LMFDB - Embedded newform 441.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	441.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:	$3.52140272914$
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:	$\Z[a_1, a_2]$
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	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} +0.585786 q^{5} +4.41421 q^{8} +O(q^{10})$	$q+2.41421 q^{2} +3.82843 q^{4} +0.585786 q^{5} +4.41421 q^{8} +1.41421 q^{10} +2.00000 q^{11} -5.41421 q^{13} +3.00000 q^{16} +6.24264 q^{17} -2.82843 q^{19} +2.24264 q^{20} +4.82843 q^{22} -3.65685 q^{23} -4.65685 q^{25} -13.0711 q^{26} +1.17157 q^{29} +6.82843 q^{31} -1.58579 q^{32} +15.0711 q^{34} -4.00000 q^{37} -6.82843 q^{38} +2.58579 q^{40} -2.24264 q^{41} -5.65685 q^{43} +7.65685 q^{44} -8.82843 q^{46} +2.82843 q^{47} -11.2426 q^{50} -20.7279 q^{52} +2.00000 q^{53} +1.17157 q^{55} +2.82843 q^{58} -6.82843 q^{59} -3.75736 q^{61} +16.4853 q^{62} -9.82843 q^{64} -3.17157 q^{65} +5.65685 q^{67} +23.8995 q^{68} +13.3137 q^{71} +5.89949 q^{73} -9.65685 q^{74} -10.8284 q^{76} +2.34315 q^{79} +1.75736 q^{80} -5.41421 q^{82} -15.3137 q^{83} +3.65685 q^{85} -13.6569 q^{86} +8.82843 q^{88} -5.75736 q^{89} -14.0000 q^{92} +6.82843 q^{94} -1.65685 q^{95} -5.41421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{11} - 8 q^{13} + 6 q^{16} + 4 q^{17} - 4 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{25} - 12 q^{26} + 8 q^{29} + 8 q^{31} - 6 q^{32} + 16 q^{34} - 8 q^{37} - 8 q^{38} + 8 q^{40} + 4 q^{41} + 4 q^{44} - 12 q^{46} - 14 q^{50} - 16 q^{52} + 4 q^{53} + 8 q^{55} - 8 q^{59} - 16 q^{61} + 16 q^{62} - 14 q^{64} - 12 q^{65} + 28 q^{68} + 4 q^{71} - 8 q^{73} - 8 q^{74} - 16 q^{76} + 16 q^{79} + 12 q^{80} - 8 q^{82} - 8 q^{83} - 4 q^{85} - 16 q^{86} + 12 q^{88} - 20 q^{89} - 28 q^{92} + 8 q^{94} + 8 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 + 4 * q^11 - 8 * q^13 + 6 * q^16 + 4 * q^17 - 4 * q^20 + 4 * q^22 + 4 * q^23 + 2 * q^25 - 12 * q^26 + 8 * q^29 + 8 * q^31 - 6 * q^32 + 16 * q^34 - 8 * q^37 - 8 * q^38 + 8 * q^40 + 4 * q^41 + 4 * q^44 - 12 * q^46 - 14 * q^50 - 16 * q^52 + 4 * q^53 + 8 * q^55 - 8 * q^59 - 16 * q^61 + 16 * q^62 - 14 * q^64 - 12 * q^65 + 28 * q^68 + 4 * q^71 - 8 * q^73 - 8 * q^74 - 16 * q^76 + 16 * q^79 + 12 * q^80 - 8 * q^82 - 8 * q^83 - 4 * q^85 - 16 * q^86 + 12 * q^88 - 20 * q^89 - 28 * q^92 + 8 * q^94 + 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	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
$8$	4.41421	1.56066
$9$	0	0
$10$	1.41421	0.447214
$11$	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	$-0.770340\pi$
$13$	−5.41421	−1.50163	−0.750816	+	0.660511i	$0.770340\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	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	$0.605177\pi$
$20$	2.24264	0.501470
$21$	0	0
$22$	4.82843	1.02942
$23$	−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
$26$	−13.0711	−2.56345
$27$	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	$0.289878\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	15.0711	2.58467
$35$	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	$0.606644\pi$
$38$	−6.82843	−1.10772
$39$	0	0
$40$	2.58579	0.408849
$41$	−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	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	7.65685	1.15431
$45$	0	0
$46$	−8.82843	−1.30168
$47$	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
$50$	−11.2426	−1.58995
$51$	0	0
$52$	−20.7279	−2.87445
$53$	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
