LMFDB - Embedded newform 7056.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(1,7056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7056.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:	$56.3424436662$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.585786 q^{5} +O(q^{10})$	$q-0.585786 q^{5} -2.00000 q^{11} +5.41421 q^{13} -6.24264 q^{17} -2.82843 q^{19} +3.65685 q^{23} -4.65685 q^{25} +1.17157 q^{29} +6.82843 q^{31} -4.00000 q^{37} +2.24264 q^{41} +5.65685 q^{43} +2.82843 q^{47} +2.00000 q^{53} +1.17157 q^{55} -6.82843 q^{59} +3.75736 q^{61} -3.17157 q^{65} -5.65685 q^{67} -13.3137 q^{71} -5.89949 q^{73} -2.34315 q^{79} -15.3137 q^{83} +3.65685 q^{85} +5.75736 q^{89} +1.65685 q^{95} +5.41421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 4 q^{11} + 8 q^{13} - 4 q^{17} - 4 q^{23} + 2 q^{25} + 8 q^{29} + 8 q^{31} - 8 q^{37} - 4 q^{41} + 4 q^{53} + 8 q^{55} - 8 q^{59} + 16 q^{61} - 12 q^{65} - 4 q^{71} + 8 q^{73} - 16 q^{79} - 8 q^{83} - 4 q^{85} + 20 q^{89} - 8 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 4 * q^11 + 8 * q^13 - 4 * q^17 - 4 * q^23 + 2 * q^25 + 8 * q^29 + 8 * q^31 - 8 * q^37 - 4 * q^41 + 4 * q^53 + 8 * q^55 - 8 * q^59 + 16 * q^61 - 12 * q^65 - 4 * q^71 + 8 * q^73 - 16 * q^79 - 8 * q^83 - 4 * q^85 + 20 * q^89 - 8 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	5.41421	1.50163	0.750816	−	0.660511i	$-0.229660\pi$
$13$	5.41421	1.50163	0.750816	+	0.660511i	$0.229660\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.24264	−1.51406	−0.757031	−	0.653379i	$-0.773351\pi$
$17$	−6.24264	−1.51406	−0.757031	+	0.653379i	$0.773351\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.65685	0.762507	0.381253	−	0.924471i	$-0.375493\pi$
$23$	3.65685	0.762507	0.381253	+	0.924471i	$0.375493\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.17157	0.217556	0.108778	−	0.994066i	$-0.465306\pi$
$29$	1.17157	0.217556	0.108778	+	0.994066i	$0.465306\pi$
$30$	0	0
$31$	6.82843	1.22642	0.613211	−	0.789919i	$-0.289878\pi$
$31$	6.82843	1.22642	0.613211	+	0.789919i	$0.289878\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.24264	0.350242	0.175121	−	0.984547i	$-0.443968\pi$
$41$	2.24264	0.350242	0.175121	+	0.984547i	$0.443968\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	1.17157	0.157975
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.82843	−0.888985	−0.444493	−	0.895782i	$-0.646616\pi$
$59$	−6.82843	−0.888985	−0.444493	+	0.895782i	$0.646616\pi$
$60$	0	0
$61$	3.75736	0.481081	0.240540	−	0.970639i	$-0.422675\pi$
$61$	3.75736	0.481081	0.240540	+	0.970639i	$0.422675\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.17157	−0.393385
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.3137	−1.58005	−0.790023	−	0.613077i	$-0.789932\pi$
$71$	−13.3137	−1.58005	−0.790023	+	0.613077i	$0.789932\pi$
$72$	0	0
$73$	−5.89949	−0.690484	−0.345242	−	0.938514i	$-0.612203\pi$
$73$	−5.89949	−0.690484	−0.345242	+	0.938514i	$0.612203\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.3137	−1.68090	−0.840449	−	0.541891i	$-0.817709\pi$
$83$	−15.3137	−1.68090	−0.840449	+	0.541891i	$0.817709\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.75736	0.610279	0.305139	−	0.952308i	$-0.401297\pi$
$89$	5.75736	0.610279	0.305139	+	0.952308i	$0.401297\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.65685	0.169990
$96$	0	0
$97$	5.41421	0.549730	0.274865	−	0.961483i	$-0.411367\pi$
$97$	5.41421	0.549730	0.274865	+	0.961483i	$0.411367\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.0711	−1.69863	−0.849317	−	0.527883i	$-0.822986\pi$
$101$	−17.0711	−1.69863	−0.849317	+	0.527883i	$0.822986\pi$
$102$	0	0
$103$	12.4853	1.23021	0.615106	−	0.788445i	$-0.289113\pi$
$103$	12.4853	1.23021	0.615106	+	0.788445i	$0.289113\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.6569	−1.12691	−0.563455	−	0.826147i	$-0.690528\pi$
$107$	−11.6569	−1.12691	−0.563455	+	0.826147i	$0.690528\pi$
$108$	0	0
