0.460133\pi$
$769$	6.92820	0.249837	0.124919	+	0.992167i	$0.460133\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	34.6410	1.24595	0.622975	−	0.782241i	$-0.285924\pi$
$773$	34.6410	1.24595	0.622975	+	0.782241i	$0.285924\pi$
$774$	0	0
$775$	−6.92820	−0.248868
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	0	0
$782$	0	0
$783$	12.0000	0.428845
$784$	0	0
$785$	−22.5167	−0.803654
$786$	0	0
$787$	−38.1051	−1.35830	−0.679150	−	0.733999i	$-0.737652\pi$
$787$	−38.1051	−1.35830	−0.679150	+	0.733999i	$0.737652\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	10.3923	0.368577
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	10.3923	0.367653
$800$	0	0
$801$	6.92820	0.244796
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	−38.1051	−1.33805	−0.669026	−	0.743239i	$-0.733288\pi$
$811$	−38.1051	−1.33805	−0.669026	+	0.743239i	$0.733288\pi$
$812$	0	0
$813$	−41.5692	−1.45790
$814$	0	0
$815$	−36.0000	−1.26102
$816$	0	0
$817$	−27.7128	−0.969549
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−41.5692	−1.45078	−0.725388	−	0.688340i	$-0.758340\pi$
$821$	−41.5692	−1.45078	−0.725388	+	0.688340i	$0.758340\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.7846	−0.722752	−0.361376	−	0.932420i	$-0.617693\pi$
$827$	−20.7846	−0.722752	−0.361376	+	0.932420i	$0.617693\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	−14.0000	−0.485655
$832$	0	0
$833$	−21.0000	−0.727607
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	13.8564	0.478947
$838$	0	0
$839$	−45.0333	−1.55472	−0.777361	−	0.629054i	$-0.783442\pi$
$839$	−45.0333	−1.55472	−0.777361	+	0.629054i	$0.783442\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	45.0333	1.55103
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	51.9615	1.78122
$852$	0	0
$853$	25.9808	0.889564	0.444782	−	0.895639i	$-0.353281\pi$
$853$	25.9808	0.889564	0.444782	+	0.895639i	$0.353281\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.7128	−0.943355	−0.471678	−	0.881771i	$-0.656351\pi$
$863$	−27.7128	−0.943355	−0.471678	+	0.881771i	$0.656351\pi$
$864$	0	0
$865$	−10.3923	−0.353349
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.92820	−0.234484
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.1244	−0.409410	−0.204705	−	0.978824i	$-0.565624\pi$
$877$	−12.1244	−0.409410	−0.204705	+	0.978824i	$0.565624\pi$
$878$	0	0
$879$	−10.3923	−0.350524
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	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
$885$	24.0000	0.806751
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.3923	0.346603
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.0526	−0.633328
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−3.46410	−0.114520
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	0	0
$921$	−34.6410	−1.14146
$922$	0	0
$923$	0	0
$924$	0	0
$925$	17.3205	0.569495
$926$	0	0
$927$	−10.0000	−0.328443
$928$	0	0
$929$	46.7654	1.53432	0.767161	−	0.641455i	$-0.221669\pi$