$58$	2.82843	0.371391
$59$	−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	$-0.577325\pi$
$61$	−3.75736	−0.481081	−0.240540	+	0.970639i	$0.577325\pi$
$62$	16.4853	2.09363
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−3.17157	−0.393385
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	23.8995	2.89824
$69$	0	0
$70$	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	$-0.387797\pi$
$73$	5.89949	0.690484	0.345242	+	0.938514i	$0.387797\pi$
$74$	−9.65685	−1.12259
$75$	0	0
$76$	−10.8284	−1.24211
$77$	0	0
$78$	0	0
$79$	2.34315	0.263624	0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	1.75736	0.196479
$81$	0	0
$82$	−5.41421	−0.597900
$83$	−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
$86$	−13.6569	−1.47266
$87$	0	0
$88$	8.82843	0.941113
$89$	−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
$92$	−14.0000	−1.45960
$93$	0	0
$94$	6.82843	0.704298
$95$	−1.65685	−0.169990
$96$	0	0
$97$	−5.41421	−0.549730	−0.274865	−	0.961483i	$-0.588633\pi$
$97$	−5.41421	−0.549730	−0.274865	+	0.961483i	$0.588633\pi$
$98$	0	0
$99$	0	0
$100$	−17.8284	−1.78284
$101$	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	$0.289113\pi$
$104$	−23.8995	−2.34354
$105$	0	0
$106$	4.82843	0.468978
$107$	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	$0.412674\pi$
$110$	2.82843	0.269680
$111$	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
$116$	4.48528	0.416448
$117$	0	0
$118$	−16.4853	−1.51759
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−9.07107	−0.821256
$123$	0	0
$124$	26.1421	2.34763
$125$	−5.65685	−0.505964
$126$	0	0
$127$	9.65685	0.856907	0.428454	−	0.903564i	$-0.359059\pi$
$127$	9.65685	0.856907	0.428454	+	0.903564i	$0.359059\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	−7.65685	−0.671551
$131$	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
$134$	13.6569	1.17977
$135$	0	0
$136$	27.5563	2.36294
$137$	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
$142$	32.1421	2.69731
$143$	−10.8284	−0.905519
$144$	0	0
$145$	0.686292	0.0569934
$146$	14.2426	1.17873
$147$	0	0
$148$	−15.3137	−1.25878
$149$	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	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−12.4853	−1.01269
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−20.2426	−1.61554	−0.807769	−	0.589499i	$-0.799325\pi$
$157$	−20.2426	−1.61554	−0.807769	+	0.589499i	$0.799325\pi$
$158$	5.65685	0.450035
$159$	0	0
$160$	−0.928932	−0.0734385
$161$	0	0
$162$	0	0
$163$	11.3137	0.886158	0.443079	−	0.896483i	$-0.353886\pi$
$163$	11.3137	0.886158	0.443079	+	0.896483i	$0.353886\pi$
$164$	−8.58579	−0.670437
$165$	0	0
$166$	−36.9706	−2.86947
$167$	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
$170$	8.82843	0.677109
$171$	0	0
$172$	−21.6569	−1.65132
$173$	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
$176$	6.00000	0.452267
$177$	0	0
$178$	−13.8995	−1.04181
$179$	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	$-0.435510\pi$
$181$	5.41421	0.402435	0.201218	+	0.979547i	$0.435510\pi$
$182$	0	0
$183$	0	0
$184$	−16.1421	−1.19001
$185$	−2.34315	−0.172272
$186$	0	0
$187$	12.4853	0.913014
$188$	10.8284	0.789744
$189$	0	0
$190$	−4.00000	−0.290191
$191$	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	$0.714141\pi$
$194$	−13.0711	−0.938448
$195$	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	$0.619479\pi$
$200$	−20.5563	−1.45355
$201$	0	0
$202$	41.2132	2.89975
$203$	0	0
$204$	0	0
$205$	−1.31371	−0.0917534
$206$	30.1421	2.10010
$207$	0	0
$208$	−16.2426	−1.12622
$209$	−5.65685	−0.391293
