LMFDB - Embedded newform 2704.2.f.b.337.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(337,2704)] 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("2704.337"); 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, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-4,0,0,0,0,0,2,0,0,0,0,0,0,0,-6,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.b.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	− 1.73205i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.46410i	0.894427i
$16$	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
$19$	3.46410i	0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.46410i	0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	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
$31$	− 3.46410i	− 0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.66025i	− 1.42374i	−0.702313	−	0.711868i	$-0.747849\pi$
$37$	− 8.66025i	− 1.42374i	0.702313	−	0.711868i	$-0.252151\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.19615i	0.811503i	0.913984	+	0.405751i	$0.132990\pi$
$41$	5.19615i	0.811503i	−0.913984	+	0.405751i	$0.867010\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	− 1.73205i	− 0.258199i
$46$	0	0
$47$	3.46410i	0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$	3.46410i	0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$	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
$57$	− 6.92820i	− 0.917663i
$58$	0	0
$59$	− 6.92820i	− 0.901975i	−0.892530	−	0.450988i	$-0.851072\pi$
$59$	− 6.92820i	− 0.901975i	0.892530	−	0.450988i	$-0.148928\pi$
$60$	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
$67$	3.46410i	0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	3.46410i	0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	−12.0000	−1.44463
$70$	0	0
$71$	− 3.46410i	− 0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$	− 3.46410i	− 0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$	0	0
$73$	1.73205i	0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$	1.73205i	0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$	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
$83$	− 13.8564i	− 1.52094i	−0.649374	−	0.760469i	$-0.724969\pi$
$83$	− 13.8564i	− 1.52094i	0.649374	−	0.760469i	$-0.275031\pi$
$84$	0	0
$85$	5.19615i	0.563602i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	6.92820i	0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$	6.92820i	0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.92820i	0.718421i
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	6.92820i	0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$	6.92820i	0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$	0	0
$99$	0	0
$100$	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
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	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
$109$	− 13.8564i	− 1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$	− 13.8564i	− 1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$	0	0
$111$	17.3205i	1.64399i
$112$	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
$115$	− 10.3923i	− 0.969087i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 10.3923i	− 0.937043i
$124$	0	0
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	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
$135$	− 6.92820i	− 0.596285i
$136$	0	0
$137$	− 15.5885i	− 1.33181i	−0.746036	−	0.665906i	$-0.768045\pi$
$137$	− 15.5885i	− 1.33181i	0.746036	−	0.665906i	$-0.231955\pi$
$138$	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
$141$	− 6.92820i	− 0.583460i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 5.19615i	− 0.431517i
$146$	0	0