$109$	5.65685	0.541828	0.270914	−	0.962604i	$-0.412674\pi$
$109$	5.65685	0.541828	0.270914	+	0.962604i	$0.412674\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.3137	−1.62874	−0.814368	−	0.580348i	$-0.802916\pi$
$113$	−17.3137	−1.62874	−0.814368	+	0.580348i	$0.802916\pi$
$114$	0	0
$115$	−2.14214	−0.199755
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−9.65685	−0.856907	−0.428454	−	0.903564i	$-0.640941\pi$
$127$	−9.65685	−0.856907	−0.428454	+	0.903564i	$0.640941\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.31371	0.639002	0.319501	−	0.947586i	$-0.396485\pi$
$131$	7.31371	0.639002	0.319501	+	0.947586i	$0.396485\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.1421	1.20824	0.604122	−	0.796892i	$-0.293524\pi$
$137$	14.1421	1.20824	0.604122	+	0.796892i	$0.293524\pi$
$138$	0	0
$139$	6.34315	0.538019	0.269009	−	0.963138i	$-0.413304\pi$
$139$	6.34315	0.538019	0.269009	+	0.963138i	$0.413304\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−10.8284	−0.905519
$144$	0	0
$145$	−0.686292	−0.0569934
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.31371	0.435316	0.217658	−	0.976025i	$-0.430158\pi$
$149$	5.31371	0.435316	0.217658	+	0.976025i	$0.430158\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	20.2426	1.61554	0.807769	−	0.589499i	$-0.200675\pi$
$157$	20.2426	1.61554	0.807769	+	0.589499i	$0.200675\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.7990	1.53209	0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990	1.53209	0.766046	+	0.642786i	$0.222221\pi$
$168$	0	0
$169$	16.3137	1.25490
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.92893	−0.526797	−0.263398	−	0.964687i	$-0.584843\pi$
$173$	−6.92893	−0.526797	−0.263398	+	0.964687i	$0.584843\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.34315	−0.623596	−0.311798	−	0.950148i	$-0.600931\pi$
$179$	−8.34315	−0.623596	−0.311798	+	0.950148i	$0.600931\pi$
$180$	0	0
$181$	−5.41421	−0.402435	−0.201218	−	0.979547i	$-0.564490\pi$
$181$	−5.41421	−0.402435	−0.201218	+	0.979547i	$0.564490\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.34315	0.172272
$186$	0	0
$187$	12.4853	0.913014
$188$	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
$193$	−17.3137	−1.24627	−0.623134	−	0.782115i	$-0.714141\pi$
$193$	−17.3137	−1.24627	−0.623134	+	0.782115i	$0.714141\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.31371	−0.0917534
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.65685	0.391293
$210$	0	0
$211$	20.9706	1.44367	0.721837	−	0.692064i	$-0.243298\pi$
$211$	20.9706	1.44367	0.721837	+	0.692064i	$0.243298\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.31371	−0.225993
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−33.7990	−2.27357
$222$	0	0
$223$	8.97056	0.600713	0.300357	−	0.953827i	$-0.402894\pi$
$223$	8.97056	0.600713	0.300357	+	0.953827i	$0.402894\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.7990	−1.04862	−0.524308	−	0.851529i	$-0.675676\pi$
$227$	−15.7990	−1.04862	−0.524308	+	0.851529i	$0.675676\pi$
$228$	0	0
$229$	8.24264	0.544689	0.272345	−	0.962200i	$-0.412201\pi$
$229$	8.24264	0.544689	0.272345	+	0.962200i	$0.412201\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.1421	−1.45058	−0.725290	−	0.688444i	$-0.758294\pi$
$233$	−22.1421	−1.45058	−0.725290	+	0.688444i	$0.758294\pi$
$234$	0	0
$235$	−1.65685	−0.108081
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.34315	−0.280935	−0.140467	−	0.990085i	$-0.544861\pi$
$239$	−4.34315	−0.280935	−0.140467	+	0.990085i	$0.544861\pi$
$240$	0	0
$241$	−7.75736	−0.499695	−0.249848	−	0.968285i	$-0.580381\pi$
$241$	−7.75736	−0.499695	−0.249848	+	0.968285i	$0.580381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−15.3137	−0.974388
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.48528	0.283108	0.141554	−	0.989931i	$-0.454790\pi$
$251$	4.48528	0.283108	0.141554	+	0.989931i	$0.454790\pi$
$252$	0	0
$253$	−7.31371	−0.459809
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.2132	1.19849	0.599243	−	0.800567i	$-0.295468\pi$