$210$	0	0
$211$	−20.9706	−1.44367	−0.721837	−	0.692064i	$-0.756702\pi$
$211$	−20.9706	−1.44367	−0.721837	+	0.692064i	$0.756702\pi$
$212$	7.65685	0.525875
$213$	0	0
$214$	28.1421	1.92376
$215$	−3.31371	−0.225993
$216$	0	0
$217$	0	0
$218$	13.6569	0.924959
$219$	0	0
$220$	4.48528	0.302398
$221$	−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
$226$	−41.7990	−2.78043
$227$	−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	$-0.587799\pi$
$229$	−8.24264	−0.544689	−0.272345	+	0.962200i	$0.587799\pi$
$230$	−5.17157	−0.341003
$231$	0	0
$232$	5.17157	0.339530
$233$	−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
$236$	−26.1421	−1.70171
$237$	0	0
$238$	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	$-0.419619\pi$
$241$	7.75736	0.499695	0.249848	+	0.968285i	$0.419619\pi$
$242$	−16.8995	−1.08634
$243$	0	0
$244$	−14.3848	−0.920891
$245$	0	0
$246$	0	0
$247$	15.3137	0.974388
$248$	30.1421	1.91403
$249$	0	0
$250$	−13.6569	−0.863735
$251$	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
$254$	23.3137	1.46283
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−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
$260$	−12.1421	−0.753023
$261$	0	0
$262$	17.6569	1.09084
$263$	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
$267$	0	0
$268$	21.6569	1.32290
$269$	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	$0.685764\pi$
$272$	18.7279	1.13555
$273$	0	0
$274$	34.1421	2.06260
$275$	−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	$0.369020\pi$
$278$	15.3137	0.918455
$279$	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	$0.581154\pi$
$284$	50.9706	3.02455
$285$	0	0
$286$	−26.1421	−1.54582
$287$	0	0
$288$	0	0
$289$	21.9706	1.29239
$290$	1.65685	0.0972938
$291$	0	0
$292$	22.5858	1.32173
$293$	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
$296$	−17.6569	−1.02628
$297$	0	0
$298$	12.8284	0.743131
$299$	19.7990	1.14501
$300$	0	0
$301$	0	0
$302$	28.9706	1.66707
$303$	0	0
$304$	−8.48528	−0.486664
$305$	−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
$310$	9.65685	0.548472
$311$	−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	$-0.331172\pi$
$313$	17.8995	1.01174	0.505870	+	0.862610i	$0.331172\pi$
$314$	−48.8701	−2.75790
$315$	0	0
$316$	8.97056	0.504634
$317$	−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
$320$	−5.75736	−0.321846
$321$	0	0
$322$	0	0
$323$	−17.6569	−0.982454
$324$	0	0
$325$	25.2132	1.39858
$326$	27.3137	1.51277
$327$	0	0
$328$	−9.89949	−0.546608
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−58.6274	−3.21760
$333$	0	0
$334$	47.7990	2.61544
$335$	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	$0.666522\pi$
$338$	39.3848	2.14225
$339$	0	0
$340$	14.0000	0.759257
$341$	13.6569	0.739560
$342$	0	0
$343$	0	0
$344$	−24.9706	−1.34632
$345$	0	0
$346$	16.7279	0.899299
$347$	−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	$-0.585357\pi$
$349$	−9.89949	−0.529908	−0.264954	+	0.964261i	$0.585357\pi$
$350$	0	0
$351$	0	0
$352$	−3.17157	−0.169045
$353$	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
$356$	−22.0416	−1.16820
$357$	0	0
$358$	20.1421	1.06454
$359$	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
$362$	13.0711	0.687000
$363$	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	$0.668170\pi$
$368$	−10.9706	−0.571880
$369$	0	0
$370$	−5.65685	−0.294086
$371$	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	$0.831077\pi$
$374$	30.1421	1.55861
$375$	0	0
$376$	12.4853	0.643879
$377$	−6.34315	−0.326689
$378$	0	0
$379$	31.3137	1.60848	0.804239	−	0.594307i	$-0.202573\pi$
