LMFDB - Embedded newform 3675.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
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:	$\Z[a_1, a_2]$
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	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.41421 q^{6} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.41421 q^{6} +4.41421 q^{8} +1.00000 q^{9} -2.00000 q^{11} -3.82843 q^{12} -5.41421 q^{13} +3.00000 q^{16} -6.24264 q^{17} +2.41421 q^{18} +2.82843 q^{19} -4.82843 q^{22} -3.65685 q^{23} -4.41421 q^{24} -13.0711 q^{26} -1.00000 q^{27} -1.17157 q^{29} -6.82843 q^{31} -1.58579 q^{32} +2.00000 q^{33} -15.0711 q^{34} +3.82843 q^{36} +4.00000 q^{37} +6.82843 q^{38} +5.41421 q^{39} -2.24264 q^{41} +5.65685 q^{43} -7.65685 q^{44} -8.82843 q^{46} -2.82843 q^{47} -3.00000 q^{48} +6.24264 q^{51} -20.7279 q^{52} +2.00000 q^{53} -2.41421 q^{54} -2.82843 q^{57} -2.82843 q^{58} -6.82843 q^{59} +3.75736 q^{61} -16.4853 q^{62} -9.82843 q^{64} +4.82843 q^{66} -5.65685 q^{67} -23.8995 q^{68} +3.65685 q^{69} -13.3137 q^{71} +4.41421 q^{72} +5.89949 q^{73} +9.65685 q^{74} +10.8284 q^{76} +13.0711 q^{78} +2.34315 q^{79} +1.00000 q^{81} -5.41421 q^{82} +15.3137 q^{83} +13.6569 q^{86} +1.17157 q^{87} -8.82843 q^{88} -5.75736 q^{89} -14.0000 q^{92} +6.82843 q^{93} -6.82843 q^{94} +1.58579 q^{96} -5.41421 q^{97} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} - 8 q^{13} + 6 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{22} + 4 q^{23} - 6 q^{24} - 12 q^{26} - 2 q^{27} - 8 q^{29} - 8 q^{31} - 6 q^{32} + 4 q^{33} - 16 q^{34} + 2 q^{36} + 8 q^{37} + 8 q^{38} + 8 q^{39} + 4 q^{41} - 4 q^{44} - 12 q^{46} - 6 q^{48} + 4 q^{51} - 16 q^{52} + 4 q^{53} - 2 q^{54} - 8 q^{59} + 16 q^{61} - 16 q^{62} - 14 q^{64} + 4 q^{66} - 28 q^{68} - 4 q^{69} - 4 q^{71} + 6 q^{72} - 8 q^{73} + 8 q^{74} + 16 q^{76} + 12 q^{78} + 16 q^{79} + 2 q^{81} - 8 q^{82} + 8 q^{83} + 16 q^{86} + 8 q^{87} - 12 q^{88} - 20 q^{89} - 28 q^{92} + 8 q^{93} - 8 q^{94} + 6 q^{96} - 8 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^8 + 2 * q^9 - 4 * q^11 - 2 * q^12 - 8 * q^13 + 6 * q^16 - 4 * q^17 + 2 * q^18 - 4 * q^22 + 4 * q^23 - 6 * q^24 - 12 * q^26 - 2 * q^27 - 8 * q^29 - 8 * q^31 - 6 * q^32 + 4 * q^33 - 16 * q^34 + 2 * q^36 + 8 * q^37 + 8 * q^38 + 8 * q^39 + 4 * q^41 - 4 * q^44 - 12 * q^46 - 6 * q^48 + 4 * q^51 - 16 * q^52 + 4 * q^53 - 2 * q^54 - 8 * q^59 + 16 * q^61 - 16 * q^62 - 14 * q^64 + 4 * q^66 - 28 * q^68 - 4 * q^69 - 4 * q^71 + 6 * q^72 - 8 * q^73 + 8 * q^74 + 16 * q^76 + 12 * q^78 + 16 * q^79 + 2 * q^81 - 8 * q^82 + 8 * q^83 + 16 * q^86 + 8 * q^87 - 12 * q^88 - 20 * q^89 - 28 * q^92 + 8 * q^93 - 8 * q^94 + 6 * q^96 - 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	−2.41421	−0.985599
$7$	0	0
$7$	0	0
$8$	4.41421	1.56066
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−3.82843	−1.10517
$13$	−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	$-0.773351\pi$
$17$	−6.24264	−1.51406	−0.757031	+	0.653379i	$0.773351\pi$
$18$	2.41421	0.569036
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	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	$0.624507\pi$
$24$	−4.41421	−0.901048
$25$	0	0
$26$	−13.0711	−2.56345
$27$	−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	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	−1.58579	−0.280330
$33$	2.00000	0.348155
$34$	−15.0711	−2.58467
$35$	0	0
$36$	3.82843	0.638071
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	6.82843	1.10772
$39$	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	$0.358044\pi$
$44$	−7.65685	−1.15431
$45$	0	0
$46$	−8.82843	−1.30168
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	−3.00000	−0.433013
$49$	0	0
$50$	0	0
$51$	6.24264	0.874145
$52$	−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	$0.456138\pi$
