LMFDB - Embedded newform 2352.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.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:	$18.7808145554$
Analytic rank:	$0$
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			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} +2.00000 q^{11} +5.41421 q^{13} -0.585786 q^{15} +6.24264 q^{17} -2.82843 q^{19} -3.65685 q^{23} -4.65685 q^{25} -1.00000 q^{27} -1.17157 q^{29} +6.82843 q^{31} -2.00000 q^{33} -4.00000 q^{37} -5.41421 q^{39} -2.24264 q^{41} +5.65685 q^{43} +0.585786 q^{45} -2.82843 q^{47} -6.24264 q^{51} -2.00000 q^{53} +1.17157 q^{55} +2.82843 q^{57} +6.82843 q^{59} +3.75736 q^{61} +3.17157 q^{65} -5.65685 q^{67} +3.65685 q^{69} +13.3137 q^{71} -5.89949 q^{73} +4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} +15.3137 q^{83} +3.65685 q^{85} +1.17157 q^{87} -5.75736 q^{89} -6.82843 q^{93} -1.65685 q^{95} +5.41421 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{11} + 8 q^{13} - 4 q^{15} + 4 q^{17} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} + 8 q^{31} - 4 q^{33} - 8 q^{37} - 8 q^{39} + 4 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{53} + 8 q^{55} + 8 q^{59} + 16 q^{61} + 12 q^{65} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} - 16 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} + 8 q^{87} - 20 q^{89} - 8 q^{93} + 8 q^{95} + 8 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9 + 4 * q^11 + 8 * q^13 - 4 * q^15 + 4 * q^17 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 + 8 * q^31 - 4 * q^33 - 8 * q^37 - 8 * q^39 + 4 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^53 + 8 * q^55 + 8 * q^59 + 16 * q^61 + 12 * q^65 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 - 16 * q^79 + 2 * q^81 + 8 * q^83 - 4 * q^85 + 8 * q^87 - 20 * q^89 - 8 * q^93 + 8 * q^95 + 8 * q^97 + 4 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	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
$15$	−0.585786	−0.151249
$16$	0	0
$17$	6.24264	1.51406	0.757031	−	0.653379i	$-0.226649\pi$
$17$	6.24264	1.51406	0.757031	+	0.653379i	$0.226649\pi$
$18$	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	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.17157	−0.217556	−0.108778	−	0.994066i	$-0.534694\pi$
$29$	−1.17157	−0.217556	−0.108778	+	0.994066i	$0.534694\pi$
$30$	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
$33$	−2.00000	−0.348155
$34$	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
$39$	−5.41421	−0.866968
$40$	0	0
$41$	−2.24264	−0.350242	−0.175121	−	0.984547i	$-0.556032\pi$
$41$	−2.24264	−0.350242	−0.175121	+	0.984547i	$0.556032\pi$
$42$	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
$45$	0.585786	0.0873239
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.24264	−0.874145
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	1.17157	0.157975
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	6.82843	0.888985	0.444493	−	0.895782i	$-0.353384\pi$
$59$	6.82843	0.888985	0.444493	+	0.895782i	$0.353384\pi$
$60$	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
$69$	3.65685	0.440234
$70$	0	0
$71$	13.3137	1.58005	0.790023	−	0.613077i	$-0.210068\pi$
$71$	13.3137	1.58005	0.790023	+	0.613077i	$0.210068\pi$
$72$	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
$75$	4.65685	0.537727
$76$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	15.3137	1.68090	0.840449	−	0.541891i	$-0.182291\pi$