$147$	−14.0000	−1.15470
$148$	0	0
$149$	− 19.0526i	− 1.56085i	−0.625252	−	0.780423i	$-0.715004\pi$
$149$	− 19.0526i	− 1.56085i	0.625252	−	0.780423i	$-0.284996\pi$
$150$	0	0
$151$	− 17.3205i	− 1.40952i	−0.709444	−	0.704761i	$-0.751054\pi$
$151$	− 17.3205i	− 1.40952i	0.709444	−	0.704761i	$-0.248946\pi$
$152$	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
$163$	− 20.7846i	− 1.62798i	−0.580881	−	0.813988i	$-0.697292\pi$
$163$	− 20.7846i	− 1.62798i	0.580881	−	0.813988i	$-0.302708\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.8564i	− 1.07224i	−0.844141	−	0.536120i	$-0.819889\pi$
$167$	− 13.8564i	− 1.07224i	0.844141	−	0.536120i	$-0.180111\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	3.46410i	0.264906i
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.8564i	1.04151i
$178$	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	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	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
$193$	5.19615i	0.374027i	0.982357	+	0.187014i	$0.0598809\pi$
$193$	5.19615i	0.374027i	−0.982357	+	0.187014i	$0.940119\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.8564i	− 0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$	− 13.8564i	− 0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	− 6.92820i	− 0.488678i
$202$	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
$213$	6.92820i	0.474713i
$214$	0	0
$215$	13.8564i	0.944999i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 3.46410i	− 0.234082i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 10.3923i	− 0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	− 10.3923i	− 0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	− 24.2487i	− 1.60944i	−0.593652	−	0.804722i	$-0.702314\pi$
$227$	− 24.2487i	− 1.60944i	0.593652	−	0.804722i	$-0.297686\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	6.00000	0.391397
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	20.7846i	1.34444i	0.740349	+	0.672222i	$0.234660\pi$
$239$	20.7846i	1.34444i	−0.740349	+	0.672222i	$0.765340\pi$
$240$	0	0
$241$	− 1.73205i	− 0.111571i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	− 1.73205i	− 0.111571i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	− 12.1244i	− 0.774597i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	27.7128i	1.75623i
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 10.3923i	− 0.650791i
$256$	0	0
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	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
$265$	5.19615i	0.319197i
$266$	0	0
$267$	− 13.8564i	− 0.847998i
$268$	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
$271$	20.7846i	1.26258i	0.775549	+	0.631288i	$0.217473\pi$
$271$	20.7846i	1.26258i	−0.775549	+	0.631288i	$0.782527\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	− 3.46410i	− 0.207390i
$280$	0	0
$281$	− 22.5167i	− 1.34323i	−0.740900	−	0.671616i	$-0.765601\pi$
$281$	− 22.5167i	− 1.34323i	0.740900	−	0.671616i	$-0.234399\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−12.0000	−0.710819
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 13.8564i	− 0.812277i
$292$	0	0
$293$	5.19615i	0.303562i	0.988414	+	0.151781i	$0.0485009\pi$
$293$	5.19615i	0.303562i	−0.988414	+	0.151781i	$0.951499\pi$
$294$	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
$305$	− 1.73205i	− 0.0991769i
$306$	0	0
$307$	17.3205i	0.988534i	0.869310	+	0.494267i	$0.164563\pi$
$307$	17.3205i	0.988534i	−0.869310	+	0.494267i	$0.835437\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	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
$317$	− 5.19615i	− 0.291845i	−0.989296	−	0.145922i	$-0.953385\pi$