$54$	−2.41421	−0.328533
$55$	0	0
$56$	0	0
$57$	−2.82843	−0.374634
$58$	−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	$-0.422675\pi$
$61$	3.75736	0.481081	0.240540	+	0.970639i	$0.422675\pi$
$62$	−16.4853	−2.09363
$63$	0	0
$64$	−9.82843	−1.22855
$65$	0	0
$66$	4.82843	0.594338
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	−23.8995	−2.89824
$69$	3.65685	0.440234
$70$	0	0
$71$	−13.3137	−1.58005	−0.790023	−	0.613077i	$-0.789932\pi$
$71$	−13.3137	−1.58005	−0.790023	+	0.613077i	$0.789932\pi$
$72$	4.41421	0.520220
$73$	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
$78$	13.0711	1.48001
$79$	2.34315	0.263624	0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.41421	−0.597900
$83$	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
$85$	0	0
$86$	13.6569	1.47266
$87$	1.17157	0.125606
$88$	−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
$93$	6.82843	0.708075
$94$	−6.82843	−0.704298
$95$	0	0
$96$	1.58579	0.161849
$97$	−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
$99$	−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	$0.177014\pi$
$102$	15.0711	1.49226
$103$	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	$0.309472\pi$
$108$	−3.82843	−0.368391
$109$	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	$-0.802916\pi$
$113$	−17.3137	−1.62874	−0.814368	+	0.580348i	$0.802916\pi$
$114$	−6.82843	−0.639541
$115$	0	0
$116$	−4.48528	−0.416448
$117$	−5.41421	−0.500544
$118$	−16.4853	−1.51759
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	9.07107	0.821256
$123$	2.24264	0.202212
$124$	−26.1421	−2.34763
$125$	0	0
$126$	0	0
$127$	−9.65685	−0.856907	−0.428454	−	0.903564i	$-0.640941\pi$
$127$	−9.65685	−0.856907	−0.428454	+	0.903564i	$0.640941\pi$
$128$	−20.5563	−1.81694
$129$	−5.65685	−0.498058
$130$	0	0
$131$	7.31371	0.639002	0.319501	−	0.947586i	$-0.396485\pi$
$131$	7.31371	0.639002	0.319501	+	0.947586i	$0.396485\pi$
$132$	7.65685	0.666444
$133$	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	$0.293524\pi$
$138$	8.82843	0.751526
$139$	−6.34315	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$140$	0	0
$141$	2.82843	0.238197
$142$	−32.1421	−2.69731
$143$	10.8284	0.905519
$144$	3.00000	0.250000
$145$	0	0
$146$	14.2426	1.17873
$147$	0	0
$148$	15.3137	1.25878
$149$	−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	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	12.4853	1.01269
$153$	−6.24264	−0.504688
$154$	0	0
$155$	0	0
$156$	20.7279	1.65956
$157$	−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
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	2.41421	0.189679
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	−8.58579	−0.670437
$165$	0	0
$166$	36.9706	2.86947
$167$	−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
$172$	21.6569	1.65132
$173$	−6.92893	−0.526797	−0.263398	−	0.964687i	$-0.584843\pi$
$173$	−6.92893	−0.526797	−0.263398	+	0.964687i	$0.584843\pi$
$174$	2.82843	0.214423
$175$	0	0
$176$	−6.00000	−0.452267
$177$	6.82843	0.513256
$178$	−13.8995	−1.04181
$179$	−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
$183$	−3.75736	−0.277752
$184$	−16.1421	−1.19001
$185$	0	0
$186$	16.4853	1.20876
$187$	12.4853	0.913014
$188$	−10.8284	−0.789744
$189$	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	$0.725741\pi$
$192$	9.82843	0.709306
$193$	17.3137	1.24627	0.623134	−	0.782115i	$-0.285859\pi$
$193$	17.3137	1.24627	0.623134	+	0.782115i	$0.285859\pi$
$194$	−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	$0.522698\pi$
$198$	−4.82843	−0.343141
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	0	0
$201$	5.65685	0.399004
$202$	41.2132	2.89975
$203$	0	0
$204$	23.8995	1.67330
$205$	0	0
$206$	30.1421	2.10010
$207$	−3.65685	−0.254169
$208$	−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