$83$	15.3137	1.68090	0.840449	+	0.541891i	$0.182291\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	0	0
$87$	1.17157	0.125606
$88$	0	0
$89$	−5.75736	−0.610279	−0.305139	−	0.952308i	$-0.598703\pi$
$89$	−5.75736	−0.610279	−0.305139	+	0.952308i	$0.598703\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.82843	−0.708075
$94$	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
$99$	2.00000	0.201008
$100$	0	0
$101$	17.0711	1.69863	0.849317	−	0.527883i	$-0.177014\pi$
$101$	17.0711	1.69863	0.849317	+	0.527883i	$0.177014\pi$
$102$	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	$-0.309472\pi$
$107$	11.6569	1.12691	0.563455	+	0.826147i	$0.309472\pi$
$108$	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
$111$	4.00000	0.379663
$112$	0	0
$113$	17.3137	1.62874	0.814368	−	0.580348i	$-0.197084\pi$
$113$	17.3137	1.62874	0.814368	+	0.580348i	$0.197084\pi$
$114$	0	0
$115$	−2.14214	−0.199755
$116$	0	0
$117$	5.41421	0.500544
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	2.24264	0.202212
$124$	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
$129$	−5.65685	−0.498058
$130$	0	0
$131$	−7.31371	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$131$	−7.31371	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.585786	−0.0504165
$136$	0	0
$137$	−14.1421	−1.20824	−0.604122	−	0.796892i	$-0.706476\pi$
$137$	−14.1421	−1.20824	−0.604122	+	0.796892i	$0.706476\pi$
$138$	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
$141$	2.82843	0.238197
$142$	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	$-0.569842\pi$
$149$	−5.31371	−0.435316	−0.217658	+	0.976025i	$0.569842\pi$
$150$	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
$153$	6.24264	0.504688
$154$	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
$159$	2.00000	0.158610
$160$	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
$165$	−1.17157	−0.0912068
$166$	0	0
$167$	−19.7990	−1.53209	−0.766046	−	0.642786i	$-0.777779\pi$
$167$	−19.7990	−1.53209	−0.766046	+	0.642786i	$0.777779\pi$
$168$	0	0
$169$	16.3137	1.25490
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	6.92893	0.526797	0.263398	−	0.964687i	$-0.415157\pi$
$173$	6.92893	0.526797	0.263398	+	0.964687i	$0.415157\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.82843	−0.513256
$178$	0	0
$179$	8.34315	0.623596	0.311798	−	0.950148i	$-0.399069\pi$
$179$	8.34315	0.623596	0.311798	+	0.950148i	$0.399069\pi$
$180$	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
$183$	−3.75736	−0.277752
$184$	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	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	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
$195$	−3.17157	−0.227121
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	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
$201$	5.65685	0.399004
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.31371	−0.0917534
$206$	0	0
$207$	−3.65685	−0.254169
$208$	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
$213$	−13.3137	−0.912240
$214$	0	0
$215$	3.31371	0.225993
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.89949	0.398651
$220$	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
$225$	−4.65685	−0.310457
$226$	0	0
$227$	15.7990	1.04862	0.524308	−	0.851529i	$-0.324324\pi$
$227$	15.7990	1.04862	0.524308	+	0.851529i	$0.324324\pi$