$317$	− 5.19615i	− 0.291845i	0.989296	−	0.145922i	$-0.0466150\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 10.3923i	− 0.578243i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.7128i	1.53252i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	27.7128i	1.52323i	0.648027	+	0.761617i	$0.275594\pi$
$331$	27.7128i	1.52323i	−0.648027	+	0.761617i	$0.724406\pi$
$332$	0	0
$333$	− 8.66025i	− 0.474579i
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	30.0000	1.62938
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	20.7846i	1.11901i
$346$	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
$349$	− 13.8564i	− 0.741716i	−0.928689	−	0.370858i	$-0.879064\pi$
$349$	− 13.8564i	− 0.741716i	0.928689	−	0.370858i	$-0.120936\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 32.9090i	− 1.75157i	−0.482704	−	0.875784i	$-0.660345\pi$
$353$	− 32.9090i	− 1.75157i	0.482704	−	0.875784i	$-0.339655\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.92820i	0.365657i	0.983145	+	0.182828i	$0.0585252\pi$
$359$	6.92820i	0.365657i	−0.983145	+	0.182828i	$0.941475\pi$
$360$	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
$369$	5.19615i	0.270501i
$370$	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
$375$	24.2487i	1.25220i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 24.2487i	− 1.24557i	−0.782392	−	0.622786i	$-0.786001\pi$
$379$	− 24.2487i	− 1.24557i	0.782392	−	0.622786i	$-0.213999\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	20.7846i	1.06204i	0.847358	+	0.531022i	$0.178192\pi$
$383$	20.7846i	1.06204i	−0.847358	+	0.531022i	$0.821808\pi$
$384$	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	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	36.0000	1.81596
$394$	0	0
$395$	6.92820i	0.348596i
$396$	0	0
$397$	− 13.8564i	− 0.695433i	−0.937600	−	0.347717i	$-0.886957\pi$
$397$	− 13.8564i	− 0.695433i	0.937600	−	0.347717i	$-0.113043\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.73205i	0.0864945i	0.999064	+	0.0432472i	$0.0137703\pi$
$401$	1.73205i	0.0864945i	−0.999064	+	0.0432472i	$0.986230\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	19.0526i	0.946729i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15.5885i	0.770800i	0.922750	+	0.385400i	$0.125936\pi$
$409$	15.5885i	0.770800i	−0.922750	+	0.385400i	$0.874064\pi$
$410$	0	0
$411$	31.1769i	1.53784i
$412$	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
$421$	15.5885i	0.759735i	0.925041	+	0.379867i	$0.124030\pi$
$421$	15.5885i	0.759735i	−0.925041	+	0.379867i	$0.875970\pi$
$422$	0	0
$423$	3.46410i	0.168430i
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 6.92820i	− 0.333720i	−0.985981	−	0.166860i	$-0.946637\pi$
$431$	− 6.92820i	− 0.333720i	0.985981	−	0.166860i	$-0.0533628\pi$
$432$	0	0
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	0	0
$435$	10.3923i	0.498273i
$436$	0	0
$437$	20.7846i	0.994263i
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	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
$447$	38.1051i	1.80231i
$448$	0	0
$449$	− 6.92820i	− 0.326962i	−0.986546	−	0.163481i	$-0.947728\pi$
$449$	− 6.92820i	− 0.326962i	0.986546	−	0.163481i	$-0.0522723\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	34.6410i	1.62758i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.73205i	− 0.0810219i	−0.999179	−	0.0405110i	$-0.987101\pi$
$457$	− 1.73205i	− 0.0810219i	0.999179	−	0.0405110i	$-0.0128986\pi$
$458$	0	0
$459$	−12.0000	−0.560112
$460$	0	0
$461$	22.5167i	1.04871i	0.851501	+	0.524353i	$0.175693\pi$
$461$	22.5167i	1.04871i	−0.851501	+	0.524353i	$0.824307\pi$
$462$	0	0
$463$	− 13.8564i	− 0.643962i	−0.946746	−	0.321981i	$-0.895651\pi$