$379$	31.3137	1.60848	0.804239	+	0.594307i	$0.202573\pi$
$380$	−6.34315	−0.325397
$381$	0	0
$382$	43.4558	2.22339
$383$	−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
$386$	−41.7990	−2.12751
$387$	0	0
$388$	−20.7279	−1.05230
$389$	−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
$394$	−4.82843	−0.243253
$395$	1.37258	0.0690621
$396$	0	0
$397$	−34.3848	−1.72572	−0.862861	−	0.505441i	$-0.831330\pi$
$397$	−34.3848	−1.72572	−0.862861	+	0.505441i	$0.831330\pi$
$398$	−24.9706	−1.25166
$399$	0	0
$400$	−13.9706	−0.698528
$401$	−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
$404$	65.3553	3.25155
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	18.5858	0.919008	0.459504	−	0.888176i	$-0.348027\pi$
$409$	18.5858	0.919008	0.459504	+	0.888176i	$0.348027\pi$
$410$	−3.17157	−0.156633
$411$	0	0
$412$	47.7990	2.35489
$413$	0	0
$414$	0	0
$415$	−8.97056	−0.440348
$416$	8.58579	0.420953
$417$	0	0
$418$	−13.6569	−0.667979
$419$	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	$0.745752\pi$
$422$	−50.6274	−2.46450
$423$	0	0
$424$	8.82843	0.428746
$425$	−29.0711	−1.41015
$426$	0	0
$427$	0	0
$428$	44.6274	2.15715
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−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	$-0.408833\pi$
$433$	11.7574	0.565023	0.282511	+	0.959264i	$0.408833\pi$
$434$	0	0
$435$	0	0
$436$	21.6569	1.03718
$437$	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	$0.819044\pi$
$440$	5.17157	0.246545
$441$	0	0
$442$	−81.5980	−3.88122
$443$	−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
$446$	21.6569	1.02548
$447$	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
$452$	−66.2843	−3.11775
$453$	0	0
$454$	−38.1421	−1.79010
$455$	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	$0.638322\pi$
$458$	−19.8995	−0.929842
$459$	0	0
$460$	−8.20101	−0.382374
$461$	−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	$-0.357510\pi$
$463$	18.6274	0.865689	0.432845	+	0.901468i	$0.357510\pi$
$464$	3.51472	0.163167
$465$	0	0
$466$	−53.4558	−2.47629
$467$	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
$470$	4.00000	0.184506
$471$	0	0
$472$	−30.1421	−1.38740
$473$	−11.3137	−0.520205
$474$	0	0
$475$	13.1716	0.604353
$476$	0	0
$477$	0	0
$478$	10.4853	0.479586
$479$	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
$482$	18.7279	0.853033
$483$	0	0
$484$	−26.7990	−1.21814
$485$	−3.17157	−0.144014
$486$	0	0
$487$	−18.6274	−0.844089	−0.422044	−	0.906575i	$-0.638687\pi$
$487$	−18.6274	−0.844089	−0.422044	+	0.906575i	$0.638687\pi$
$488$	−16.5858	−0.750803
$489$	0	0
$490$	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
$494$	36.9706	1.66338
$495$	0	0
$496$	20.4853	0.919816
$497$	0	0
$498$	0	0
$499$	−19.3137	−0.864600	−0.432300	−	0.901730i	$-0.642298\pi$
$499$	−19.3137	−0.864600	−0.432300	+	0.901730i	$0.642298\pi$
$500$	−21.6569	−0.968524
$501$	0	0
$502$	10.8284	0.483296
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	−17.6569	−0.784943
$507$	0	0
$508$	36.9706	1.64030
$509$	−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
$512$	−31.2426	−1.38074
$513$	0	0
$514$	−46.3848	−2.04594
$515$	7.31371	0.322281
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	0	0
$520$	−14.0000	−0.613941
$521$	−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	$0.398468\pi$
$524$	28.0000	1.22319
$525$	0	0
$526$	41.7990	1.82252
$527$	42.6274	1.85688
$528$	0	0
$529$	−9.62742	−0.418583
$530$	2.82843	0.122859
$531$	0	0
$532$	0	0
$533$	12.1421	0.525934
$534$	0	0
$535$	6.82843	0.295219