$228$	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	$-0.241706\pi$
$233$	22.1421	1.45058	0.725290	+	0.688444i	$0.241706\pi$
$234$	0	0
$235$	−1.65685	−0.108081
$236$	0	0
$237$	2.34315	0.152204
$238$	0	0
$239$	4.34315	0.280935	0.140467	−	0.990085i	$-0.455139\pi$
$239$	4.34315	0.280935	0.140467	+	0.990085i	$0.455139\pi$
$240$	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
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−15.3137	−0.974388
$248$	0	0
$249$	−15.3137	−0.970467
$250$	0	0
$251$	−4.48528	−0.283108	−0.141554	−	0.989931i	$-0.545210\pi$
$251$	−4.48528	−0.283108	−0.141554	+	0.989931i	$0.545210\pi$
$252$	0	0
$253$	−7.31371	−0.459809
$254$	0	0
$255$	−3.65685	−0.229001
$256$	0	0
$257$	−19.2132	−1.19849	−0.599243	−	0.800567i	$-0.704532\pi$
$257$	−19.2132	−1.19849	−0.599243	+	0.800567i	$0.704532\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.17157	−0.0725185
$262$	0	0
$263$	17.3137	1.06761	0.533805	−	0.845608i	$-0.320762\pi$
$263$	17.3137	1.06761	0.533805	+	0.845608i	$0.320762\pi$
$264$	0	0
$265$	−1.17157	−0.0719691
$266$	0	0
$267$	5.75736	0.352345
$268$	0	0
$269$	10.7279	0.654093	0.327046	−	0.945008i	$-0.393947\pi$
$269$	10.7279	0.654093	0.327046	+	0.945008i	$0.393947\pi$
$270$	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
$279$	6.82843	0.408807
$280$	0	0
$281$	−16.4853	−0.983429	−0.491715	−	0.870756i	$-0.663630\pi$
$281$	−16.4853	−0.983429	−0.491715	+	0.870756i	$0.663630\pi$
$282$	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
$285$	1.65685	0.0981436
$286$	0	0
$287$	0	0
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	−5.41421	−0.317387
$292$	0	0
$293$	19.4142	1.13419	0.567095	−	0.823652i	$-0.308067\pi$
$293$	19.4142	1.13419	0.567095	+	0.823652i	$0.308067\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	−19.7990	−1.14501
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−17.0711	−0.980707
$304$	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
$309$	−12.4853	−0.710263
$310$	0	0
$311$	22.1421	1.25557	0.627783	−	0.778389i	$-0.283963\pi$
$311$	22.1421	1.25557	0.627783	+	0.778389i	$0.283963\pi$
$312$	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	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	−2.34315	−0.131191
$320$	0	0
$321$	−11.6569	−0.650622
$322$	0	0
$323$	−17.6569	−0.982454
$324$	0	0
$325$	−25.2132	−1.39858
$326$	0	0
$327$	−5.65685	−0.312825
$328$	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
$333$	−4.00000	−0.219199
$334$	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
$339$	−17.3137	−0.940352
$340$	0	0
$341$	13.6569	0.739560
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.14214	0.115329
$346$	0	0
$347$	−10.6863	−0.573670	−0.286835	−	0.957980i	$-0.592603\pi$
$347$	−10.6863	−0.573670	−0.286835	+	0.957980i	$0.592603\pi$
$348$	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
$351$	−5.41421	−0.288989
$352$	0	0
$353$	10.7279	0.570990	0.285495	−	0.958380i	$-0.407842\pi$
$353$	10.7279	0.570990	0.285495	+	0.958380i	$0.407842\pi$
$354$	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	$-0.400470\pi$
$359$	11.6569	0.615225	0.307613	+	0.951512i	$0.400470\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	7.00000	0.367405
$364$	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
$369$	−2.24264	−0.116747
$370$	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