$257$	19.2132	1.19849	0.599243	+	0.800567i	$0.295468\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.3137	−1.06761	−0.533805	−	0.845608i	$-0.679238\pi$
$263$	−17.3137	−1.06761	−0.533805	+	0.845608i	$0.679238\pi$
$264$	0	0
$265$	−1.17157	−0.0719691
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.7279	−0.654093	−0.327046	−	0.945008i	$-0.606053\pi$
$269$	−10.7279	−0.654093	−0.327046	+	0.945008i	$0.606053\pi$
$270$	0	0
$271$	−18.1421	−1.10206	−0.551028	−	0.834487i	$-0.685764\pi$
$271$	−18.1421	−1.10206	−0.551028	+	0.834487i	$0.685764\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.31371	0.561638
$276$	0	0
$277$	13.3137	0.799943	0.399972	−	0.916528i	$-0.369020\pi$
$277$	13.3137	0.799943	0.399972	+	0.916528i	$0.369020\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.4853	0.983429	0.491715	−	0.870756i	$-0.336370\pi$
$281$	16.4853	0.983429	0.491715	+	0.870756i	$0.336370\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.4142	−1.13419	−0.567095	−	0.823652i	$-0.691933\pi$
$293$	−19.4142	−1.13419	−0.567095	+	0.823652i	$0.691933\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	19.7990	1.14501
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.20101	−0.126029
$306$	0	0
$307$	−1.85786	−0.106034	−0.0530170	−	0.998594i	$-0.516884\pi$
$307$	−1.85786	−0.106034	−0.0530170	+	0.998594i	$0.516884\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.1421	−1.25557	−0.627783	−	0.778389i	$-0.716037\pi$
$311$	−22.1421	−1.25557	−0.627783	+	0.778389i	$0.716037\pi$
$312$	0	0
$313$	−17.8995	−1.01174	−0.505870	−	0.862610i	$-0.668828\pi$
$313$	−17.8995	−1.01174	−0.505870	+	0.862610i	$0.668828\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	−2.34315	−0.131191
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.6569	0.982454
$324$	0	0
$325$	−25.2132	−1.39858
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.31371	0.181047
$336$	0	0
$337$	−18.3431	−0.999215	−0.499607	−	0.866252i	$-0.666522\pi$
$337$	−18.3431	−0.999215	−0.499607	+	0.866252i	$0.666522\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.6569	−0.739560
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.6863	0.573670	0.286835	−	0.957980i	$-0.407397\pi$
$347$	10.6863	0.573670	0.286835	+	0.957980i	$0.407397\pi$
$348$	0	0
$349$	9.89949	0.529908	0.264954	−	0.964261i	$-0.414643\pi$
$349$	9.89949	0.529908	0.264954	+	0.964261i	$0.414643\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.7279	−0.570990	−0.285495	−	0.958380i	$-0.592158\pi$
$353$	−10.7279	−0.570990	−0.285495	+	0.958380i	$0.592158\pi$
$354$	0	0
$355$	7.79899	0.413927
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.6569	−0.615225	−0.307613	−	0.951512i	$-0.599530\pi$
$359$	−11.6569	−0.615225	−0.307613	+	0.951512i	$0.599530\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.45584	0.180887
$366$	0	0
$367$	−19.3137	−1.00817	−0.504084	−	0.863655i	$-0.668170\pi$
$367$	−19.3137	−1.00817	−0.504084	+	0.863655i	$0.668170\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−33.3137	−1.72492	−0.862459	−	0.506127i	$-0.831077\pi$
$373$	−33.3137	−1.72492	−0.862459	+	0.506127i	$0.831077\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.34315	0.326689
$378$	0	0
$379$	−31.3137	−1.60848	−0.804239	−	0.594307i	$-0.797427\pi$
$379$	−31.3137	−1.60848	−0.804239	+	0.594307i	$0.797427\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−29.6569	−1.51539	−0.757697	−	0.652606i	$-0.773676\pi$
$383$	−29.6569	−1.51539	−0.757697	+	0.652606i	$0.773676\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.1421	−0.514227	−0.257113	−	0.966381i	$-0.582771\pi$
$389$	−10.1421	−0.514227	−0.257113	+	0.966381i	$0.582771\pi$
$390$	0	0
$391$	−22.8284	−1.15448
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.37258	0.0690621
$396$	0	0
$397$	34.3848	1.72572	0.862861	−	0.505441i	$-0.168670\pi$
$397$	34.3848	1.72572	0.862861	+	0.505441i	$0.168670\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.1421	−1.10573	−0.552863	−	0.833272i	$-0.686465\pi$