$536$	24.9706	1.07856
$537$	0	0
$538$	25.8995	1.11661
$539$	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	$0.536439\pi$
$542$	−43.7990	−1.88133
$543$	0	0
$544$	−9.89949	−0.424437
$545$	3.31371	0.141944
$546$	0	0
$547$	−3.02944	−0.129529	−0.0647647	−	0.997901i	$-0.520630\pi$
$547$	−3.02944	−0.129529	−0.0647647	+	0.997901i	$0.520630\pi$
$548$	54.1421	2.31284
$549$	0	0
$550$	−22.4853	−0.958776
$551$	−3.31371	−0.141169
$552$	0	0
$553$	0	0
$554$	32.1421	1.36559
$555$	0	0
$556$	24.2843	1.02988
$557$	−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
$562$	39.7990	1.67882
$563$	−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
$566$	−20.4853	−0.861061
$567$	0	0
$568$	58.7696	2.46592
$569$	−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	$-0.251284\pi$
$571$	33.6569	1.40850	0.704248	+	0.709954i	$0.251284\pi$
$572$	−41.4558	−1.73336
$573$	0	0
$574$	0	0
$575$	17.0294	0.710177
$576$	0	0
$577$	14.1005	0.587012	0.293506	−	0.955957i	$-0.405178\pi$
$577$	14.1005	0.587012	0.293506	+	0.955957i	$0.405178\pi$
$578$	53.0416	2.20624
$579$	0	0
$580$	2.62742	0.109098
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	26.0416	1.07761
$585$	0	0
$586$	46.8701	1.93618
$587$	−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
$590$	−9.65685	−0.397566
$591$	0	0
$592$	−12.0000	−0.493197
$593$	−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
$596$	20.3431	0.833288
$597$	0	0
$598$	47.7990	1.95465
$599$	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	$-0.493969\pi$
$601$	0.928932	0.0378919	0.0189460	+	0.999821i	$0.493969\pi$
$602$	0	0
$603$	0	0
$604$	45.9411	1.86932
$605$	−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	$0.294424\pi$
$608$	4.48528	0.181902
$609$	0	0
$610$	−5.31371	−0.215146
$611$	−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	$0.314019\pi$
$614$	−4.48528	−0.181011
$615$	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	$0.468150\pi$
$620$	15.3137	0.615013
$621$	0	0
$622$	−53.4558	−2.14338
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	43.2132	1.72715
$627$	0	0
$628$	−77.4975	−3.09249
$629$	−24.9706	−0.995642
$630$	0	0
$631$	0.686292	0.0273208	0.0136604	−	0.999907i	$-0.495652\pi$
$631$	0.686292	0.0273208	0.0136604	+	0.999907i	$0.495652\pi$
$632$	10.3431	0.411428
$633$	0	0
$634$	−24.1421	−0.958807
$635$	5.65685	0.224485
$636$	0	0
$637$	0	0
$638$	5.65685	0.223957
$639$	0	0
$640$	−12.0416	−0.475987
$641$	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
$646$	−42.6274	−1.67715
$647$	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
$650$	60.8701	2.38752
$651$	0	0
$652$	43.3137	1.69630
$653$	−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
$656$	−6.72792	−0.262681
$657$	0	0
$658$	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	$-0.546947\pi$
$661$	−7.55635	−0.293908	−0.146954	+	0.989143i	$0.546947\pi$
$662$	−9.65685	−0.375324
$663$	0	0
$664$	−67.5980	−2.62331
$665$	0	0
$666$	0	0
$667$	−4.28427	−0.165888
$668$	75.7990	2.93275
$669$	0	0
$670$	8.00000	0.309067
$671$	−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	$0.495789\pi$
$674$	−44.2843	−1.70577
$675$	0	0
$676$	62.4558	2.40215
$677$	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
$680$	16.1421	0.619023
$681$	0	0
$682$	32.9706	1.26251
$683$	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
$687$	0	0
$688$	−16.9706	−0.646997
$689$	−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	$0.353755\pi$
$692$	26.5269	1.00840
$693$	0	0
$694$	−25.7990	−0.979316
$695$	3.71573	0.140946
$696$	0	0
$697$	−14.0000	−0.530288
