LMFDB - Embedded newform 9464.2.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,1,0,0,0,-4,0,9,0,-1,0,0,0,4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.183064.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 6x + 8 $ x^4 - x^3 - 10x^2 + 6x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.669601$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.669601 q^{3} +4.30411 q^{5} -1.00000 q^{7} -2.55163 q^{9} +0.669601 q^{11} -2.88203 q^{15} +6.22124 q^{17} -6.30411 q^{19} +0.669601 q^{21} -5.63451 q^{23} +13.5253 q^{25} +3.71738 q^{27} -1.42207 q^{29} -4.09167 q^{31} -0.448365 q^{33} -4.30411 q^{35} -2.66960 q^{37} -7.05658 q^{41} -10.5253 q^{43} -10.9825 q^{45} -4.09167 q^{47} +1.00000 q^{49} -4.16574 q^{51} +5.42207 q^{53} +2.88203 q^{55} +4.22124 q^{57} +7.26901 q^{59} +2.28262 q^{61} +2.55163 q^{63} -2.21243 q^{67} +3.77287 q^{69} -5.56044 q^{71} -11.8557 q^{73} -9.05658 q^{75} -0.669601 q^{77} +0.312910 q^{79} +5.16574 q^{81} +3.03509 q^{83} +26.7769 q^{85} +0.952222 q^{87} -4.12076 q^{89} +2.73979 q^{93} -27.1336 q^{95} -7.43088 q^{97} -1.70858 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 4 q^{7} + 9 q^{9} - q^{11} + 4 q^{15} + 2 q^{17} - 8 q^{19} - q^{21} - 9 q^{23} + 14 q^{25} + 7 q^{27} - 4 q^{29} - 11 q^{31} - 21 q^{33} - 7 q^{37} - 13 q^{41} - 2 q^{43} - 12 q^{45} - 11 q^{47}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q + q^3 - 4 * q^7 + 9 * q^9 - q^11 + 4 * q^15 + 2 * q^17 - 8 * q^19 - q^21 - 9 * q^23 + 14 * q^25 + 7 * q^27 - 4 * q^29 - 11 * q^31 - 21 * q^33 - 7 * q^37 - 13 * q^41 - 2 * q^43 - 12 * q^45 - 11 * q^47 + 4 * q^49 - 28 * q^51 + 20 * q^53 - 4 * q^55 - 6 * q^57 + 2 * q^59 + 17 * q^61 - 9 * q^63 + 3 * q^67 - 27 * q^69 + 8 * q^71 - 11 * q^73 - 21 * q^75 + q^77 - 27 * q^79 + 32 * q^81 + 22 * q^83 + 8 * q^85 + 8 * q^87 - 10 * q^89 + 27 * q^93 - 34 * q^95 - 17 * q^97 - 10 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.669601	−0.386594	−0.193297	−	0.981140i	$-0.561918\pi$
$3$	−0.669601	−0.386594	−0.193297	+	0.981140i	$0.561918\pi$
$4$	0	0
$5$	4.30411	1.92486	0.962428	−	0.271538i	$-0.0875322\pi$
$5$	4.30411	1.92486	0.962428	+	0.271538i	$0.0875322\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.55163	−0.850545
$10$	0	0
$11$	0.669601	0.201892	0.100946	−	0.994892i	$-0.467813\pi$
$11$	0.669601	0.201892	0.100946	+	0.994892i	$0.467813\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.88203	−0.744138
$16$	0	0
$17$	6.22124	1.50887	0.754436	−	0.656374i	$-0.227911\pi$
$17$	6.22124	1.50887	0.754436	+	0.656374i	$0.227911\pi$
$18$	0	0
$19$	−6.30411	−1.44626	−0.723131	−	0.690711i	$-0.757298\pi$