$401$	−22.1421	−1.10573	−0.552863	+	0.833272i	$0.686465\pi$
$402$	0	0
$403$	36.9706	1.84163
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−18.5858	−0.919008	−0.459504	−	0.888176i	$-0.651973\pi$
$409$	−18.5858	−0.919008	−0.459504	+	0.888176i	$0.651973\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.97056	0.440348
$416$	0	0
$417$	0	0
$418$	0	0
$419$	38.8284	1.89689	0.948446	−	0.316938i	$-0.102655\pi$
$419$	38.8284	1.89689	0.948446	+	0.316938i	$0.102655\pi$
$420$	0	0
$421$	−28.6274	−1.39521	−0.697607	−	0.716480i	$-0.745752\pi$
$421$	−28.6274	−1.39521	−0.697607	+	0.716480i	$0.745752\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.0711	1.41015
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.97056	0.335760	0.167880	−	0.985807i	$-0.446308\pi$
$431$	6.97056	0.335760	0.167880	+	0.985807i	$0.446308\pi$
$432$	0	0
$433$	−11.7574	−0.565023	−0.282511	−	0.959264i	$-0.591167\pi$
$433$	−11.7574	−0.565023	−0.282511	+	0.959264i	$0.591167\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.3431	−0.494780
$438$	0	0
$439$	−35.3137	−1.68543	−0.842716	−	0.538359i	$-0.819044\pi$
$439$	−35.3137	−1.68543	−0.842716	+	0.538359i	$0.819044\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.02944	0.0489100	0.0244550	−	0.999701i	$-0.492215\pi$
$443$	1.02944	0.0489100	0.0244550	+	0.999701i	$0.492215\pi$
$444$	0	0
$445$	−3.37258	−0.159876
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.3137	−0.817084	−0.408542	−	0.912739i	$-0.633963\pi$
$449$	−17.3137	−0.817084	−0.408542	+	0.912739i	$0.633963\pi$
$450$	0	0
$451$	−4.48528	−0.211204
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.4142	0.904210	0.452105	−	0.891965i	$-0.350673\pi$
$461$	19.4142	0.904210	0.452105	+	0.891965i	$0.350673\pi$
$462$	0	0
$463$	−18.6274	−0.865689	−0.432845	−	0.901468i	$-0.642490\pi$
$463$	−18.6274	−0.865689	−0.432845	+	0.901468i	$0.642490\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.7990	1.84168	0.920839	−	0.389943i	$-0.127505\pi$
$467$	39.7990	1.84168	0.920839	+	0.389943i	$0.127505\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.3137	−0.520205
$474$	0	0
$475$	13.1716	0.604353
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.1421	1.37723	0.688615	−	0.725127i	$-0.258219\pi$
$479$	30.1421	1.37723	0.688615	+	0.725127i	$0.258219\pi$
$480$	0	0
$481$	−21.6569	−0.987468
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.17157	−0.144014
$486$	0	0
$487$	18.6274	0.844089	0.422044	−	0.906575i	$-0.361313\pi$
$487$	18.6274	0.844089	0.422044	+	0.906575i	$0.361313\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.9706	1.75872	0.879358	−	0.476160i	$-0.157972\pi$
$491$	38.9706	1.75872	0.879358	+	0.476160i	$0.157972\pi$
$492$	0	0
$493$	−7.31371	−0.329393
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.3137	0.864600	0.432300	−	0.901730i	$-0.357702\pi$
$499$	19.3137	0.864600	0.432300	+	0.901730i	$0.357702\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	0	0
$508$	0	0
$509$	25.5563	1.13277	0.566383	−	0.824142i	$-0.308342\pi$
$509$	25.5563	1.13277	0.566383	+	0.824142i	$0.308342\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.31371	−0.322281
$516$	0	0
$517$	−5.65685	−0.248788
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.5858	1.42761	0.713805	−	0.700345i	$-0.246970\pi$
$521$	32.5858	1.42761	0.713805	+	0.700345i	$0.246970\pi$
$522$	0	0
$523$	14.3431	0.627182	0.313591	−	0.949558i	$-0.398468\pi$
$523$	14.3431	0.627182	0.313591	+	0.949558i	$0.398468\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.6274	−1.85688
$528$	0	0
$529$	−9.62742	−0.418583
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.1421	0.525934
$534$	0	0
$535$	6.82843	0.295219
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.31371	−0.228454	−0.114227	−	0.993455i	$-0.536439\pi$
$541$	−5.31371	−0.228454	−0.114227	+	0.993455i	$0.536439\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.31371	−0.141944