$698$	−23.8995	−0.904609
$699$	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
$704$	−19.6569	−0.740846
$705$	0	0
$706$	25.8995	0.974740
$707$	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	$0.375616\pi$
$710$	18.8284	0.706618
$711$	0	0
$712$	−25.4142	−0.952438
$713$	−24.9706	−0.935155
$714$	0	0
$715$	−6.34315	−0.237220
$716$	31.9411	1.19370
$717$	0	0
$718$	28.1421	1.05026
$719$	−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
$722$	−26.5563	−0.988325
$723$	0	0
$724$	20.7279	0.770347
$725$	−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
$730$	8.34315	0.308794
$731$	−35.3137	−1.30612
$732$	0	0
$733$	9.69848	0.358222	0.179111	−	0.983829i	$-0.442678\pi$
$733$	9.69848	0.358222	0.179111	+	0.983829i	$0.442678\pi$
$734$	−46.6274	−1.72105
$735$	0	0
$736$	5.79899	0.213754
$737$	11.3137	0.416746
$738$	0	0
$739$	27.3137	1.00475	0.502376	−	0.864650i	$-0.332460\pi$
$739$	27.3137	1.00475	0.502376	+	0.864650i	$0.332460\pi$
$740$	−8.97056	−0.329764
$741$	0	0
$742$	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
$746$	−80.4264	−2.94462
$747$	0	0
$748$	47.7990	1.74770
$749$	0	0
$750$	0	0
$751$	2.34315	0.0855026	0.0427513	−	0.999086i	$-0.486388\pi$
$751$	2.34315	0.0855026	0.0427513	+	0.999086i	$0.486388\pi$
$752$	8.48528	0.309426
$753$	0	0
$754$	−15.3137	−0.557692
$755$	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	$0.260094\pi$
$758$	75.5980	2.74584
$759$	0	0
$760$	−7.31371	−0.265296
$761$	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
$764$	68.9117	2.49314
$765$	0	0
$766$	−71.5980	−2.58694
$767$	36.9706	1.33493
$768$	0	0
$769$	29.6985	1.07095	0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985	1.07095	0.535477	+	0.844550i	$0.320132\pi$
$770$	0	0
$771$	0	0
$772$	−66.2843	−2.38562
$773$	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
$776$	−23.8995	−0.857942
$777$	0	0
$778$	−24.4853	−0.877840
$779$	6.34315	0.227267
$780$	0	0
$781$	26.6274	0.952804
$782$	−55.1127	−1.97083
$783$	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	$0.819376\pi$
$788$	−7.65685	−0.272764
$789$	0	0
$790$	3.31371	0.117896
$791$	0	0
$792$	0	0
$793$	20.3431	0.722406
$794$	−83.0122	−2.94599
$795$	0	0
$796$	−39.5980	−1.40351
$797$	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
$800$	7.38478	0.261091
$801$	0	0
$802$	−53.4558	−1.88759
$803$	11.7990	0.416377
$804$	0	0
$805$	0	0
$806$	−89.2548	−3.14387
$807$	0	0
$808$	75.3553	2.65099
$809$	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
$814$	−19.3137	−0.676945
$815$	6.62742	0.232148
$816$	0	0
$817$	16.0000	0.559769
$818$	44.8701	1.56884
$819$	0	0
$820$	−5.02944	−0.175636
$821$	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	$-0.643326\pi$
$823$	−24.9706	−0.870419	−0.435210	+	0.900329i	$0.643326\pi$
$824$	55.1127	1.91994
$825$	0	0
$826$	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	$-0.641281\pi$
$829$	−24.7279	−0.858836	−0.429418	+	0.903106i	$0.641281\pi$
$830$	−21.6569	−0.751720
$831$	0	0
$832$	53.2132	1.84484
$833$	0	0
$834$	0	0
$835$	11.5980	0.401365
$836$	−21.6569	−0.749018
$837$	0	0
$838$	93.7401	3.23820
$839$	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
$842$	−69.1127	−2.38178
$843$	0	0
$844$	−80.2843	−2.76350
$845$	9.55635	0.328748
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	−70.1838	−2.40728
$851$	14.6274	0.501421
$852$	0	0
$853$	−49.4975	−1.69476	−0.847381	−	0.530986i	$-0.821822\pi$
$853$	−49.4975	−1.69476	−0.847381	+	0.530986i	$0.821822\pi$
$854$	0	0
$855$	0	0
$856$	51.4558	1.75872
$857$	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	$0.535611\pi$