$929$	46.7654	1.53432	0.767161	+	0.641455i	$0.221669\pi$
$930$	0	0
$931$	24.2487	0.794719
$932$	0	0
$933$	60.0000	1.96431
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	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
$943$	31.1769	1.01526
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.3205	−0.562841	−0.281420	−	0.959585i	$-0.590806\pi$
$947$	−17.3205	−0.562841	−0.281420	+	0.959585i	$0.590806\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−10.3923	−0.336994
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−31.1769	−1.00886
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	9.00000	0.289720
$966$	0	0
$967$	58.8897	1.89377	0.946883	−	0.321578i	$-0.104213\pi$
$967$	58.8897	1.89377	0.946883	+	0.321578i	$0.104213\pi$
$968$	0	0
$969$	20.7846	0.667698
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.3013	1.38533	0.692665	−	0.721259i	$-0.256436\pi$
$977$	43.3013	1.38533	0.692665	+	0.721259i	$0.256436\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	13.8564	0.442401
$982$	0	0
$983$	−51.9615	−1.65732	−0.828658	−	0.559756i	$-0.810895\pi$
$983$	−51.9615	−1.65732	−0.828658	+	0.559756i	$0.810895\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	55.4256	1.75888
$994$	0	0
$995$	−3.46410	−0.109819
$996$	0	0
$997$	17.0000	0.538395	0.269198	−	0.963085i	$-0.413241\pi$
$997$	17.0000	0.538395	0.269198	+	0.963085i	$0.413241\pi$
$998$	0	0
$999$	−34.6410	−1.09599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.a.o.1.2		2
4.3	odd	2		169.2.a.a.1.1		2
12.11	even	2		1521.2.a.k.1.2		2
13.2	odd	12		208.2.w.b.17.1		2
13.5	odd	4		2704.2.f.b.337.1		2
13.7	odd	12		208.2.w.b.49.1		2
13.8	odd	4		2704.2.f.b.337.2		2
13.12	even	2	inner	2704.2.a.o.1.1		2
20.19	odd	2		4225.2.a.v.1.2		2
28.27	even	2		8281.2.a.q.1.1		2
39.2	even	12		1872.2.by.d.433.1		2
39.20	even	12		1872.2.by.d.1297.1		2
52.3	odd	6		169.2.c.a.22.2		4
52.7	even	12		13.2.e.a.10.1	yes	2
52.11	even	12		169.2.e.a.147.1		2
52.15	even	12		13.2.e.a.4.1	✓	2
52.19	even	12		169.2.e.a.23.1		2
52.23	odd	6		169.2.c.a.22.1		4
52.31	even	4		169.2.b.a.168.2		2
52.35	odd	6		169.2.c.a.146.2		4
52.43	odd	6		169.2.c.a.146.1		4
52.47	even	4		169.2.b.a.168.1		2
52.51	odd	2		169.2.a.a.1.2		2
104.59	even	12		832.2.w.d.257.1		2
104.67	even	12		832.2.w.d.641.1		2
104.85	odd	12		832.2.w.a.257.1		2
104.93	odd	12		832.2.w.a.641.1		2
156.47	odd	4		1521.2.b.a.1351.2		2
156.59	odd	12		117.2.q.c.10.1		2
156.83	odd	4		1521.2.b.a.1351.1		2
156.119	odd	12		117.2.q.c.82.1		2
156.155	even	2		1521.2.a.k.1.1		2
260.7	odd	12		325.2.m.a.49.1		4
260.59	even	12		325.2.n.a.101.1		2
260.67	odd	12		325.2.m.a.199.2		4
260.119	even	12		325.2.n.a.251.1		2
260.163	odd	12		325.2.m.a.49.2		4
260.223	odd	12		325.2.m.a.199.1		4
260.259	odd	2		4225.2.a.v.1.1		2
364.59	odd	12		637.2.k.c.569.1		2
364.67	even	12		637.2.u.c.30.1		2
364.111	odd	12		637.2.q.a.491.1		2
364.163	even	12		637.2.u.c.361.1		2
364.171	odd	12		637.2.u.b.30.1		2
364.215	odd	12		637.2.u.b.361.1		2
364.223	odd	12		637.2.q.a.589.1		2