$213$	13.3137	0.912240
$214$	28.1421	1.92376
$215$	0	0
$216$	−4.41421	−0.300349
$217$	0	0
$218$	13.6569	0.924959
$219$	−5.89949	−0.398651
$220$	0	0
$221$	33.7990	2.27357
$222$	−9.65685	−0.648126
$223$	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	$-0.324324\pi$
$227$	15.7990	1.04862	0.524308	+	0.851529i	$0.324324\pi$
$228$	−10.8284	−0.717130
$229$	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
$232$	−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	$0.758294\pi$
$234$	−13.0711	−0.854482
$235$	0	0
$236$	−26.1421	−1.70171
$237$	−2.34315	−0.152204
$238$	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	$0.580381\pi$
$242$	−16.8995	−1.08634
$243$	−1.00000	−0.0641500
$244$	14.3848	0.920891
$245$	0	0
$246$	5.41421	0.345198
$247$	−15.3137	−0.974388
$248$	−30.1421	−1.91403
$249$	−15.3137	−0.970467
$250$	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
$254$	−23.3137	−1.46283
$255$	0	0
$256$	−29.9706	−1.87316
$257$	19.2132	1.19849	0.599243	−	0.800567i	$-0.295468\pi$
$257$	19.2132	1.19849	0.599243	+	0.800567i	$0.295468\pi$
$258$	−13.6569	−0.850239
$259$	0	0
$260$	0	0
$261$	−1.17157	−0.0725185
$262$	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	$0.320762\pi$
$264$	8.82843	0.543352
$265$	0	0
$266$	0	0
$267$	5.75736	0.352345
$268$	−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	$-0.314236\pi$
$271$	18.1421	1.10206	0.551028	+	0.834487i	$0.314236\pi$
$272$	−18.7279	−1.13555
$273$	0	0
$274$	34.1421	2.06260
$275$	0	0
$276$	14.0000	0.842701
$277$	−13.3137	−0.799943	−0.399972	−	0.916528i	$-0.630980\pi$
$277$	−13.3137	−0.799943	−0.399972	+	0.916528i	$0.630980\pi$
$278$	−15.3137	−0.918455
$279$	−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	$0.663630\pi$
$282$	6.82843	0.406627
$283$	−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
$288$	−1.58579	−0.0934434
$289$	21.9706	1.29239
$290$	0	0
$291$	5.41421	0.317387
$292$	22.5858	1.32173
$293$	−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
$295$	0	0
$296$	17.6569	1.02628
$297$	2.00000	0.116052
$298$	−12.8284	−0.743131
$299$	19.7990	1.14501
$300$	0	0
$301$	0	0
$302$	28.9706	1.66707
$303$	−17.0711	−0.980707
$304$	8.48528	0.486664
$305$	0	0
$306$	−15.0711	−0.861556
$307$	−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	$-0.716037\pi$
$311$	−22.1421	−1.25557	−0.627783	+	0.778389i	$0.716037\pi$
$312$	23.8995	1.35304
$313$	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	$0.590609\pi$
$318$	−4.82843	−0.270765
$319$	2.34315	0.131191
$320$	0	0
$321$	−11.6569	−0.650622
$322$	0	0
$323$	−17.6569	−0.982454
$324$	3.82843	0.212690
$325$	0	0
$326$	−27.3137	−1.51277
$327$	−5.65685	−0.312825
$328$	−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
$333$	4.00000	0.219199
$334$	−47.7990	−2.61544
$335$	0	0
$336$	0	0
$337$	18.3431	0.999215	0.499607	−	0.866252i	$-0.333478\pi$
$337$	18.3431	0.999215	0.499607	+	0.866252i	$0.333478\pi$
$338$	39.3848	2.14225
$339$	17.3137	0.940352
$340$	0	0
$341$	13.6569	0.739560
$342$	6.82843	0.369239
$343$	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	$0.592603\pi$
$348$	4.48528	0.240436
$349$	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
$352$	3.17157	0.169045
$353$	−10.7279	−0.570990	−0.285495	−	0.958380i	$-0.592158\pi$
$353$	−10.7279	−0.570990	−0.285495	+	0.958380i	$0.592158\pi$
$354$	16.4853	0.876183
$355$	0	0
$356$	−22.0416	−1.16820
$357$	0	0
$358$	−20.1421	−1.06454
$359$	−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
$362$	−13.0711	−0.687000
$363$	7.00000	0.367405
$364$	0	0
$365$	0	0
$366$	−9.07107	−0.474152
$367$	−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
$369$	−2.24264	−0.116747
$370$	0	0
$371$	0	0
$372$	26.1421	1.35541
$373$	33.3137	1.72492	0.862459	−	0.506127i	$-0.168923\pi$
$373$	33.3137	1.72492	0.862459	+	0.506127i	$0.168923\pi$