$463$	− 13.8564i	− 0.643962i	0.946746	−	0.321981i	$-0.104349\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	26.0000	1.19802
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.92820i	0.317888i
$476$	0	0
$477$	−3.00000	−0.137361
$478$	0	0
$479$	24.2487i	1.10795i	0.832533	+	0.553976i	$0.186890\pi$
$479$	24.2487i	1.10795i	−0.832533	+	0.553976i	$0.813110\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	6.92820i	0.313947i	0.987603	+	0.156973i	$0.0501737\pi$
$487$	6.92820i	0.313947i	−0.987603	+	0.156973i	$0.949826\pi$
$488$	0	0
$489$	41.5692i	1.87983i
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 31.1769i	− 1.39567i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	− 31.1769i	− 1.39567i	0.716258	−	0.697835i	$-0.245853\pi$
$500$	0	0
$501$	27.7128i	1.23812i
$502$	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
$505$	5.19615i	0.231226i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 19.0526i	− 0.844490i	−0.906482	−	0.422245i	$-0.861242\pi$
$509$	− 19.0526i	− 0.844490i	0.906482	−	0.422245i	$-0.138758\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	13.8564i	0.611775i
$514$	0	0
$515$	− 17.3205i	− 0.763233i
$516$	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
$527$	10.3923i	0.452696i
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	− 6.92820i	− 0.300658i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	10.3923i	0.449299i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	29.4449i	1.26593i	0.774179	+	0.632967i	$0.218163\pi$
$541$	29.4449i	1.26593i	−0.774179	+	0.632967i	$0.781837\pi$
$542$	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
$551$	10.3923i	0.442727i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	30.0000	1.27343
$556$	0	0
$557$	15.5885i	0.660504i	0.943893	+	0.330252i	$0.107134\pi$
$557$	15.5885i	0.660504i	−0.943893	+	0.330252i	$0.892866\pi$
$558$	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
$565$	25.9808i	1.09302i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	36.0000	1.50392
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	− 19.0526i	− 0.793168i	−0.917998	−	0.396584i	$-0.870195\pi$
$577$	− 19.0526i	− 0.793168i	0.917998	−	0.396584i	$-0.129805\pi$
$578$	0	0
$579$	− 10.3923i	− 0.431889i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 20.7846i	− 0.857873i	−0.903335	−	0.428936i	$-0.858888\pi$
$587$	− 20.7846i	− 0.857873i	0.903335	−	0.428936i	$-0.141112\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	27.7128i	1.13995i
$592$	0	0
$593$	− 25.9808i	− 1.06690i	−0.845831	−	0.533451i	$-0.820895\pi$
$593$	− 25.9808i	− 1.06690i	0.845831	−	0.533451i	$-0.179105\pi$
$594$	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
$603$	3.46410i	0.141069i
$604$	0	0
$605$	− 19.0526i	− 0.774597i
$606$	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
$613$	12.1244i	0.489698i	0.969561	+	0.244849i	$0.0787384\pi$
$613$	12.1244i	0.489698i	−0.969561	+	0.244849i	$0.921262\pi$
$614$	0	0
$615$	−18.0000	−0.725830
$616$	0	0
$617$	− 22.5167i	− 0.906487i	−0.891387	−	0.453243i	$-0.850267\pi$
$617$	− 22.5167i	− 0.906487i	0.891387	−	0.453243i	$-0.149733\pi$
$618$	0	0
$619$	20.7846i	0.835404i	0.908584	+	0.417702i	$0.137164\pi$
$619$	20.7846i	0.835404i	−0.908584	+	0.417702i	$0.862836\pi$
$620$	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
$629$	25.9808i	1.03592i
$630$	0	0
$631$	48.4974i	1.93065i	0.261048	+	0.965326i	$0.415932\pi$
$631$	48.4974i	1.93065i	−0.261048	+	0.965326i	$0.584068\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	− 3.46410i	− 0.137469i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 3.46410i	− 0.137038i
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	− 13.8564i	− 0.546443i	−0.961951	−	0.273222i	$-0.911911\pi$