$546$	0	0
$547$	3.02944	0.129529	0.0647647	−	0.997901i	$-0.479370\pi$
$547$	3.02944	0.129529	0.0647647	+	0.997901i	$0.479370\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.31371	−0.141169
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	0	0
$559$	30.6274	1.29540
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.82843	−0.287784	−0.143892	−	0.989593i	$-0.545962\pi$
$563$	−6.82843	−0.287784	−0.143892	+	0.989593i	$0.545962\pi$
$564$	0	0
$565$	10.1421	0.426683
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.485281	−0.0203441	−0.0101720	−	0.999948i	$-0.503238\pi$
$569$	−0.485281	−0.0203441	−0.0101720	+	0.999948i	$0.503238\pi$
$570$	0	0
$571$	−33.6569	−1.40850	−0.704248	−	0.709954i	$-0.748716\pi$
$571$	−33.6569	−1.40850	−0.704248	+	0.709954i	$0.748716\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−17.0294	−0.710177
$576$	0	0
$577$	−14.1005	−0.587012	−0.293506	−	0.955957i	$-0.594822\pi$
$577$	−14.1005	−0.587012	−0.293506	+	0.955957i	$0.594822\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.1716	−0.708747	−0.354373	−	0.935104i	$-0.615306\pi$
$587$	−17.1716	−0.708747	−0.354373	+	0.935104i	$0.615306\pi$
$588$	0	0
$589$	−19.3137	−0.795807
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.0711	0.865285	0.432643	−	0.901566i	$-0.357581\pi$
$593$	21.0711	0.865285	0.432643	+	0.901566i	$0.357581\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.00000	−0.0817178	−0.0408589	−	0.999165i	$-0.513009\pi$
$599$	−2.00000	−0.0817178	−0.0408589	+	0.999165i	$0.513009\pi$
$600$	0	0
$601$	−0.928932	−0.0378919	−0.0189460	−	0.999821i	$-0.506031\pi$
$601$	−0.928932	−0.0378919	−0.0189460	+	0.999821i	$0.506031\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.10051	0.166709
$606$	0	0
$607$	29.6569	1.20373	0.601867	−	0.798596i	$-0.294424\pi$
$607$	29.6569	1.20373	0.601867	+	0.798596i	$0.294424\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15.3137	0.619526
$612$	0	0
$613$	27.3137	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$613$	27.3137	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.51472	0.302531	0.151266	−	0.988493i	$-0.451665\pi$
$617$	7.51472	0.302531	0.151266	+	0.988493i	$0.451665\pi$
$618$	0	0
$619$	4.97056	0.199784	0.0998919	−	0.994998i	$-0.468150\pi$
$619$	4.97056	0.199784	0.0998919	+	0.994998i	$0.468150\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.9706	0.995642
$630$	0	0
$631$	−0.686292	−0.0273208	−0.0136604	−	0.999907i	$-0.504348\pi$
$631$	−0.686292	−0.0273208	−0.0136604	+	0.999907i	$0.504348\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.65685	0.224485
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.17157	0.204265	0.102132	−	0.994771i	$-0.467433\pi$
$641$	5.17157	0.204265	0.102132	+	0.994771i	$0.467433\pi$
$642$	0	0
$643$	−50.4264	−1.98862	−0.994312	−	0.106510i	$-0.966033\pi$
$643$	−50.4264	−1.98862	−0.994312	+	0.106510i	$0.966033\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.1716	0.832340	0.416170	−	0.909287i	$-0.363372\pi$
$647$	21.1716	0.832340	0.416170	+	0.909287i	$0.363372\pi$
$648$	0	0
$649$	13.6569	0.536078
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.5147	−0.763670	−0.381835	−	0.924231i	$-0.624708\pi$
$653$	−19.5147	−0.763670	−0.381835	+	0.924231i	$0.624708\pi$
$654$	0	0
$655$	−4.28427	−0.167400
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.3137	−0.518628	−0.259314	−	0.965793i	$-0.583497\pi$
$659$	−13.3137	−0.518628	−0.259314	+	0.965793i	$0.583497\pi$
$660$	0	0
$661$	7.55635	0.293908	0.146954	−	0.989143i	$-0.453053\pi$
$661$	7.55635	0.293908	0.146954	+	0.989143i	$0.453053\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.28427	0.165888
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.51472	−0.290102
$672$	0	0
$673$	0.686292	0.0264546	0.0132273	−	0.999913i	$-0.495789\pi$
$673$	0.686292	0.0264546	0.0132273	+	0.999913i	$0.495789\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.5858	−1.09864	−0.549321	−	0.835612i	$-0.685113\pi$