$374$	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	$0.202573\pi$
$380$	0	0
$381$	9.65685	0.494736
$382$	−43.4558	−2.22339
$383$	29.6569	1.51539	0.757697	−	0.652606i	$-0.226324\pi$
$383$	29.6569	1.51539	0.757697	+	0.652606i	$0.226324\pi$
$384$	20.5563	1.04901
$385$	0	0
$386$	41.7990	2.12751
$387$	5.65685	0.287554
$388$	−20.7279	−1.05230
$389$	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
$394$	−4.82843	−0.243253
$395$	0	0
$396$	−7.65685	−0.384771
$397$	−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
$400$	0	0
$401$	22.1421	1.10573	0.552863	−	0.833272i	$-0.313535\pi$
$401$	22.1421	1.10573	0.552863	+	0.833272i	$0.313535\pi$
$402$	13.6569	0.681142
$403$	36.9706	1.84163
$404$	65.3553	3.25155
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	27.5563	1.36424
$409$	−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
$412$	47.7990	2.35489
$413$	0	0
$414$	−8.82843	−0.433894
$415$	0	0
$416$	8.58579	0.420953
$417$	6.34315	0.310625
$418$	−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
$423$	−2.82843	−0.137523
$424$	8.82843	0.428746
$425$	0	0
$426$	32.1421	1.55729
$427$	0	0
$428$	44.6274	2.15715
$429$	−10.8284	−0.522801
$430$	0	0
$431$	6.97056	0.335760	0.167880	−	0.985807i	$-0.446308\pi$
$431$	6.97056	0.335760	0.167880	+	0.985807i	$0.446308\pi$
$432$	−3.00000	−0.144338
$433$	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
$438$	−14.2426	−0.680540
$439$	35.3137	1.68543	0.842716	−	0.538359i	$-0.180956\pi$
$439$	35.3137	1.68543	0.842716	+	0.538359i	$0.180956\pi$
$440$	0	0
$441$	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	$0.507785\pi$
$444$	−15.3137	−0.726756
$445$	0	0
$446$	21.6569	1.02548
$447$	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
$452$	−66.2843	−3.11775
$453$	−12.0000	−0.563809
$454$	38.1421	1.79010
$455$	0	0
$456$	−12.4853	−0.584677
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	19.8995	0.929842
$459$	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	$0.642490\pi$
$464$	−3.51472	−0.163167
$465$	0	0
$466$	−53.4558	−2.47629
$467$	−39.7990	−1.84168	−0.920839	−	0.389943i	$-0.872495\pi$
$467$	−39.7990	−1.84168	−0.920839	+	0.389943i	$0.872495\pi$
$468$	−20.7279	−0.958149
$469$	0	0
$470$	0	0
$471$	20.2426	0.932732
$472$	−30.1421	−1.38740
$473$	−11.3137	−0.520205
$474$	−5.65685	−0.259828
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	−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
$485$	0	0
$486$	−2.41421	−0.109511
$487$	18.6274	0.844089	0.422044	−	0.906575i	$-0.361313\pi$
$487$	18.6274	0.844089	0.422044	+	0.906575i	$0.361313\pi$
$488$	16.5858	0.750803
$489$	11.3137	0.511624
$490$	0	0
$491$	38.9706	1.75872	0.879358	−	0.476160i	$-0.157972\pi$
$491$	38.9706	1.75872	0.879358	+	0.476160i	$0.157972\pi$
$492$	8.58579	0.387077
$493$	7.31371	0.329393
$494$	−36.9706	−1.66338
$495$	0	0
$496$	−20.4853	−0.919816
$497$	0	0
$498$	−36.9706	−1.65669
$499$	−19.3137	−0.864600	−0.432300	−	0.901730i	$-0.642298\pi$
$499$	−19.3137	−0.864600	−0.432300	+	0.901730i	$0.642298\pi$
$500$	0	0
$501$	19.7990	0.884554
$502$	10.8284	0.483296
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	17.6569	0.784943
$507$	−16.3137	−0.724517
$508$	−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
$513$	−2.82843	−0.124878
$514$	46.3848	2.04594
$515$	0	0
$516$	−21.6569	−0.953390
$517$	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	$0.753030\pi$
$522$	−2.82843	−0.123797
$523$	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
$528$	6.00000	0.261116
$529$	−9.62742	−0.418583
$530$	0	0
$531$	−6.82843	−0.296328
$532$	0	0
$533$	12.1421	0.525934
$534$	13.8995	0.601490
$535$	0	0
$536$	−24.9706	−1.07856
$537$	8.34315	0.360033
$538$	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
$543$	5.41421	0.232346
$544$	9.89949	0.424437
$545$	0	0