$375$	5.65685	0.292119
$376$	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
$381$	9.65685	0.494736
$382$	0	0
$383$	29.6569	1.51539	0.757697	−	0.652606i	$-0.226324\pi$
$383$	29.6569	1.51539	0.757697	+	0.652606i	$0.226324\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.65685	0.287554
$388$	0	0
$389$	10.1421	0.514227	0.257113	−	0.966381i	$-0.417229\pi$
$389$	10.1421	0.514227	0.257113	+	0.966381i	$0.417229\pi$
$390$	0	0
$391$	−22.8284	−1.15448
$392$	0	0
$393$	7.31371	0.368928
$394$	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	$-0.313535\pi$
$401$	22.1421	1.10573	0.552863	+	0.833272i	$0.313535\pi$
$402$	0	0
$403$	36.9706	1.84163
$404$	0	0
$405$	0.585786	0.0291080
$406$	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
$411$	14.1421	0.697580
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.97056	0.440348
$416$	0	0
$417$	−6.34315	−0.310625
$418$	0	0
$419$	−38.8284	−1.89689	−0.948446	−	0.316938i	$-0.897345\pi$
$419$	−38.8284	−1.89689	−0.948446	+	0.316938i	$0.897345\pi$
$420$	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
$423$	−2.82843	−0.137523
$424$	0	0
$425$	−29.0711	−1.41015
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−10.8284	−0.522801
$430$	0	0
$431$	−6.97056	−0.335760	−0.167880	−	0.985807i	$-0.553692\pi$
$431$	−6.97056	−0.335760	−0.167880	+	0.985807i	$0.553692\pi$
$432$	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
$435$	0.686292	0.0329052
$436$	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	$-0.507785\pi$
$443$	−1.02944	−0.0489100	−0.0244550	+	0.999701i	$0.507785\pi$
$444$	0	0
$445$	−3.37258	−0.159876
$446$	0	0
$447$	5.31371	0.251330
$448$	0	0
$449$	17.3137	0.817084	0.408542	−	0.912739i	$-0.366037\pi$
$449$	17.3137	0.817084	0.408542	+	0.912739i	$0.366037\pi$
$450$	0	0
$451$	−4.48528	−0.211204
$452$	0	0
$453$	12.0000	0.563809
$454$	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
$459$	−6.24264	−0.291382
$460$	0	0
$461$	−19.4142	−0.904210	−0.452105	−	0.891965i	$-0.649327\pi$
$461$	−19.4142	−0.904210	−0.452105	+	0.891965i	$0.649327\pi$
$462$	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
$465$	−4.00000	−0.185496
$466$	0	0
$467$	−39.7990	−1.84168	−0.920839	−	0.389943i	$-0.872495\pi$
$467$	−39.7990	−1.84168	−0.920839	+	0.389943i	$0.872495\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−20.2426	−0.932732
$472$	0	0
$473$	11.3137	0.520205
$474$	0	0
$475$	13.1716	0.604353
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−30.1421	−1.37723	−0.688615	−	0.725127i	$-0.741781\pi$
$479$	−30.1421	−1.37723	−0.688615	+	0.725127i	$0.741781\pi$
$480$	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
$489$	11.3137	0.511624
$490$	0	0
$491$	−38.9706	−1.75872	−0.879358	−	0.476160i	$-0.842028\pi$
$491$	−38.9706	−1.75872	−0.879358	+	0.476160i	$0.842028\pi$
$492$	0	0
$493$	−7.31371	−0.329393
$494$	0	0
$495$	1.17157	0.0526583
$496$	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
$501$	19.7990	0.884554
$502$	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
$507$	−16.3137	−0.724517
$508$	0	0
$509$	−25.5563	−1.13277	−0.566383	−	0.824142i	$-0.691658\pi$
$509$	−25.5563	−1.13277	−0.566383	+	0.824142i	$0.691658\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.82843	0.124878
$514$	0	0
$515$	7.31371	0.322281
$516$	0	0
$517$	−5.65685	−0.248788
$518$	0	0
$519$	−6.92893	−0.304146
$520$	0	0