$677$	−28.5858	−1.09864	−0.549321	+	0.835612i	$0.685113\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.34315	−0.319242	−0.159621	−	0.987178i	$-0.551027\pi$
$683$	−8.34315	−0.319242	−0.159621	+	0.987178i	$0.551027\pi$
$684$	0	0
$685$	−8.28427	−0.316526
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.8284	0.412530
$690$	0	0
$691$	23.3137	0.886895	0.443448	−	0.896300i	$-0.353755\pi$
$691$	23.3137	0.886895	0.443448	+	0.896300i	$0.353755\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.71573	−0.140946
$696$	0	0
$697$	−14.0000	−0.530288
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.8284	0.862218	0.431109	−	0.902300i	$-0.358122\pi$
$701$	22.8284	0.862218	0.431109	+	0.902300i	$0.358122\pi$
$702$	0	0
$703$	11.3137	0.426705
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.2843	0.761792	0.380896	−	0.924618i	$-0.375616\pi$
$709$	20.2843	0.761792	0.380896	+	0.924618i	$0.375616\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	24.9706	0.935155
$714$	0	0
$715$	6.34315	0.237220
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.9411	−0.967441	−0.483720	−	0.875223i	$-0.660715\pi$
$719$	−25.9411	−0.967441	−0.483720	+	0.875223i	$0.660715\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.45584	−0.202625
$726$	0	0
$727$	4.48528	0.166350	0.0831749	−	0.996535i	$-0.473494\pi$
$727$	4.48528	0.166350	0.0831749	+	0.996535i	$0.473494\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−35.3137	−1.30612
$732$	0	0
$733$	−9.69848	−0.358222	−0.179111	−	0.983829i	$-0.557322\pi$
$733$	−9.69848	−0.358222	−0.179111	+	0.983829i	$0.557322\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.3137	0.416746
$738$	0	0
$739$	−27.3137	−1.00475	−0.502376	−	0.864650i	$-0.667540\pi$
$739$	−27.3137	−1.00475	−0.502376	+	0.864650i	$0.667540\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.0294	−0.624749	−0.312375	−	0.949959i	$-0.601124\pi$
$743$	−17.0294	−0.624749	−0.312375	+	0.949959i	$0.601124\pi$
$744$	0	0
$745$	−3.11270	−0.114040
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.34315	−0.0855026	−0.0427513	−	0.999086i	$-0.513612\pi$
$751$	−2.34315	−0.0855026	−0.0427513	+	0.999086i	$0.513612\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.02944	0.255827
$756$	0	0
$757$	37.6569	1.36866	0.684331	−	0.729172i	$-0.260094\pi$
$757$	37.6569	1.36866	0.684331	+	0.729172i	$0.260094\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.5269	−1.68660	−0.843300	−	0.537444i	$-0.819390\pi$
$761$	−46.5269	−1.68660	−0.843300	+	0.537444i	$0.819390\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.9706	−1.33493
$768$	0	0
$769$	−29.6985	−1.07095	−0.535477	−	0.844550i	$-0.679868\pi$
$769$	−29.6985	−1.07095	−0.535477	+	0.844550i	$0.679868\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21.5563	−0.775328	−0.387664	−	0.921801i	$-0.626718\pi$
$773$	−21.5563	−0.775328	−0.387664	+	0.921801i	$0.626718\pi$
$774$	0	0
$775$	−31.7990	−1.14225
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.34315	−0.227267
$780$	0	0
$781$	26.6274	0.952804
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−11.8579	−0.423225
$786$	0	0
$787$	−47.3137	−1.68655	−0.843276	−	0.537481i	$-0.819376\pi$
$787$	−47.3137	−1.68655	−0.843276	+	0.537481i	$0.819376\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20.3431	0.722406
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.3848	−1.00544	−0.502720	−	0.864449i	$-0.667667\pi$
$797$	−28.3848	−1.00544	−0.502720	+	0.864449i	$0.667667\pi$
$798$	0	0
$799$	−17.6569	−0.624655
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.7990	0.416377
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	47.9411	1.68552	0.842760	−	0.538289i	$-0.180929\pi$
$809$	47.9411	1.68552	0.842760	+	0.538289i	$0.180929\pi$
$810$	0	0
$811$	6.34315	0.222738	0.111369	−	0.993779i	$-0.464476\pi$
$811$	6.34315	0.222738	0.111369	+	0.993779i	$0.464476\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.62742	0.232148