$546$	0	0
$547$	3.02944	0.129529	0.0647647	−	0.997901i	$-0.479370\pi$
$547$	3.02944	0.129529	0.0647647	+	0.997901i	$0.479370\pi$
$548$	54.1421	2.31284
$549$	3.75736	0.160360
$550$	0	0
$551$	−3.31371	−0.141169
$552$	16.1421	0.687055
$553$	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	$0.685688\pi$
$558$	−16.4853	−0.697878
$559$	−30.6274	−1.29540
$560$	0	0
$561$	−12.4853	−0.527129
$562$	−39.7990	−1.67882
$563$	6.82843	0.287784	0.143892	−	0.989593i	$-0.454038\pi$
$563$	6.82843	0.287784	0.143892	+	0.989593i	$0.454038\pi$
$564$	10.8284	0.455959
$565$	0	0
$566$	−20.4853	−0.861061
$567$	0	0
$568$	−58.7696	−2.46592
$569$	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	$-0.251284\pi$
$571$	33.6569	1.40850	0.704248	+	0.709954i	$0.251284\pi$
$572$	41.4558	1.73336
$573$	18.0000	0.751961
$574$	0	0
$575$	0	0
$576$	−9.82843	−0.409518
$577$	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
$579$	−17.3137	−0.719533
$580$	0	0
$581$	0	0
$582$	13.0711	0.541813
$583$	−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	$-0.384694\pi$
$587$	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
$592$	12.0000	0.493197
$593$	21.0711	0.865285	0.432643	−	0.901566i	$-0.357581\pi$
$593$	21.0711	0.865285	0.432643	+	0.901566i	$0.357581\pi$
$594$	4.82843	0.198113
$595$	0	0
$596$	−20.3431	−0.833288
$597$	−10.3431	−0.423317
$598$	47.7990	1.95465
$599$	−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
$603$	−5.65685	−0.230365
$604$	45.9411	1.86932
$605$	0	0
$606$	−41.2132	−1.67417
$607$	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
$610$	0	0
$611$	15.3137	0.619526
$612$	−23.8995	−0.966080
$613$	−27.3137	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$613$	−27.3137	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$614$	−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	$0.451665\pi$
$618$	−30.1421	−1.21249
$619$	−4.97056	−0.199784	−0.0998919	−	0.994998i	$-0.531850\pi$
$619$	−4.97056	−0.199784	−0.0998919	+	0.994998i	$0.531850\pi$
$620$	0	0
$621$	3.65685	0.146745
$622$	−53.4558	−2.14338
$623$	0	0
$624$	16.2426	0.650226
$625$	0	0
$626$	43.2132	1.72715
$627$	5.65685	0.225913
$628$	−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
$633$	20.9706	0.833505
$634$	−24.1421	−0.958807
$635$	0	0
$636$	−7.65685	−0.303614
$637$	0	0
$638$	5.65685	0.223957
$639$	−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	$0.532567\pi$
$642$	−28.1421	−1.11068
$643$	−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	$-0.636628\pi$
$647$	−21.1716	−0.832340	−0.416170	+	0.909287i	$0.636628\pi$
$648$	4.41421	0.173407
$649$	13.6569	0.536078
$650$	0	0
$651$	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	$0.624708\pi$
$654$	−13.6569	−0.534025
$655$	0	0
$656$	−6.72792	−0.262681
$657$	5.89949	0.230161
$658$	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	$0.453053\pi$
$662$	−9.65685	−0.375324
$663$	−33.7990	−1.31264
$664$	67.5980	2.62331
$665$	0	0
$666$	9.65685	0.374196
$667$	4.28427	0.165888
$668$	−75.7990	−2.93275
$669$	−8.97056	−0.346822
$670$	0	0
$671$	−7.51472	−0.290102
$672$	0	0
$673$	−0.686292	−0.0264546	−0.0132273	−	0.999913i	$-0.504211\pi$
$673$	−0.686292	−0.0264546	−0.0132273	+	0.999913i	$0.504211\pi$
$674$	44.2843	1.70577
$675$	0	0
$676$	62.4558	2.40215
$677$	−28.5858	−1.09864	−0.549321	−	0.835612i	$-0.685113\pi$
$677$	−28.5858	−1.09864	−0.549321	+	0.835612i	$0.685113\pi$
$678$	41.7990	1.60528
$679$	0	0
$680$	0	0
$681$	−15.7990	−0.605419
$682$	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	$0.448973\pi$
$684$	10.8284	0.414035
$685$	0	0
$686$	0	0
$687$	−8.24264	−0.314476
$688$	16.9706	0.646997
$689$	−10.8284	−0.412530
$690$	0	0
$691$	−23.3137	−0.886895	−0.443448	−	0.896300i	$-0.646245\pi$
$691$	−23.3137	−0.886895	−0.443448	+	0.896300i	$0.646245\pi$
$692$	−26.5269	−1.00840