$860$	−12.6863	−0.432599
$861$	0	0
$862$	−16.8284	−0.573179
$863$	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
$866$	28.3848	0.964554
$867$	0	0
$868$	0	0
$869$	4.68629	0.158972
$870$	0	0
$871$	−30.6274	−1.03777
$872$	24.9706	0.845610
$873$	0	0
$874$	24.9706	0.844642
$875$	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	$0.438821\pi$
$878$	−85.2548	−2.87721
$879$	0	0
$880$	3.51472	0.118481
$881$	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	$-0.652004\pi$
$883$	−27.3137	−0.919179	−0.459590	+	0.888131i	$0.652004\pi$
$884$	−129.397	−4.35209
$885$	0	0
$886$	−2.48528	−0.0834947
$887$	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
$890$	−8.14214	−0.272925
$891$	0	0
$892$	34.3431	1.14989
$893$	−8.00000	−0.267710
$894$	0	0
$895$	4.88730	0.163364
$896$	0	0
$897$	0	0
$898$	−41.7990	−1.39485
$899$	8.00000	0.266815
$900$	0	0
$901$	12.4853	0.415945
$902$	−10.8284	−0.360547
$903$	0	0
$904$	−76.4264	−2.54190
$905$	3.17157	0.105427
$906$	0	0
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	−60.4853	−2.00727
$909$	0	0
$910$	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
$914$	−43.4558	−1.43739
$915$	0	0
$916$	−31.5563	−1.04265
$917$	0	0
$918$	0	0
$919$	48.2843	1.59275	0.796376	−	0.604802i	$-0.206748\pi$
$919$	48.2843	1.59275	0.796376	+	0.604802i	$0.206748\pi$
$920$	−9.45584	−0.311750
$921$	0	0
$922$	−46.8701	−1.54358
$923$	−72.0833	−2.37265
$924$	0	0
$925$	18.6274	0.612466
$926$	44.9706	1.47782
$927$	0	0
$928$	−1.85786	−0.0609874
$929$	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
$932$	−84.7696	−2.77672
$933$	0	0
$934$	96.0833	3.14394
$935$	7.31371	0.239184
$936$	0	0
$937$	−33.4142	−1.09159	−0.545797	−	0.837917i	$-0.683773\pi$
$937$	−33.4142	−1.09159	−0.545797	+	0.837917i	$0.683773\pi$
$938$	0	0
$939$	0	0
$940$	6.34315	0.206891
$941$	−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
$944$	−20.4853	−0.666739
$945$	0	0
$946$	−27.3137	−0.888045
$947$	−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
$950$	31.7990	1.03170
$951$	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
$956$	16.6274	0.537769
$957$	0	0
$958$	72.7696	2.35108
$959$	0	0
$960$	0	0
$961$	15.6274	0.504110
$962$	52.2843	1.68571
$963$	0	0
$964$	29.6985	0.956524
$965$	−10.1421	−0.326487
$966$	0	0
$967$	22.3431	0.718507	0.359254	−	0.933240i	$-0.383031\pi$
$967$	22.3431	0.718507	0.359254	+	0.933240i	$0.383031\pi$
$968$	−30.8995	−0.993147
$969$	0	0
$970$	−7.65685	−0.245847
$971$	−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
$974$	−44.9706	−1.44095
$975$	0	0
$976$	−11.2721	−0.360810
$977$	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
$982$	−94.0833	−3.00232
$983$	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
$986$	17.6569	0.562309
$987$	0	0
$988$	58.6274	1.86519
$989$	20.6863	0.657786
$990$	0	0
$991$	20.9706	0.666152	0.333076	−	0.942900i	$-0.391913\pi$
$991$	20.9706	0.666152	0.333076	+	0.942900i	$0.391913\pi$
$992$	−10.8284	−0.343803
$993$	0	0
$994$	0	0
$995$	−6.05887	−0.192079
$996$	0	0
$997$	10.3848	0.328889	0.164445	−	0.986386i	$-0.447417\pi$
$997$	10.3848	0.328889	0.164445	+	0.986386i	$0.447417\pi$
$998$	−46.6274	−1.47597
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.a.j.1.2		2
3.2	odd	2		147.2.a.d.1.1	✓	2
4.3	odd	2		7056.2.a.cv.1.1		2
7.2	even	3		441.2.e.f.361.1		4
7.3	odd	6		441.2.e.g.226.1		4
7.4	even	3		441.2.e.f.226.1		4
7.5	odd	6		441.2.e.g.361.1		4
7.6	odd	2		441.2.a.i.1.2		2
12.11	even	2		2352.2.a.be.1.2		2
15.14	odd	2		3675.2.a.bf.1.2		2
21.2	odd	6		147.2.e.e.67.2		4