$643$	− 13.8564i	− 0.546443i	0.961951	−	0.273222i	$-0.0880892\pi$
$644$	0	0
$645$	− 27.7128i	− 1.09119i
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	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
$655$	31.1769i	1.21818i
$656$	0	0
$657$	1.73205i	0.0675737i
$658$	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
$661$	46.7654i	1.81896i	0.415745	+	0.909481i	$0.363521\pi$
$661$	46.7654i	1.81896i	−0.415745	+	0.909481i	$0.636479\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.0000	0.696963
$668$	0	0
$669$	20.7846i	0.803579i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	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
$681$	48.4974i	1.85843i
$682$	0	0
$683$	− 24.2487i	− 0.927851i	−0.885874	−	0.463926i	$-0.846441\pi$
$683$	− 24.2487i	− 0.927851i	0.885874	−	0.463926i	$-0.153559\pi$
$684$	0	0
$685$	−27.0000	−1.03162
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	13.8564i	0.527123i	0.964643	+	0.263561i	$0.0848971\pi$
$691$	13.8564i	0.527123i	−0.964643	+	0.263561i	$0.915103\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 6.92820i	− 0.262802i
$696$	0	0
$697$	− 15.5885i	− 0.590455i
$698$	0	0
$699$	−12.0000	−0.453882
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	30.0000	1.13147
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 5.19615i	− 0.195146i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	− 5.19615i	− 0.195146i	0.995228	−	0.0975728i	$-0.0311079\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	− 20.7846i	− 0.778390i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 41.5692i	− 1.55243i
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.46410i	0.128831i
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	12.1244i	0.447823i	0.974609	+	0.223912i	$0.0718827\pi$
$733$	12.1244i	0.447823i	−0.974609	+	0.223912i	$0.928117\pi$
$734$	0	0
$735$	24.2487i	0.894427i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 20.7846i	− 0.764574i	−0.924044	−	0.382287i	$-0.875137\pi$
$739$	− 20.7846i	− 0.764574i	0.924044	−	0.382287i	$-0.124863\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 34.6410i	− 1.27086i	−0.772160	−	0.635428i	$-0.780824\pi$
$743$	− 34.6410i	− 1.27086i	0.772160	−	0.635428i	$-0.219176\pi$
$744$	0	0
$745$	−33.0000	−1.20903
$746$	0	0
$747$	− 13.8564i	− 0.506979i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	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
$761$	34.6410i	1.25574i	0.778320	+	0.627868i	$0.216072\pi$
$761$	34.6410i	1.25574i	−0.778320	+	0.627868i	$0.783928\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.19615i	0.187867i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 6.92820i	− 0.249837i	−0.992167	−	0.124919i	$-0.960133\pi$
$769$	− 6.92820i	− 0.249837i	0.992167	−	0.124919i	$-0.0398670\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	− 34.6410i	− 1.24595i	−0.782241	−	0.622975i	$-0.785924\pi$
$773$	− 34.6410i	− 1.24595i	0.782241	−	0.622975i	$-0.214076\pi$
$774$	0	0
$775$	− 6.92820i	− 0.248868i
$776$	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
$785$	22.5167i	0.803654i
$786$	0	0
$787$	− 38.1051i	− 1.35830i	−0.733999	−	0.679150i	$-0.762348\pi$
$787$	− 38.1051i	− 1.35830i	0.733999	−	0.679150i	$-0.237652\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 10.3923i	− 0.368577i
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	− 10.3923i	− 0.367653i
$800$	0	0
$801$	6.92820i	0.244796i
$802$	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
$811$	38.1051i	1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$811$	38.1051i	1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$812$	0	0