$693$	0	0
$694$	−25.7990	−0.979316
$695$	0	0
$696$	5.17157	0.196028
$697$	14.0000	0.530288
$698$	23.8995	0.904609
$699$	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	$0.641878\pi$
$702$	13.0711	0.493336
$703$	11.3137	0.426705
$704$	19.6569	0.740846
$705$	0	0
$706$	−25.8995	−0.974740
$707$	0	0
$708$	26.1421	0.982482
$709$	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
$712$	−25.4142	−0.952438
$713$	24.9706	0.935155
$714$	0	0
$715$	0	0
$716$	−31.9411	−1.19370
$717$	4.34315	0.162198
$718$	−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
$723$	7.75736	0.288499
$724$	−20.7279	−0.770347
$725$	0	0
$726$	16.8995	0.627199
$727$	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
$732$	−14.3848	−0.531677
$733$	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
$738$	−5.41421	−0.199300
$739$	27.3137	1.00475	0.502376	−	0.864650i	$-0.332460\pi$
$739$	27.3137	1.00475	0.502376	+	0.864650i	$0.332460\pi$
$740$	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	$0.398876\pi$
$744$	30.1421	1.10506
$745$	0	0
$746$	80.4264	2.94462
$747$	15.3137	0.560299
$748$	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
$753$	−4.48528	−0.163453
$754$	15.3137	0.557692
$755$	0	0
$756$	0	0
$757$	−37.6569	−1.36866	−0.684331	−	0.729172i	$-0.739906\pi$
$757$	−37.6569	−1.36866	−0.684331	+	0.729172i	$0.739906\pi$
$758$	75.5980	2.74584
$759$	−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	$0.180610\pi$
$762$	23.3137	0.844567
$763$	0	0
$764$	−68.9117	−2.49314
$765$	0	0
$766$	71.5980	2.58694
$767$	36.9706	1.33493
$768$	29.9706	1.08147
$769$	−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
$772$	66.2843	2.38562
$773$	−21.5563	−0.775328	−0.387664	−	0.921801i	$-0.626718\pi$
$773$	−21.5563	−0.775328	−0.387664	+	0.921801i	$0.626718\pi$
$774$	13.6569	0.490885
$775$	0	0
$776$	−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
$783$	1.17157	0.0418686
$784$	0	0
$785$	0	0
$786$	−17.6569	−0.629799
$787$	−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
$789$	−17.3137	−0.616384
$790$	0	0
$791$	0	0
$792$	−8.82843	−0.313704
$793$	−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	$-0.667667\pi$
$797$	−28.3848	−1.00544	−0.502720	+	0.864449i	$0.667667\pi$
$798$	0	0
$799$	17.6569	0.624655
$800$	0	0
$801$	−5.75736	−0.203426
$802$	53.4558	1.88759
$803$	−11.7990	−0.416377
$804$	21.6569	0.763778
$805$	0	0
$806$	89.2548	3.14387
$807$	−10.7279	−0.377641
$808$	75.3553	2.65099
$809$	−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	$-0.535524\pi$
$811$	−6.34315	−0.222738	−0.111369	+	0.993779i	$0.535524\pi$
$812$	0	0
$813$	−18.1421	−0.636272
$814$	−19.3137	−0.676945
$815$	0	0
$816$	18.7279	0.655608
$817$	16.0000	0.559769
$818$	−44.8701	−1.56884
$819$	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	$0.697467\pi$
$822$	−34.1421	−1.19084
$823$	24.9706	0.870419	0.435210	−	0.900329i	$-0.356674\pi$
$823$	24.9706	0.870419	0.435210	+	0.900329i	$0.356674\pi$
$824$	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	$0.717720\pi$
$828$	−14.0000	−0.486534
$829$	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
$832$	53.2132	1.84484
$833$	0	0
$834$	15.3137	0.530270
$835$	0	0
$836$	−21.6569	−0.749018
$837$	6.82843	0.236025
$838$	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
$843$	16.4853	0.567783
$844$	−80.2843	−2.76350
$845$	0	0
$846$	−6.82843	−0.234766
$847$	0	0
$848$	6.00000	0.206041
$849$	8.48528	0.291214
$850$	0	0
$851$	−14.6274	−0.501421
$852$	50.9706	1.74622
$853$	−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	$-0.568962\pi$
$857$	−12.5858	−0.429922	−0.214961	+	0.976623i	$0.568962\pi$
$858$	−26.1421	−0.892478
$859$	6.54416	0.223284	0.111642	−	0.993749i	$-0.464389\pi$
$859$	6.54416	0.223284	0.111642	+	0.993749i	$0.464389\pi$
$860$	0	0