$19$	−6.30411	−1.44626	−0.723131	+	0.690711i	$0.757298\pi$
$20$	0	0
$21$	0.669601	0.146119
$22$	0	0
$23$	−5.63451	−1.17488	−0.587438	−	0.809269i	$-0.699863\pi$
$23$	−5.63451	−1.17488	−0.587438	+	0.809269i	$0.699863\pi$
$24$	0	0
$25$	13.5253	2.70507
$26$	0	0
$27$	3.71738	0.715410
$28$	0	0
$29$	−1.42207	−0.264072	−0.132036	−	0.991245i	$-0.542152\pi$
$29$	−1.42207	−0.264072	−0.132036	+	0.991245i	$0.542152\pi$
$30$	0	0
$31$	−4.09167	−0.734886	−0.367443	−	0.930046i	$-0.619767\pi$
$31$	−4.09167	−0.734886	−0.367443	+	0.930046i	$0.619767\pi$
$32$	0	0
$33$	−0.448365	−0.0780504
$34$	0	0
$35$	−4.30411	−0.727527
$36$	0	0
$37$	−2.66960	−0.438880	−0.219440	−	0.975626i	$-0.570423\pi$
$37$	−2.66960	−0.438880	−0.219440	+	0.975626i	$0.570423\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.05658	−1.10205	−0.551026	−	0.834488i	$-0.685764\pi$
$41$	−7.05658	−1.10205	−0.551026	+	0.834488i	$0.685764\pi$
$42$	0	0
$43$	−10.5253	−1.60510	−0.802550	−	0.596585i	$-0.796524\pi$
$43$	−10.5253	−1.60510	−0.802550	+	0.596585i	$0.796524\pi$
$44$	0	0
$45$	−10.9825	−1.63718
$46$	0	0
$47$	−4.09167	−0.596832	−0.298416	−	0.954436i	$-0.596458\pi$
$47$	−4.09167	−0.596832	−0.298416	+	0.954436i	$0.596458\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.16574	−0.583321
$52$	0	0
$53$	5.42207	0.744779	0.372390	−	0.928077i	$-0.378539\pi$
$53$	5.42207	0.744779	0.372390	+	0.928077i	$0.378539\pi$
$54$	0	0
$55$	2.88203	0.388613
$56$	0	0
$57$	4.22124	0.559116
$58$	0	0
$59$	7.26901	0.946345	0.473172	−	0.880970i	$-0.343109\pi$
$59$	7.26901	0.946345	0.473172	+	0.880970i	$0.343109\pi$
$60$	0	0
$61$	2.28262	0.292260	0.146130	−	0.989265i	$-0.453318\pi$
$61$	2.28262	0.292260	0.146130	+	0.989265i	$0.453318\pi$
$62$	0	0
$63$	2.55163	0.321476
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.21243	−0.270292	−0.135146	−	0.990826i	$-0.543150\pi$
$67$	−2.21243	−0.270292	−0.135146	+	0.990826i	$0.543150\pi$
$68$	0	0
$69$	3.77287	0.454200
$70$	0	0
$71$	−5.56044	−0.659902	−0.329951	−	0.943998i	$-0.607032\pi$
$71$	−5.56044	−0.659902	−0.329951	+	0.943998i	$0.607032\pi$
$72$	0	0
$73$	−11.8557	−1.38761	−0.693805	−	0.720163i	$-0.744067\pi$
$73$	−11.8557	−1.38761	−0.693805	+	0.720163i	$0.744067\pi$
$74$	0	0
$75$	−9.05658	−1.04576
$76$	0	0
$77$	−0.669601	−0.0763081
$78$	0	0
$79$	0.312910	0.0352051	0.0176026	−	0.999845i	$-0.494397\pi$
$79$	0.312910	0.0352051	0.0176026	+	0.999845i	$0.494397\pi$
$80$	0	0
$81$	5.16574	0.573972
$82$	0	0
$83$	3.03509	0.333145	0.166572	−	0.986029i	$-0.446730\pi$
$83$	3.03509	0.333145	0.166572	+	0.986029i	$0.446730\pi$
$84$	0	0
$85$	26.7769	2.90436
$86$	0	0