21.5	even	6		147.2.e.d.67.2		4
21.11	odd	6		147.2.e.e.79.2		4
21.17	even	6		147.2.e.d.79.2		4
21.20	even	2		147.2.a.e.1.1	yes	2
24.5	odd	2		9408.2.a.ef.1.1		2
24.11	even	2		9408.2.a.dq.1.1		2
28.27	even	2		7056.2.a.cf.1.2		2
84.11	even	6		2352.2.q.bb.961.1		4
84.23	even	6		2352.2.q.bb.1537.1		4
84.47	odd	6		2352.2.q.bd.1537.2		4
84.59	odd	6		2352.2.q.bd.961.2		4
84.83	odd	2		2352.2.a.bc.1.1		2
105.104	even	2		3675.2.a.bd.1.2		2
168.83	odd	2		9408.2.a.dt.1.2		2
168.125	even	2		9408.2.a.di.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.a.d.1.1	✓	2	3.2	odd	2
147.2.a.e.1.1	yes	2	21.20	even	2
147.2.e.d.67.2		4	21.5	even	6
147.2.e.d.79.2		4	21.17	even	6
147.2.e.e.67.2		4	21.2	odd	6
147.2.e.e.79.2		4	21.11	odd	6
441.2.a.i.1.2		2	7.6	odd	2
441.2.a.j.1.2		2	1.1	even	1	trivial
441.2.e.f.226.1		4	7.4	even	3
441.2.e.f.361.1		4	7.2	even	3
441.2.e.g.226.1		4	7.3	odd	6
441.2.e.g.361.1		4	7.5	odd	6
2352.2.a.bc.1.1		2	84.83	odd	2
2352.2.a.be.1.2		2	12.11	even	2
2352.2.q.bb.961.1		4	84.11	even	6
2352.2.q.bb.1537.1		4	84.23	even	6
2352.2.q.bd.961.2		4	84.59	odd	6
2352.2.q.bd.1537.2		4	84.47	odd	6
3675.2.a.bd.1.2		2	105.104	even	2
3675.2.a.bf.1.2		2	15.14	odd	2
7056.2.a.cf.1.2		2	28.27	even	2
7056.2.a.cv.1.1		2	4.3	odd	2
9408.2.a.di.1.2		2	168.125	even	2
9408.2.a.dq.1.1		2	24.11	even	2
9408.2.a.dt.1.2		2	168.83	odd	2
9408.2.a.ef.1.1		2	24.5	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} +0.585786 q^{5} +4.41421 q^{8} +O(q^{10})\)	\(q+2.41421 q^{2} +3.82843 q^{4} +0.585786 q^{5} +4.41421 q^{8} +1.41421 q^{10} +2.00000 q^{11} -5.41421 q^{13} +3.00000 q^{16} +6.24264 q^{17} -2.82843 q^{19} +2.24264 q^{20} +4.82843 q^{22} -3.65685 q^{23} -4.65685 q^{25} -13.0711 q^{26} +1.17157 q^{29} +6.82843 q^{31} -1.58579 q^{32} +15.0711 q^{34} -4.00000 q^{37} -6.82843 q^{38} +2.58579 q^{40} -2.24264 q^{41} -5.65685 q^{43} +7.65685 q^{44} -8.82843 q^{46} +2.82843 q^{47} -11.2426 q^{50} -20.7279 q^{52} +2.00000 q^{53} +1.17157 q^{55} +2.82843 q^{58} -6.82843 q^{59} -3.75736 q^{61} +16.4853 q^{62} -9.82843 q^{64} -3.17157 q^{65} +5.65685 q^{67} +23.8995 q^{68} +13.3137 q^{71} +5.89949 q^{73} -9.65685 q^{74} -10.8284 q^{76} +2.34315 q^{79} +1.75736 q^{80} -5.41421 q^{82} -15.3137 q^{83} +3.65685 q^{85} -13.6569 q^{86} +8.82843 q^{88} -5.75736 q^{89} -14.0000 q^{92} +6.82843 q^{94} -1.65685 q^{95} -5.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{11} - 8 q^{13} + 6 q^{16} + 4 q^{17} - 4 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{25} - 12 q^{26} + 8 q^{29} + 8 q^{31} - 6 q^{32} + 16 q^{34} - 8 q^{37} - 8 q^{38} + 8 q^{40} + 4 q^{41} + 4 q^{44} - 12 q^{46} - 14 q^{50} - 16 q^{52} + 4 q^{53} + 8 q^{55} - 8 q^{59} - 16 q^{61} + 16 q^{62} - 14 q^{64} - 12 q^{65} + 28 q^{68} + 4 q^{71} - 8 q^{73} - 8 q^{74} - 16 q^{76} + 16 q^{79} + 12 q^{80} - 8 q^{82} - 8 q^{83} - 4 q^{85} - 16 q^{86} + 12 q^{88} - 20 q^{89} - 28 q^{92} + 8 q^{94} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 + 4 * q^11 - 8 * q^13 + 6 * q^16 + 4 * q^17 - 4 * q^20 + 4 * q^22 + 4 * q^23 + 2 * q^25 - 12 * q^26 + 8 * q^29 + 8 * q^31 - 6 * q^32 + 16 * q^34 - 8 * q^37 - 8 * q^38 + 8 * q^40 + 4 * q^41 + 4 * q^44 - 12 * q^46 - 14 * q^50 - 16 * q^52 + 4 * q^53 + 8 * q^55 - 8 * q^59 - 16 * q^61 + 16 * q^62 - 14 * q^64 - 12 * q^65 + 28 * q^68 + 4 * q^71 - 8 * q^73 - 8 * q^74 - 16 * q^76 + 16 * q^79 + 12 * q^80 - 8 * q^82 - 8 * q^83 - 4 * q^85 - 16 * q^86 + 12 * q^88 - 20 * q^89 - 28 * q^92 + 8 * q^94 + 8 * q^95 - 8 * q^97

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