$861$	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	$0.471173\pi$
$864$	1.58579	0.0539496
$865$	0	0
$866$	28.3848	0.964554
$867$	−21.9706	−0.746159
$868$	0	0
$869$	−4.68629	−0.158972
$870$	0	0
$871$	30.6274	1.03777
$872$	24.9706	0.845610
$873$	−5.41421	−0.183243
$874$	−24.9706	−0.844642
$875$	0	0
$876$	−22.5858	−0.763103
$877$	−11.3137	−0.382037	−0.191018	−	0.981586i	$-0.561179\pi$
$877$	−11.3137	−0.382037	−0.191018	+	0.981586i	$0.561179\pi$
$878$	85.2548	2.87721
$879$	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	$0.347996\pi$
$884$	129.397	4.35209
$885$	0	0
$886$	−2.48528	−0.0834947
$887$	−2.82843	−0.0949693	−0.0474846	−	0.998872i	$-0.515121\pi$
$887$	−2.82843	−0.0949693	−0.0474846	+	0.998872i	$0.515121\pi$
$888$	−17.6569	−0.592525
$889$	0	0
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	34.3431	1.14989
$893$	−8.00000	−0.267710
$894$	12.8284	0.429047
$895$	0	0
$896$	0	0
$897$	−19.7990	−0.661069
$898$	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
$905$	0	0
$906$	−28.9706	−0.962482
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	60.4853	2.00727
$909$	17.0711	0.566212
$910$	0	0
$911$	−34.9706	−1.15863	−0.579313	−	0.815105i	$-0.696679\pi$
$911$	−34.9706	−1.15863	−0.579313	+	0.815105i	$0.696679\pi$
$912$	−8.48528	−0.280976
$913$	−30.6274	−1.01362
$914$	43.4558	1.43739
$915$	0	0
$916$	31.5563	1.04265
$917$	0	0
$918$	15.0711	0.497419
$919$	48.2843	1.59275	0.796376	−	0.604802i	$-0.206748\pi$
$919$	48.2843	1.59275	0.796376	+	0.604802i	$0.206748\pi$
$920$	0	0
$921$	1.85786	0.0612187
$922$	−46.8701	−1.54358
$923$	72.0833	2.37265
$924$	0	0
$925$	0	0
$926$	−44.9706	−1.47782
$927$	12.4853	0.410070
$928$	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
$933$	22.1421	0.724901
$934$	−96.0833	−3.14394
$935$	0	0
$936$	−23.8995	−0.781179
$937$	−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
$939$	−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	$0.537511\pi$
$942$	48.8701	1.59227
$943$	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	$0.833465\pi$
$948$	−8.97056	−0.291350
$949$	−31.9411	−1.03685
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	4.82843	0.156326
$955$	0	0
$956$	−16.6274	−0.537769
$957$	−2.34315	−0.0757431
$958$	72.7696	2.35108
$959$	0	0
$960$	0	0
$961$	15.6274	0.504110
$962$	−52.2843	−1.68571
$963$	11.6569	0.375637
$964$	−29.6985	−0.956524
$965$	0	0
$966$	0	0
$967$	−22.3431	−0.718507	−0.359254	−	0.933240i	$-0.616969\pi$
$967$	−22.3431	−0.718507	−0.359254	+	0.933240i	$0.616969\pi$
$968$	−30.8995	−0.993147
$969$	17.6569	0.567220
$970$	0	0
$971$	−5.37258	−0.172414	−0.0862072	−	0.996277i	$-0.527475\pi$
$971$	−5.37258	−0.172414	−0.0862072	+	0.996277i	$0.527475\pi$
$972$	−3.82843	−0.122797
$973$	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	$0.358810\pi$
$978$	27.3137	0.873396
$979$	11.5147	0.368012
$980$	0	0
$981$	5.65685	0.180609
$982$	94.0833	3.00232
$983$	−37.2548	−1.18824	−0.594122	−	0.804375i	$-0.702501\pi$
$983$	−37.2548	−1.18824	−0.594122	+	0.804375i	$0.702501\pi$
$984$	9.89949	0.315584
$985$	0	0
$986$	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
$993$	4.00000	0.126936
$994$	0	0
$995$	0	0
$996$	−58.6274	−1.85768
$997$	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
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bd.1.2		2
5.4	even	2		147.2.a.e.1.1	yes	2
7.6	odd	2		3675.2.a.bf.1.2		2
15.14	odd	2		441.2.a.i.1.2		2
20.19	odd	2		2352.2.a.bc.1.1		2
35.4	even	6		147.2.e.d.79.2		4
35.9	even	6		147.2.e.d.67.2		4
35.19	odd	6		147.2.e.e.67.2		4
35.24	odd	6		147.2.e.e.79.2		4
35.34	odd	2		147.2.a.d.1.1	✓	2
40.19	odd	2		9408.2.a.dt.1.2		2
40.29	even	2		9408.2.a.di.1.2		2
60.59	even	2		7056.2.a.cf.1.2		2
105.44	odd	6		441.2.e.g.361.1		4
105.59	even	6		441.2.e.f.226.1		4
105.74	odd	6		441.2.e.g.226.1		4
105.89	even	6		441.2.e.f.361.1		4