$813$	− 41.5692i	− 1.45790i
$814$	0	0
$815$	−36.0000	−1.26102
$816$	0	0
$817$	− 27.7128i	− 0.969549i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.5692i	1.45078i	0.688340	+	0.725388i	$0.258340\pi$
$821$	41.5692i	1.45078i	−0.688340	+	0.725388i	$0.741660\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 20.7846i	− 0.722752i	−0.932420	−	0.361376i	$-0.882307\pi$
$827$	− 20.7846i	− 0.722752i	0.932420	−	0.361376i	$-0.117693\pi$
$828$	0	0
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	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
$837$	− 13.8564i	− 0.478947i
$838$	0	0
$839$	− 45.0333i	− 1.55472i	−0.629054	−	0.777361i	$-0.716558\pi$
$839$	− 45.0333i	− 1.55472i	0.629054	−	0.777361i	$-0.283442\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	45.0333i	1.55103i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	− 51.9615i	− 1.78122i
$852$	0	0
$853$	25.9808i	0.889564i	0.895639	+	0.444782i	$0.146719\pi$
$853$	25.9808i	0.889564i	−0.895639	+	0.444782i	$0.853281\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	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
$863$	27.7128i	0.943355i	0.881771	+	0.471678i	$0.156351\pi$
$863$	27.7128i	0.943355i	−0.881771	+	0.471678i	$0.843649\pi$
$864$	0	0
$865$	− 10.3923i	− 0.353349i
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.92820i	0.234484i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.1244i	0.409410i	0.978824	+	0.204705i	$0.0656236\pi$
$877$	12.1244i	0.409410i	−0.978824	+	0.204705i	$0.934376\pi$
$878$	0	0
$879$	− 10.3923i	− 0.350524i
$880$	0	0
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	0	0
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	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
$899$	− 10.3923i	− 0.346603i
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 19.0526i	− 0.633328i
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	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
$915$	3.46410i	0.114520i
$916$	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
$921$	− 34.6410i	− 1.14146i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 17.3205i	− 0.569495i
$926$	0	0
$927$	10.0000	0.328443
$928$	0	0
$929$	− 46.7654i	− 1.53432i	−0.641455	−	0.767161i	$-0.721669\pi$
$929$	− 46.7654i	− 1.53432i	0.641455	−	0.767161i	$-0.278331\pi$
$930$	0	0
$931$	24.2487i	0.794719i
$932$	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
$941$	20.7846i	0.677559i	0.940866	+	0.338779i	$0.110014\pi$
$941$	20.7846i	0.677559i	−0.940866	+	0.338779i	$0.889986\pi$
$942$	0	0
$943$	31.1769i	1.01526i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 17.3205i	− 0.562841i	−0.959585	−	0.281420i	$-0.909194\pi$
$947$	− 17.3205i	− 0.562841i	0.959585	−	0.281420i	$-0.0908056\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	10.3923i	0.336994i
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	31.1769i	1.00886i
$956$	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
$967$	− 58.8897i	− 1.89377i	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	− 58.8897i	− 1.89377i	0.321578	−	0.946883i	$-0.395787\pi$
$968$	0	0
$969$	20.7846i	0.667698i
$970$	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
$977$	− 43.3013i	− 1.38533i	−0.721259	−	0.692665i	$-0.756436\pi$
$977$	− 43.3013i	− 1.38533i	0.721259	−	0.692665i	$-0.243564\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 13.8564i	− 0.442401i
$982$	0	0
$983$	− 51.9615i	− 1.65732i	−0.559756	−	0.828658i	$-0.689105\pi$
$983$	− 51.9615i	− 1.65732i	0.559756	−	0.828658i	$-0.310895\pi$
$984$	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
$993$	− 55.4256i	− 1.75888i
$994$	0	0
$995$	− 3.46410i	− 0.109819i