$521$	−32.5858	−1.42761	−0.713805	−	0.700345i	$-0.753030\pi$
$521$	−32.5858	−1.42761	−0.713805	+	0.700345i	$0.753030\pi$
$522$	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
$531$	6.82843	0.296328
$532$	0	0
$533$	−12.1421	−0.525934
$534$	0	0
$535$	6.82843	0.295219
$536$	0	0
$537$	−8.34315	−0.360033
$538$	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
$543$	5.41421	0.232346
$544$	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
$549$	3.75736	0.160360
$550$	0	0
$551$	3.31371	0.141169
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.34315	0.0994610
$556$	0	0
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	0	0
$559$	30.6274	1.29540
$560$	0	0
$561$	−12.4853	−0.527129
$562$	0	0
$563$	6.82843	0.287784	0.143892	−	0.989593i	$-0.454038\pi$
$563$	6.82843	0.287784	0.143892	+	0.989593i	$0.454038\pi$
$564$	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	$-0.496762\pi$
$569$	0.485281	0.0203441	0.0101720	+	0.999948i	$0.496762\pi$
$570$	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
$573$	−18.0000	−0.751961
$574$	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
$579$	17.3137	0.719533
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	3.17157	0.131128
$586$	0	0
$587$	17.1716	0.708747	0.354373	−	0.935104i	$-0.384694\pi$
$587$	17.1716	0.708747	0.354373	+	0.935104i	$0.384694\pi$
$588$	0	0
$589$	−19.3137	−0.795807
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	−21.0711	−0.865285	−0.432643	−	0.901566i	$-0.642419\pi$
$593$	−21.0711	−0.865285	−0.432643	+	0.901566i	$0.642419\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.3431	0.423317
$598$	0	0
$599$	2.00000	0.0817178	0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000	0.0817178	0.0408589	+	0.999165i	$0.486991\pi$
$600$	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
$603$	−5.65685	−0.230365
$604$	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
$615$	1.31371	0.0529738
$616$	0	0
$617$	−7.51472	−0.302531	−0.151266	−	0.988493i	$-0.548335\pi$
$617$	−7.51472	−0.302531	−0.151266	+	0.988493i	$0.548335\pi$
$618$	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
$621$	3.65685	0.146745
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	5.65685	0.225913
$628$	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
$633$	−20.9706	−0.833505
$634$	0	0
$635$	−5.65685	−0.224485
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.3137	0.526682
$640$	0	0
$641$	−5.17157	−0.204265	−0.102132	−	0.994771i	$-0.532567\pi$
$641$	−5.17157	−0.204265	−0.102132	+	0.994771i	$0.532567\pi$
$642$	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
$645$	−3.31371	−0.130477
$646$	0	0
$647$	−21.1716	−0.832340	−0.416170	−	0.909287i	$-0.636628\pi$
$647$	−21.1716	−0.832340	−0.416170	+	0.909287i	$0.636628\pi$
$648$	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	$-0.375292\pi$
$653$	19.5147	0.763670	0.381835	+	0.924231i	$0.375292\pi$
$654$	0	0
$655$	−4.28427	−0.167400
$656$	0	0
$657$	−5.89949	−0.230161
$658$	0	0
$659$	13.3137	0.518628	0.259314	−	0.965793i	$-0.416503\pi$
$659$	13.3137	0.518628	0.259314	+	0.965793i	$0.416503\pi$
$660$	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
$663$	−33.7990	−1.31264
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.28427	0.165888
$668$	0	0
$669$	−8.97056	−0.346822
$670$	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
$675$	4.65685	0.179242
$676$	0	0