$87$	0.952222	0.102089
$88$	0	0
$89$	−4.12076	−0.436800	−0.218400	−	0.975859i	$-0.570084\pi$
$89$	−4.12076	−0.436800	−0.218400	+	0.975859i	$0.570084\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.73979	0.284103
$94$	0	0
$95$	−27.1336	−2.78384
$96$	0	0
$97$	−7.43088	−0.754491	−0.377246	−	0.926113i	$-0.623129\pi$
$97$	−7.43088	−0.754491	−0.377246	+	0.926113i	$0.623129\pi$
$98$	0	0
$99$	−1.70858	−0.171718
$100$	0	0
$101$	−11.0945	−1.10394	−0.551970	−	0.833864i	$-0.686124\pi$
$101$	−11.0945	−1.10394	−0.551970	+	0.833864i	$0.686124\pi$
$102$	0	0
$103$	−4.42487	−0.435995	−0.217998	−	0.975949i	$-0.569952\pi$
$103$	−4.42487	−0.435995	−0.217998	+	0.975949i	$0.569952\pi$
$104$	0	0
$105$	2.88203	0.281258
$106$	0	0
$107$	−16.6082	−1.60558	−0.802788	−	0.596264i	$-0.796651\pi$
$107$	−16.6082	−1.60558	−0.802788	+	0.596264i	$0.796651\pi$
$108$	0	0
$109$	1.79637	0.172061	0.0860305	−	0.996293i	$-0.472582\pi$
$109$	1.79637	0.172061	0.0860305	+	0.996293i	$0.472582\pi$
$110$	0	0
$111$	1.78757	0.169668
$112$	0	0
$113$	7.63451	0.718194	0.359097	−	0.933300i	$-0.383085\pi$
$113$	7.63451	0.718194	0.359097	+	0.933300i	$0.383085\pi$
$114$	0	0
$115$	−24.2515	−2.26147
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.22124	−0.570300
$120$	0	0
$121$	−10.5516	−0.959240
$122$	0	0
$123$	4.72509	0.426047
$124$	0	0
$125$	36.6940	3.28201
$126$	0	0
$127$	20.3432	1.80517	0.902584	−	0.430515i	$-0.141668\pi$
$127$	20.3432	1.80517	0.902584	+	0.430515i	$0.141668\pi$
$128$	0	0
$129$	7.04778	0.620522
$130$	0	0
$131$	21.6589	1.89235	0.946174	−	0.323660i	$-0.104913\pi$
$131$	21.6589	1.89235	0.946174	+	0.323660i	$0.104913\pi$
$132$	0	0
$133$	6.30411	0.546635
$134$	0	0
$135$	16.0000	1.37706
$136$	0	0
$137$	−8.71629	−0.744683	−0.372341	−	0.928096i	$-0.621445\pi$
$137$	−8.71629	−0.744683	−0.372341	+	0.928096i	$0.621445\pi$
$138$	0	0
$139$	12.6461	1.07263	0.536314	−	0.844018i	$-0.319816\pi$
$139$	12.6461	1.07263	0.536314	+	0.844018i	$0.319816\pi$
$140$	0	0
$141$	2.73979	0.230732
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.12076	−0.508301
$146$	0	0
$147$	−0.669601	−0.0552277
$148$	0	0
$149$	−15.7203	−1.28786	−0.643928	−	0.765086i	$-0.722696\pi$
$149$	−15.7203	−1.28786	−0.643928	+	0.765086i	$0.722696\pi$
$150$	0	0
$151$	−2.88203	−0.234537	−0.117268	−	0.993100i	$-0.537414\pi$
$151$	−2.88203	−0.234537	−0.117268	+	0.993100i	$0.537414\pi$
$152$	0	0
$153$	−15.8743	−1.28336
$154$	0	0
$155$	−17.6110	−1.41455
$156$	0	0
$157$	−17.0419	−1.36009	−0.680045	−	0.733170i	$-0.738040\pi$
$157$	−17.0419	−1.36009	−0.680045	+	0.733170i	$0.738040\pi$
$158$	0	0
$159$	−3.63062	−0.287927
$160$	0	0