$996$	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
$999$	− 34.6410i	− 1.09599i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.b.337.1		2
4.3	odd	2		169.2.b.a.168.2		2
12.11	even	2		1521.2.b.a.1351.1		2
13.3	even	3		208.2.w.b.17.1		2
13.4	even	6		208.2.w.b.49.1		2
13.5	odd	4		2704.2.a.o.1.1		2
13.8	odd	4		2704.2.a.o.1.2		2
13.12	even	2	inner	2704.2.f.b.337.2		2
39.17	odd	6		1872.2.by.d.1297.1		2
39.29	odd	6		1872.2.by.d.433.1		2
52.3	odd	6		13.2.e.a.4.1	✓	2
52.7	even	12		169.2.c.a.146.2		4
52.11	even	12		169.2.c.a.22.2		4
52.15	even	12		169.2.c.a.22.1		4
52.19	even	12		169.2.c.a.146.1		4
52.23	odd	6		169.2.e.a.147.1		2
52.31	even	4		169.2.a.a.1.2		2
52.35	odd	6		169.2.e.a.23.1		2
52.43	odd	6		13.2.e.a.10.1	yes	2
52.47	even	4		169.2.a.a.1.1		2
52.51	odd	2		169.2.b.a.168.1		2
104.3	odd	6		832.2.w.d.641.1		2
104.29	even	6		832.2.w.a.641.1		2
104.43	odd	6		832.2.w.d.257.1		2
104.69	even	6		832.2.w.a.257.1		2
156.47	odd	4		1521.2.a.k.1.2		2
156.83	odd	4		1521.2.a.k.1.1		2
156.95	even	6		117.2.q.c.10.1		2
156.107	even	6		117.2.q.c.82.1		2
156.155	even	2		1521.2.b.a.1351.2		2
260.3	even	12		325.2.m.a.199.1		4
260.43	even	12		325.2.m.a.49.2		4
260.99	even	4		4225.2.a.v.1.2		2
260.107	even	12		325.2.m.a.199.2		4
260.147	even	12		325.2.m.a.49.1		4
260.159	odd	6		325.2.n.a.251.1		2
260.199	odd	6		325.2.n.a.101.1		2
260.239	even	4		4225.2.a.v.1.1		2
364.3	even	6		637.2.u.b.30.1		2
364.55	even	6		637.2.q.a.589.1		2
364.83	odd	4		8281.2.a.q.1.2		2
364.95	odd	6		637.2.k.a.569.1		2
364.107	odd	6		637.2.k.a.459.1		2
364.159	even	6		637.2.k.c.459.1		2
364.199	even	6		637.2.k.c.569.1		2
364.251	even	6		637.2.q.a.491.1		2
364.263	odd	6		637.2.u.c.30.1		2
364.303	odd	6		637.2.u.c.361.1		2
364.307	odd	4		8281.2.a.q.1.1		2
364.355	even	6		637.2.u.b.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.2.e.a.4.1	✓	2	52.3	odd	6
13.2.e.a.10.1	yes	2	52.43	odd	6
117.2.q.c.10.1		2	156.95	even	6
117.2.q.c.82.1		2	156.107	even	6
169.2.a.a.1.1		2	52.47	even	4
169.2.a.a.1.2		2	52.31	even	4
169.2.b.a.168.1		2	52.51	odd	2
169.2.b.a.168.2		2	4.3	odd	2
169.2.c.a.22.1		4	52.15	even	12
169.2.c.a.22.2		4	52.11	even	12
169.2.c.a.146.1		4	52.19	even	12
169.2.c.a.146.2		4	52.7	even	12
169.2.e.a.23.1		2	52.35	odd	6
169.2.e.a.147.1		2	52.23	odd	6
208.2.w.b.17.1		2	13.3	even	3
208.2.w.b.49.1		2	13.4	even	6
325.2.m.a.49.1		4	260.147	even	12
325.2.m.a.49.2		4	260.43	even	12
325.2.m.a.199.1		4	260.3	even	12
325.2.m.a.199.2		4	260.107	even	12
325.2.n.a.101.1		2	260.199	odd	6
325.2.n.a.251.1		2	260.159	odd	6
637.2.k.a.459.1		2	364.107	odd	6
637.2.k.a.569.1		2	364.95	odd	6
637.2.k.c.459.1		2	364.159	even	6
637.2.k.c.569.1		2	364.199	even	6
637.2.q.a.491.1		2	364.251	even	6
637.2.q.a.589.1		2	364.55	even	6
637.2.u.b.30.1		2	364.3	even	6
637.2.u.b.361.1		2	364.355	even	6
637.2.u.c.30.1		2	364.263	odd	6
637.2.u.c.361.1		2	364.303	odd	6
832.2.w.a.257.1		2	104.69	even	6
832.2.w.a.641.1		2	104.29	even	6
832.2.w.d.257.1		2	104.43	odd	6
832.2.w.d.641.1		2	104.3	odd	6
1521.2.a.k.1.1		2	156.83	odd	4
1521.2.a.k.1.2		2	156.47	odd	4
1521.2.b.a.1351.1		2	12.11	even	2
1521.2.b.a.1351.2		2	156.155	even	2
1872.2.by.d.433.1		2	39.29	odd	6
1872.2.by.d.1297.1		2	39.17	odd	6
2704.2.a.o.1.1		2	13.5	odd	4
2704.2.a.o.1.2		2	13.8	odd	4
2704.2.f.b.337.1		2	1.1	even	1	trivial
2704.2.f.b.337.2		2	13.12	even	2	inner
4225.2.a.v.1.1		2	260.239	even	4
4225.2.a.v.1.2		2	260.99	even	4
8281.2.a.q.1.1		2	364.307	odd	4
8281.2.a.q.1.2		2	364.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

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

Introduction

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