$677$	28.5858	1.09864	0.549321	−	0.835612i	$-0.314887\pi$
$677$	28.5858	1.09864	0.549321	+	0.835612i	$0.314887\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15.7990	−0.605419
$682$	0	0
$683$	8.34315	0.319242	0.159621	−	0.987178i	$-0.448973\pi$
$683$	8.34315	0.319242	0.159621	+	0.987178i	$0.448973\pi$
$684$	0	0
$685$	−8.28427	−0.316526
$686$	0	0
$687$	−8.24264	−0.314476
$688$	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
$699$	−22.1421	−0.837492
$700$	0	0
$701$	−22.8284	−0.862218	−0.431109	−	0.902300i	$-0.641878\pi$
$701$	−22.8284	−0.862218	−0.431109	+	0.902300i	$0.641878\pi$
$702$	0	0
$703$	11.3137	0.426705
$704$	0	0
$705$	1.65685	0.0624007
$706$	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
$711$	−2.34315	−0.0878748
$712$	0	0
$713$	−24.9706	−0.935155
$714$	0	0
$715$	6.34315	0.237220
$716$	0	0
$717$	−4.34315	−0.162198
$718$	0	0
$719$	25.9411	0.967441	0.483720	−	0.875223i	$-0.339285\pi$
$719$	25.9411	0.967441	0.483720	+	0.875223i	$0.339285\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.75736	0.288499
$724$	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
$729$	1.00000	0.0370370
$730$	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
$741$	15.3137	0.562563
$742$	0	0
$743$	17.0294	0.624749	0.312375	−	0.949959i	$-0.398876\pi$
$743$	17.0294	0.624749	0.312375	+	0.949959i	$0.398876\pi$
$744$	0	0
$745$	−3.11270	−0.114040
$746$	0	0
$747$	15.3137	0.560299
$748$	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
$753$	4.48528	0.163453
$754$	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
$759$	7.31371	0.265471
$760$	0	0
$761$	46.5269	1.68660	0.843300	−	0.537444i	$-0.180610\pi$
$761$	46.5269	1.68660	0.843300	+	0.537444i	$0.180610\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.65685	0.132214
$766$	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
$771$	19.2132	0.691947
$772$	0	0
$773$	21.5563	0.775328	0.387664	−	0.921801i	$-0.373282\pi$
$773$	21.5563	0.775328	0.387664	+	0.921801i	$0.373282\pi$
$774$	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
$783$	1.17157	0.0418686
$784$	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
$789$	−17.3137	−0.616384
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20.3431	0.722406
$794$	0	0
$795$	1.17157	0.0415514
$796$	0	0
$797$	28.3848	1.00544	0.502720	−	0.864449i	$-0.332333\pi$
$797$	28.3848	1.00544	0.502720	+	0.864449i	$0.332333\pi$
$798$	0	0
$799$	−17.6569	−0.624655
$800$	0	0
$801$	−5.75736	−0.203426
$802$	0	0
$803$	−11.7990	−0.416377
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.7279	−0.377641
$808$	0	0
$809$	−47.9411	−1.68552	−0.842760	−	0.538289i	$-0.819071\pi$
$809$	−47.9411	−1.68552	−0.842760	+	0.538289i	$0.819071\pi$
$810$	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
$813$	18.1421	0.636272
$814$	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	$-0.697467\pi$
$821$	−33.3137	−1.16266	−0.581328	+	0.813669i	$0.697467\pi$
$822$	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
$825$	9.31371	0.324262
$826$	0	0
$827$	−36.3431	−1.26378	−0.631888	−	0.775060i	$-0.717720\pi$
$827$	−36.3431	−1.26378	−0.631888	+	0.775060i	$0.717720\pi$
$828$	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
$831$	−13.3137	−0.461847
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−11.5980	−0.401365
$836$	0	0
$837$	−6.82843	−0.236025
$838$	0	0
$839$	−45.1716	−1.55950	−0.779748	−	0.626094i	$-0.784653\pi$