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.3137	1.16266	0.581328	−	0.813669i	$-0.302533\pi$
$821$	33.3137	1.16266	0.581328	+	0.813669i	$0.302533\pi$
$822$	0	0
$823$	24.9706	0.870419	0.435210	−	0.900329i	$-0.356674\pi$
$823$	24.9706	0.870419	0.435210	+	0.900329i	$0.356674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.3431	1.26378	0.631888	−	0.775060i	$-0.282280\pi$
$827$	36.3431	1.26378	0.631888	+	0.775060i	$0.282280\pi$
$828$	0	0
$829$	24.7279	0.858836	0.429418	−	0.903106i	$-0.358719\pi$
$829$	24.7279	0.858836	0.429418	+	0.903106i	$0.358719\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−11.5980	−0.401365
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45.1716	1.55950	0.779748	−	0.626094i	$-0.215347\pi$
$839$	45.1716	1.55950	0.779748	+	0.626094i	$0.215347\pi$
$840$	0	0
$841$	−27.6274	−0.952670
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.55635	−0.328748
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.6274	−0.501421
$852$	0	0
$853$	49.4975	1.69476	0.847381	−	0.530986i	$-0.178178\pi$
$853$	49.4975	1.69476	0.847381	+	0.530986i	$0.178178\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.5858	−0.429922	−0.214961	−	0.976623i	$-0.568962\pi$
$857$	−12.5858	−0.429922	−0.214961	+	0.976623i	$0.568962\pi$
$858$	0	0
$859$	−6.54416	−0.223284	−0.111642	−	0.993749i	$-0.535611\pi$
$859$	−6.54416	−0.223284	−0.111642	+	0.993749i	$0.535611\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.31371	−0.180881	−0.0904404	−	0.995902i	$-0.528827\pi$
$863$	−5.31371	−0.180881	−0.0904404	+	0.995902i	$0.528827\pi$
$864$	0	0
$865$	4.05887	0.138006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.68629	0.158972
$870$	0	0
$871$	−30.6274	−1.03777
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.3137	0.382037	0.191018	−	0.981586i	$-0.438821\pi$
$877$	11.3137	0.382037	0.191018	+	0.981586i	$0.438821\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.2426	−1.01890	−0.509450	−	0.860500i	$-0.670151\pi$
$881$	−30.2426	−1.01890	−0.509450	+	0.860500i	$0.670151\pi$
$882$	0	0
$883$	27.3137	0.919179	0.459590	−	0.888131i	$-0.347996\pi$
$883$	27.3137	0.919179	0.459590	+	0.888131i	$0.347996\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.82843	0.0949693	0.0474846	−	0.998872i	$-0.484879\pi$
$887$	2.82843	0.0949693	0.0474846	+	0.998872i	$0.484879\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	4.88730	0.163364
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	−12.4853	−0.415945
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.17157	0.105427
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−34.9706	−1.15863	−0.579313	−	0.815105i	$-0.696679\pi$
$911$	−34.9706	−1.15863	−0.579313	+	0.815105i	$0.696679\pi$
$912$	0	0
$913$	30.6274	1.01362
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−48.2843	−1.59275	−0.796376	−	0.604802i	$-0.793252\pi$
$919$	−48.2843	−1.59275	−0.796376	+	0.604802i	$0.793252\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−72.0833	−2.37265
$924$	0	0
$925$	18.6274	0.612466
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.21320	−0.105422	−0.0527109	−	0.998610i	$-0.516786\pi$
$929$	−3.21320	−0.105422	−0.0527109	+	0.998610i	$0.516786\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.31371	−0.239184
$936$	0	0
$937$	33.4142	1.09159	0.545797	−	0.837917i	$-0.316227\pi$
$937$	33.4142	1.09159	0.545797	+	0.837917i	$0.316227\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.21320	0.235144	0.117572	−	0.993064i	$-0.462489\pi$
$941$	7.21320	0.235144	0.117572	+	0.993064i	$0.462489\pi$
$942$	0	0
$943$	8.20101	0.267062
$944$	0	0
$945$	0	0
$946$	0	0
$947$	53.3137	1.73246	0.866231	−	0.499643i	$-0.166535\pi$
$947$	53.3137	1.73246	0.866231	+	0.499643i	$0.166535\pi$
$948$	0	0
$949$	−31.9411	−1.03685
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	10.5442	0.341201
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	0	0
$964$	0	0
$965$	10.1421	0.326487
$966$	0	0
$967$	−22.3431	−0.718507	−0.359254	−	0.933240i	$-0.616969\pi$