$161$	5.63451	0.444061
$162$	0	0
$163$	−15.4524	−1.21032	−0.605161	−	0.796103i	$-0.706891\pi$
$163$	−15.4524	−1.21032	−0.605161	+	0.796103i	$0.706891\pi$
$164$	0	0
$165$	−1.92981	−0.150236
$166$	0	0
$167$	22.0156	1.70362	0.851809	−	0.523853i	$-0.175506\pi$
$167$	22.0156	1.70362	0.851809	+	0.523853i	$0.175506\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	16.0858	1.23011
$172$	0	0
$173$	12.3344	0.937767	0.468883	−	0.883260i	$-0.344656\pi$
$173$	12.3344	0.937767	0.468883	+	0.883260i	$0.344656\pi$
$174$	0	0
$175$	−13.5253	−1.02242
$176$	0	0
$177$	−4.86734	−0.365851
$178$	0	0
$179$	−0.743671	−0.0555845	−0.0277923	−	0.999614i	$-0.508848\pi$
$179$	−0.743671	−0.0555845	−0.0277923	+	0.999614i	$0.508848\pi$
$180$	0	0
$181$	−13.7729	−1.02373	−0.511865	−	0.859066i	$-0.671045\pi$
$181$	−13.7729	−1.02373	−0.511865	+	0.859066i	$0.671045\pi$
$182$	0	0
$183$	−1.52844	−0.112986
$184$	0	0
$185$	−11.4902	−0.844780
$186$	0	0
$187$	4.16574	0.304629
$188$	0	0
$189$	−3.71738	−0.270400
$190$	0	0
$191$	−17.5983	−1.27337	−0.636685	−	0.771124i	$-0.719695\pi$
$191$	−17.5983	−1.27337	−0.636685	+	0.771124i	$0.719695\pi$
$192$	0	0
$193$	−8.66080	−0.623418	−0.311709	−	0.950178i	$-0.600901\pi$
$193$	−8.66080	−0.623418	−0.311709	+	0.950178i	$0.600901\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.6122	−1.53981	−0.769903	−	0.638161i	$-0.779696\pi$
$197$	−21.6122	−1.53981	−0.769903	+	0.638161i	$0.779696\pi$
$198$	0	0
$199$	−4.40458	−0.312233	−0.156116	−	0.987739i	$-0.549897\pi$
$199$	−4.40458	−0.312233	−0.156116	+	0.987739i	$0.549897\pi$
$200$	0	0
$201$	1.48145	0.104493
$202$	0	0
$203$	1.42207	0.0998100
$204$	0	0
$205$	−30.3723	−2.12129
$206$	0	0
$207$	14.3772	0.999285
$208$	0	0
$209$	−4.22124	−0.291989
$210$	0	0
$211$	−13.7944	−0.949643	−0.474821	−	0.880082i	$-0.657487\pi$
$211$	−13.7944	−0.949643	−0.474821	+	0.880082i	$0.657487\pi$
$212$	0	0
$213$	3.72327	0.255114
$214$	0	0
$215$	−45.3022	−3.08959
$216$	0	0
$217$	4.09167	0.277761
$218$	0	0
$219$	7.93861	0.536442
$220$	0	0
$221$	0	0
$222$	0	0
$223$	27.5146	1.84252	0.921258	−	0.388951i	$-0.127162\pi$
$223$	27.5146	1.84252	0.921258	+	0.388951i	$0.127162\pi$
$224$	0	0
$225$	−34.5117	−2.30078
$226$	0	0
$227$	−20.4425	−1.35681	−0.678407	−	0.734686i	$-0.737329\pi$
$227$	−20.4425	−1.35681	−0.678407	+	0.734686i	$0.737329\pi$
$228$	0	0
$229$	13.7115	0.906080	0.453040	−	0.891490i	$-0.350339\pi$
$229$	13.7115	0.906080	0.453040	+	0.891490i	$0.350339\pi$
$230$	0	0
$231$	0.448365	0.0295003
$232$	0	0
$233$	0.531237	0.0348025	0.0174013	−	0.999849i	$-0.494461\pi$
$233$	0.531237	0.0348025	0.0174013	+	0.999849i	$0.494461\pi$
$234$	0	0