$839$	−45.1716	−1.55950	−0.779748	+	0.626094i	$0.784653\pi$
$840$	0	0
$841$	−27.6274	−0.952670
$842$	0	0
$843$	16.4853	0.567783
$844$	0	0
$845$	9.55635	0.328748
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.48528	0.291214
$850$	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
$855$	−1.65685	−0.0566632
$856$	0	0
$857$	12.5858	0.429922	0.214961	−	0.976623i	$-0.431038\pi$
$857$	12.5858	0.429922	0.214961	+	0.976623i	$0.431038\pi$
$858$	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	$-0.471173\pi$
$863$	5.31371	0.180881	0.0904404	+	0.995902i	$0.471173\pi$
$864$	0	0
$865$	4.05887	0.138006
$866$	0	0
$867$	−21.9706	−0.746159
$868$	0	0
$869$	−4.68629	−0.158972
$870$	0	0
$871$	−30.6274	−1.03777
$872$	0	0
$873$	5.41421	0.183243
$874$	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
$879$	−19.4142	−0.654825
$880$	0	0
$881$	30.2426	1.01890	0.509450	−	0.860500i	$-0.329849\pi$
$881$	30.2426	1.01890	0.509450	+	0.860500i	$0.329849\pi$
$882$	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
$885$	−4.00000	−0.134459
$886$	0	0
$887$	−2.82843	−0.0949693	−0.0474846	−	0.998872i	$-0.515121\pi$
$887$	−2.82843	−0.0949693	−0.0474846	+	0.998872i	$0.515121\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	4.88730	0.163364
$896$	0	0
$897$	19.7990	0.661069
$898$	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
$909$	17.0711	0.566212
$910$	0	0
$911$	34.9706	1.15863	0.579313	−	0.815105i	$-0.303321\pi$
$911$	34.9706	1.15863	0.579313	+	0.815105i	$0.303321\pi$
$912$	0	0
$913$	30.6274	1.01362
$914$	0	0
$915$	−2.20101	−0.0727632
$916$	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
$921$	1.85786	0.0612187
$922$	0	0
$923$	72.0833	2.37265
$924$	0	0
$925$	18.6274	0.612466
$926$	0	0
$927$	12.4853	0.410070
$928$	0	0
$929$	3.21320	0.105422	0.0527109	−	0.998610i	$-0.483214\pi$
$929$	3.21320	0.105422	0.0527109	+	0.998610i	$0.483214\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−22.1421	−0.724901
$934$	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
$939$	17.8995	0.584128
$940$	0	0
$941$	−7.21320	−0.235144	−0.117572	−	0.993064i	$-0.537511\pi$
$941$	−7.21320	−0.235144	−0.117572	+	0.993064i	$0.537511\pi$
$942$	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	$-0.833465\pi$
$947$	−53.3137	−1.73246	−0.866231	+	0.499643i	$0.833465\pi$
$948$	0	0
$949$	−31.9411	−1.03685
$950$	0	0
$951$	−10.0000	−0.324272
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	10.5442	0.341201
$956$	0	0
$957$	2.34315	0.0757431
$958$	0	0
$959$	0	0
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	11.6569	0.375637
$964$	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
$969$	17.6569	0.567220
$970$	0	0
$971$	5.37258	0.172414	0.0862072	−	0.996277i	$-0.472525\pi$
$971$	5.37258	0.172414	0.0862072	+	0.996277i	$0.472525\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	25.2132	0.807469
$976$	0	0
$977$	−26.8284	−0.858317	−0.429159	−	0.903229i	$-0.641190\pi$
$977$	−26.8284	−0.858317	−0.429159	+	0.903229i	$0.641190\pi$
$978$	0	0
$979$	−11.5147	−0.368012
$980$	0	0
$981$	5.65685	0.180609
$982$	0	0
$983$	−37.2548	−1.18824	−0.594122	−	0.804375i	$-0.702501\pi$
$983$	−37.2548	−1.18824	−0.594122	+	0.804375i	$0.702501\pi$
$984$	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
$993$	−4.00000	−0.126936
$994$	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