$967$	−22.3431	−0.718507	−0.359254	+	0.933240i	$0.616969\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.37258	−0.172414	−0.0862072	−	0.996277i	$-0.527475\pi$
$971$	−5.37258	−0.172414	−0.0862072	+	0.996277i	$0.527475\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.8284	0.858317	0.429159	−	0.903229i	$-0.358810\pi$
$977$	26.8284	0.858317	0.429159	+	0.903229i	$0.358810\pi$
$978$	0	0
$979$	−11.5147	−0.368012
$980$	0	0
$981$	0	0
$982$	0	0
$983$	37.2548	1.18824	0.594122	−	0.804375i	$-0.297499\pi$
$983$	37.2548	1.18824	0.594122	+	0.804375i	$0.297499\pi$
$984$	0	0
$985$	1.17157	0.0373294
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.6863	0.657786
$990$	0	0
$991$	−20.9706	−0.666152	−0.333076	−	0.942900i	$-0.608087\pi$
$991$	−20.9706	−0.666152	−0.333076	+	0.942900i	$0.608087\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.05887	0.192079
$996$	0	0
$997$	−10.3848	−0.328889	−0.164445	−	0.986386i	$-0.552583\pi$
$997$	−10.3848	−0.328889	−0.164445	+	0.986386i	$0.552583\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.a.cf.1.2		2
3.2	odd	2		2352.2.a.bc.1.1		2
4.3	odd	2		441.2.a.i.1.2		2
7.6	odd	2		7056.2.a.cv.1.1		2
12.11	even	2		147.2.a.e.1.1	yes	2
21.2	odd	6		2352.2.q.bd.1537.2		4
21.5	even	6		2352.2.q.bb.1537.1		4
21.11	odd	6		2352.2.q.bd.961.2		4
21.17	even	6		2352.2.q.bb.961.1		4
21.20	even	2		2352.2.a.be.1.2		2
24.5	odd	2		9408.2.a.dt.1.2		2
24.11	even	2		9408.2.a.di.1.2		2
28.3	even	6		441.2.e.f.226.1		4
28.11	odd	6		441.2.e.g.226.1		4
28.19	even	6		441.2.e.f.361.1		4
28.23	odd	6		441.2.e.g.361.1		4
28.27	even	2		441.2.a.j.1.2		2
60.59	even	2		3675.2.a.bd.1.2		2
84.11	even	6		147.2.e.d.79.2		4
84.23	even	6		147.2.e.d.67.2		4
84.47	odd	6		147.2.e.e.67.2		4
84.59	odd	6		147.2.e.e.79.2		4
84.83	odd	2		147.2.a.d.1.1	✓	2
168.83	odd	2		9408.2.a.ef.1.1		2
168.125	even	2		9408.2.a.dq.1.1		2
420.419	odd	2		3675.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.a.d.1.1	✓	2	84.83	odd	2
147.2.a.e.1.1	yes	2	12.11	even	2
147.2.e.d.67.2		4	84.23	even	6
147.2.e.d.79.2		4	84.11	even	6
147.2.e.e.67.2		4	84.47	odd	6
147.2.e.e.79.2		4	84.59	odd	6
441.2.a.i.1.2		2	4.3	odd	2
441.2.a.j.1.2		2	28.27	even	2
441.2.e.f.226.1		4	28.3	even	6
441.2.e.f.361.1		4	28.19	even	6
441.2.e.g.226.1		4	28.11	odd	6
441.2.e.g.361.1		4	28.23	odd	6
2352.2.a.bc.1.1		2	3.2	odd	2
2352.2.a.be.1.2		2	21.20	even	2
2352.2.q.bb.961.1		4	21.17	even	6
2352.2.q.bb.1537.1		4	21.5	even	6
2352.2.q.bd.961.2		4	21.11	odd	6
2352.2.q.bd.1537.2		4	21.2	odd	6
3675.2.a.bd.1.2		2	60.59	even	2
3675.2.a.bf.1.2		2	420.419	odd	2
7056.2.a.cf.1.2		2	1.1	even	1	trivial
7056.2.a.cv.1.1		2	7.6	odd	2
9408.2.a.di.1.2		2	24.11	even	2
9408.2.a.dq.1.1		2	168.125	even	2
9408.2.a.dt.1.2		2	24.5	odd	2
9408.2.a.ef.1.1		2	168.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.a (trivial)

\(f(q)\)	\(=\)	\(q-0.585786 q^{5} +O(q^{10})\)	\(q-0.585786 q^{5} -2.00000 q^{11} +5.41421 q^{13} -6.24264 q^{17} -2.82843 q^{19} +3.65685 q^{23} -4.65685 q^{25} +1.17157 q^{29} +6.82843 q^{31} -4.00000 q^{37} +2.24264 q^{41} +5.65685 q^{43} +2.82843 q^{47} +2.00000 q^{53} +1.17157 q^{55} -6.82843 q^{59} +3.75736 q^{61} -3.17157 q^{65} -5.65685 q^{67} -13.3137 q^{71} -5.89949 q^{73} -2.34315 q^{79} -15.3137 q^{83} +3.65685 q^{85} +5.75736 q^{89} +1.65685 q^{95} +5.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 4 q^{11} + 8 q^{13} - 4 q^{17} - 4 q^{23} + 2 q^{25} + 8 q^{29} + 8 q^{31} - 8 q^{37} - 4 q^{41} + 4 q^{53} + 8 q^{55} - 8 q^{59} + 16 q^{61} - 12 q^{65} - 4 q^{71} + 8 q^{73} - 16 q^{79} - 8 q^{83} - 4 q^{85} + 20 q^{89} - 8 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 4 * q^11 + 8 * q^13 - 4 * q^17 - 4 * q^23 + 2 * q^25 + 8 * q^29 + 8 * q^31 - 8 * q^37 - 4 * q^41 + 4 * q^53 + 8 * q^55 - 8 * q^59 + 16 * q^61 - 12 * q^65 - 4 * q^71 + 8 * q^73 - 16 * q^79 - 8 * q^83 - 4 * q^85 + 20 * q^89 - 8 * q^95 + 8 * q^97

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