364.275	even	12		637.2.k.a.459.1		2
364.319	even	12		637.2.k.a.569.1		2
364.327	odd	12		637.2.k.c.459.1		2
364.363	even	2		8281.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.2.e.a.4.1	✓	2	52.15	even	12
13.2.e.a.10.1	yes	2	52.7	even	12
117.2.q.c.10.1		2	156.59	odd	12
117.2.q.c.82.1		2	156.119	odd	12
169.2.a.a.1.1		2	4.3	odd	2
169.2.a.a.1.2		2	52.51	odd	2
169.2.b.a.168.1		2	52.47	even	4
169.2.b.a.168.2		2	52.31	even	4
169.2.c.a.22.1		4	52.23	odd	6
169.2.c.a.22.2		4	52.3	odd	6
169.2.c.a.146.1		4	52.43	odd	6
169.2.c.a.146.2		4	52.35	odd	6
169.2.e.a.23.1		2	52.19	even	12
169.2.e.a.147.1		2	52.11	even	12
208.2.w.b.17.1		2	13.2	odd	12
208.2.w.b.49.1		2	13.7	odd	12
325.2.m.a.49.1		4	260.7	odd	12
325.2.m.a.49.2		4	260.163	odd	12
325.2.m.a.199.1		4	260.223	odd	12
325.2.m.a.199.2		4	260.67	odd	12
325.2.n.a.101.1		2	260.59	even	12
325.2.n.a.251.1		2	260.119	even	12
637.2.k.a.459.1		2	364.275	even	12
637.2.k.a.569.1		2	364.319	even	12
637.2.k.c.459.1		2	364.327	odd	12
637.2.k.c.569.1		2	364.59	odd	12
637.2.q.a.491.1		2	364.111	odd	12
637.2.q.a.589.1		2	364.223	odd	12
637.2.u.b.30.1		2	364.171	odd	12
637.2.u.b.361.1		2	364.215	odd	12
637.2.u.c.30.1		2	364.67	even	12
637.2.u.c.361.1		2	364.163	even	12
832.2.w.a.257.1		2	104.85	odd	12
832.2.w.a.641.1		2	104.93	odd	12
832.2.w.d.257.1		2	104.59	even	12
832.2.w.d.641.1		2	104.67	even	12
1521.2.a.k.1.1		2	156.155	even	2
1521.2.a.k.1.2		2	12.11	even	2
1521.2.b.a.1351.1		2	156.83	odd	4
1521.2.b.a.1351.2		2	156.47	odd	4
1872.2.by.d.433.1		2	39.2	even	12
1872.2.by.d.1297.1		2	39.20	even	12
2704.2.a.o.1.1		2	13.12	even	2	inner
2704.2.a.o.1.2		2	1.1	even	1	trivial
2704.2.f.b.337.1		2	13.5	odd	4
2704.2.f.b.337.2		2	13.8	odd	4
4225.2.a.v.1.1		2	260.259	odd	2
4225.2.a.v.1.2		2	20.19	odd	2
8281.2.a.q.1.1		2	28.27	even	2
8281.2.a.q.1.2		2	364.363	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +1.73205 q^{5} +1.00000 q^{9} -3.46410 q^{15} +3.00000 q^{17} -3.46410 q^{19} -6.00000 q^{23} -2.00000 q^{25} +4.00000 q^{27} +3.00000 q^{29} +3.46410 q^{31} -8.66025 q^{37} -5.19615 q^{41} +8.00000 q^{43} +1.73205 q^{45} +3.46410 q^{47} -7.00000 q^{49} -6.00000 q^{51} -3.00000 q^{53} +6.92820 q^{57} -6.92820 q^{59} +1.00000 q^{61} -3.46410 q^{67} +12.0000 q^{69} +3.46410 q^{71} +1.73205 q^{73} +4.00000 q^{75} -4.00000 q^{79} -11.0000 q^{81} +13.8564 q^{83} +5.19615 q^{85} -6.00000 q^{87} +6.92820 q^{89} -6.92820 q^{93} -6.00000 q^{95} -6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{9} + 6 q^{17} - 12 q^{23} - 4 q^{25} + 8 q^{27} + 6 q^{29} + 16 q^{43} - 14 q^{49} - 12 q^{51} - 6 q^{53} + 2 q^{61} + 24 q^{69} + 8 q^{75} - 8 q^{79} - 22 q^{81} - 12 q^{87} - 12 q^{95}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^9 + 6 * q^17 - 12 * q^23 - 4 * q^25 + 8 * q^27 + 6 * q^29 + 16 * q^43 - 14 * q^49 - 12 * q^51 - 6 * q^53 + 2 * q^61 + 24 * q^69 + 8 * q^75 - 8 * q^79 - 22 * q^81 - 12 * q^87 - 12 * q^95

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