105.104	even	2		441.2.a.j.1.2		2
140.19	even	6		2352.2.q.bb.1537.1		4
140.39	odd	6		2352.2.q.bd.961.2		4
140.59	even	6		2352.2.q.bb.961.1		4
140.79	odd	6		2352.2.q.bd.1537.2		4
140.139	even	2		2352.2.a.be.1.2		2
280.69	odd	2		9408.2.a.ef.1.1		2
280.139	even	2		9408.2.a.dq.1.1		2
420.419	odd	2		7056.2.a.cv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.a.d.1.1	✓	2	35.34	odd	2
147.2.a.e.1.1	yes	2	5.4	even	2
147.2.e.d.67.2		4	35.9	even	6
147.2.e.d.79.2		4	35.4	even	6
147.2.e.e.67.2		4	35.19	odd	6
147.2.e.e.79.2		4	35.24	odd	6
441.2.a.i.1.2		2	15.14	odd	2
441.2.a.j.1.2		2	105.104	even	2
441.2.e.f.226.1		4	105.59	even	6
441.2.e.f.361.1		4	105.89	even	6
441.2.e.g.226.1		4	105.74	odd	6
441.2.e.g.361.1		4	105.44	odd	6
2352.2.a.bc.1.1		2	20.19	odd	2
2352.2.a.be.1.2		2	140.139	even	2
2352.2.q.bb.961.1		4	140.59	even	6
2352.2.q.bb.1537.1		4	140.19	even	6
2352.2.q.bd.961.2		4	140.39	odd	6
2352.2.q.bd.1537.2		4	140.79	odd	6
3675.2.a.bd.1.2		2	1.1	even	1	trivial
3675.2.a.bf.1.2		2	7.6	odd	2
7056.2.a.cf.1.2		2	60.59	even	2
7056.2.a.cv.1.1		2	420.419	odd	2
9408.2.a.di.1.2		2	40.29	even	2
9408.2.a.dq.1.1		2	280.139	even	2
9408.2.a.dt.1.2		2	40.19	odd	2
9408.2.a.ef.1.1		2	280.69	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:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.41421 q^{6} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.41421 q^{6} +4.41421 q^{8} +1.00000 q^{9} -2.00000 q^{11} -3.82843 q^{12} -5.41421 q^{13} +3.00000 q^{16} -6.24264 q^{17} +2.41421 q^{18} +2.82843 q^{19} -4.82843 q^{22} -3.65685 q^{23} -4.41421 q^{24} -13.0711 q^{26} -1.00000 q^{27} -1.17157 q^{29} -6.82843 q^{31} -1.58579 q^{32} +2.00000 q^{33} -15.0711 q^{34} +3.82843 q^{36} +4.00000 q^{37} +6.82843 q^{38} +5.41421 q^{39} -2.24264 q^{41} +5.65685 q^{43} -7.65685 q^{44} -8.82843 q^{46} -2.82843 q^{47} -3.00000 q^{48} +6.24264 q^{51} -20.7279 q^{52} +2.00000 q^{53} -2.41421 q^{54} -2.82843 q^{57} -2.82843 q^{58} -6.82843 q^{59} +3.75736 q^{61} -16.4853 q^{62} -9.82843 q^{64} +4.82843 q^{66} -5.65685 q^{67} -23.8995 q^{68} +3.65685 q^{69} -13.3137 q^{71} +4.41421 q^{72} +5.89949 q^{73} +9.65685 q^{74} +10.8284 q^{76} +13.0711 q^{78} +2.34315 q^{79} +1.00000 q^{81} -5.41421 q^{82} +15.3137 q^{83} +13.6569 q^{86} +1.17157 q^{87} -8.82843 q^{88} -5.75736 q^{89} -14.0000 q^{92} +6.82843 q^{93} -6.82843 q^{94} +1.58579 q^{96} -5.41421 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} - 8 q^{13} + 6 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{22} + 4 q^{23} - 6 q^{24} - 12 q^{26} - 2 q^{27} - 8 q^{29} - 8 q^{31} - 6 q^{32} + 4 q^{33} - 16 q^{34} + 2 q^{36} + 8 q^{37} + 8 q^{38} + 8 q^{39} + 4 q^{41} - 4 q^{44} - 12 q^{46} - 6 q^{48} + 4 q^{51} - 16 q^{52} + 4 q^{53} - 2 q^{54} - 8 q^{59} + 16 q^{61} - 16 q^{62} - 14 q^{64} + 4 q^{66} - 28 q^{68} - 4 q^{69} - 4 q^{71} + 6 q^{72} - 8 q^{73} + 8 q^{74} + 16 q^{76} + 12 q^{78} + 16 q^{79} + 2 q^{81} - 8 q^{82} + 8 q^{83} + 16 q^{86} + 8 q^{87} - 12 q^{88} - 20 q^{89} - 28 q^{92} + 8 q^{93} - 8 q^{94} + 6 q^{96} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^8 + 2 * q^9 - 4 * q^11 - 2 * q^12 - 8 * q^13 + 6 * q^16 - 4 * q^17 + 2 * q^18 - 4 * q^22 + 4 * q^23 - 6 * q^24 - 12 * q^26 - 2 * q^27 - 8 * q^29 - 8 * q^31 - 6 * q^32 + 4 * q^33 - 16 * q^34 + 2 * q^36 + 8 * q^37 + 8 * q^38 + 8 * q^39 + 4 * q^41 - 4 * q^44 - 12 * q^46 - 6 * q^48 + 4 * q^51 - 16 * q^52 + 4 * q^53 - 2 * q^54 - 8 * q^59 + 16 * q^61 - 16 * q^62 - 14 * q^64 + 4 * q^66 - 28 * q^68 - 4 * q^69 - 4 * q^71 + 6 * q^72 - 8 * q^73 + 8 * q^74 + 16 * q^76 + 12 * q^78 + 16 * q^79 + 2 * q^81 - 8 * q^82 + 8 * q^83 + 16 * q^86 + 8 * q^87 - 12 * q^88 - 20 * q^89 - 28 * q^92 + 8 * q^93 - 8 * q^94 + 6 * q^96 - 8 * q^97 - 4 * q^99

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