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.a.bc.1.1		2
3.2	odd	2		7056.2.a.cf.1.2		2
4.3	odd	2		147.2.a.e.1.1	yes	2
7.2	even	3		2352.2.q.bd.1537.2		4
7.3	odd	6		2352.2.q.bb.961.1		4
7.4	even	3		2352.2.q.bd.961.2		4
7.5	odd	6		2352.2.q.bb.1537.1		4
7.6	odd	2		2352.2.a.be.1.2		2
8.3	odd	2		9408.2.a.di.1.2		2
8.5	even	2		9408.2.a.dt.1.2		2
12.11	even	2		441.2.a.i.1.2		2
20.19	odd	2		3675.2.a.bd.1.2		2
21.20	even	2		7056.2.a.cv.1.1		2
28.3	even	6		147.2.e.e.79.2		4
28.11	odd	6		147.2.e.d.79.2		4
28.19	even	6		147.2.e.e.67.2		4
28.23	odd	6		147.2.e.d.67.2		4
28.27	even	2		147.2.a.d.1.1	✓	2
56.13	odd	2		9408.2.a.dq.1.1		2
56.27	even	2		9408.2.a.ef.1.1		2
84.11	even	6		441.2.e.g.226.1		4
84.23	even	6		441.2.e.g.361.1		4
84.47	odd	6		441.2.e.f.361.1		4
84.59	odd	6		441.2.e.f.226.1		4
84.83	odd	2		441.2.a.j.1.2		2
140.139	even	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	28.27	even	2
147.2.a.e.1.1	yes	2	4.3	odd	2
147.2.e.d.67.2		4	28.23	odd	6
147.2.e.d.79.2		4	28.11	odd	6
147.2.e.e.67.2		4	28.19	even	6
147.2.e.e.79.2		4	28.3	even	6
441.2.a.i.1.2		2	12.11	even	2
441.2.a.j.1.2		2	84.83	odd	2
441.2.e.f.226.1		4	84.59	odd	6
441.2.e.f.361.1		4	84.47	odd	6
441.2.e.g.226.1		4	84.11	even	6
441.2.e.g.361.1		4	84.23	even	6
2352.2.a.bc.1.1		2	1.1	even	1	trivial
2352.2.a.be.1.2		2	7.6	odd	2
2352.2.q.bb.961.1		4	7.3	odd	6
2352.2.q.bb.1537.1		4	7.5	odd	6
2352.2.q.bd.961.2		4	7.4	even	3
2352.2.q.bd.1537.2		4	7.2	even	3
3675.2.a.bd.1.2		2	20.19	odd	2
3675.2.a.bf.1.2		2	140.139	even	2
7056.2.a.cf.1.2		2	3.2	odd	2
7056.2.a.cv.1.1		2	21.20	even	2
9408.2.a.di.1.2		2	8.3	odd	2
9408.2.a.dq.1.1		2	56.13	odd	2
9408.2.a.dt.1.2		2	8.5	even	2
9408.2.a.ef.1.1		2	56.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} +2.00000 q^{11} +5.41421 q^{13} -0.585786 q^{15} +6.24264 q^{17} -2.82843 q^{19} -3.65685 q^{23} -4.65685 q^{25} -1.00000 q^{27} -1.17157 q^{29} +6.82843 q^{31} -2.00000 q^{33} -4.00000 q^{37} -5.41421 q^{39} -2.24264 q^{41} +5.65685 q^{43} +0.585786 q^{45} -2.82843 q^{47} -6.24264 q^{51} -2.00000 q^{53} +1.17157 q^{55} +2.82843 q^{57} +6.82843 q^{59} +3.75736 q^{61} +3.17157 q^{65} -5.65685 q^{67} +3.65685 q^{69} +13.3137 q^{71} -5.89949 q^{73} +4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} +15.3137 q^{83} +3.65685 q^{85} +1.17157 q^{87} -5.75736 q^{89} -6.82843 q^{93} -1.65685 q^{95} +5.41421 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{11} + 8 q^{13} - 4 q^{15} + 4 q^{17} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} + 8 q^{31} - 4 q^{33} - 8 q^{37} - 8 q^{39} + 4 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{53} + 8 q^{55} + 8 q^{59} + 16 q^{61} + 12 q^{65} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} - 16 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} + 8 q^{87} - 20 q^{89} - 8 q^{93} + 8 q^{95} + 8 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9 + 4 * q^11 + 8 * q^13 - 4 * q^15 + 4 * q^17 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 + 8 * q^31 - 4 * q^33 - 8 * q^37 - 8 * q^39 + 4 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^53 + 8 * q^55 + 8 * q^59 + 16 * q^61 + 12 * q^65 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 - 16 * q^79 + 2 * q^81 + 8 * q^83 - 4 * q^85 + 8 * q^87 - 20 * q^89 - 8 * q^93 + 8 * q^95 + 8 * q^97 + 4 * q^99

Introduction

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