$235$	−17.6110	−1.14882
$236$	0	0
$237$	−0.209525	−0.0136101
$238$	0	0
$239$	−2.55563	−0.165310	−0.0826551	−	0.996578i	$-0.526340\pi$
$239$	−2.55563	−0.165310	−0.0826551	+	0.996578i	$0.526340\pi$
$240$	0	0
$241$	8.48746	0.546725	0.273363	−	0.961911i	$-0.411864\pi$
$241$	8.48746	0.546725	0.273363	+	0.961911i	$0.411864\pi$
$242$	0	0
$243$	−14.6111	−0.937304
$244$	0	0
$245$	4.30411	0.274979
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2.03230	−0.128792
$250$	0	0
$251$	−6.73979	−0.425412	−0.212706	−	0.977116i	$-0.568228\pi$
$251$	−6.73979	−0.425412	−0.212706	+	0.977116i	$0.568228\pi$
$252$	0	0
$253$	−3.77287	−0.237198
$254$	0	0
$255$	−17.9298	−1.12281
$256$	0	0
$257$	−3.37709	−0.210657	−0.105328	−	0.994437i	$-0.533589\pi$
$257$	−3.37709	−0.210657	−0.105328	+	0.994437i	$0.533589\pi$
$258$	0	0
$259$	2.66960	0.165881
$260$	0	0
$261$	3.62861	0.224605
$262$	0	0
$263$	−18.7263	−1.15471	−0.577356	−	0.816492i	$-0.695916\pi$
$263$	−18.7263	−1.15471	−0.577356	+	0.816492i	$0.695916\pi$
$264$	0	0
$265$	23.3372	1.43359
$266$	0	0
$267$	2.75926	0.168864
$268$	0	0
$269$	29.4289	1.79431	0.897155	−	0.441716i	$-0.145630\pi$
$269$	29.4289	1.79431	0.897155	+	0.441716i	$0.145630\pi$
$270$	0	0
$271$	−0.707487	−0.0429768	−0.0214884	−	0.999769i	$-0.506840\pi$
$271$	−0.707487	−0.0429768	−0.0214884	+	0.999769i	$0.506840\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.05658	0.546132
$276$	0	0
$277$	18.2136	1.09435	0.547176	−	0.837018i	$-0.315703\pi$
$277$	18.2136	1.09435	0.547176	+	0.837018i	$0.315703\pi$
$278$	0	0
$279$	10.4405	0.625054
$280$	0	0
$281$	−26.1161	−1.55795	−0.778977	−	0.627052i	$-0.784261\pi$
$281$	−26.1161	−1.55795	−0.778977	+	0.627052i	$0.784261\pi$
$282$	0	0
$283$	31.2631	1.85840	0.929200	−	0.369578i	$-0.120498\pi$
$283$	31.2631	1.85840	0.929200	+	0.369578i	$0.120498\pi$
$284$	0	0
$285$	18.1687	1.07622
$286$	0	0
$287$	7.05658	0.416537
$288$	0	0
$289$	21.7038	1.27669
$290$	0	0
$291$	4.97572	0.291682
$292$	0	0
$293$	−24.6238	−1.43854	−0.719269	−	0.694732i	$-0.755523\pi$
$293$	−24.6238	−1.43854	−0.719269	+	0.694732i	$0.755523\pi$
$294$	0	0
$295$	31.2866	1.82158
$296$	0	0
$297$	2.48916	0.144436
$298$	0	0
$299$	0	0
$300$	0	0
$301$	10.5253	0.606671
$302$	0	0
$303$	7.42886	0.426777
$304$	0	0
$305$	9.82465	0.562558
$306$	0	0
$307$	−31.6433	−1.80598	−0.902990	−	0.429663i	$-0.858633\pi$
$307$	−31.6433	−1.80598	−0.902990	+	0.429663i	$0.858633\pi$
$308$	0	0
$309$	2.96289	0.168553
$310$	0	0
$311$	−18.3520	−1.04065	−0.520323	−	0.853969i	$-0.674189\pi$
$311$	−18.3520	−1.04065	−0.520323	+	0.853969i	$0.674189\pi$
$312$	0	0
$313$	8.71629	0.492674	0.246337	−	0.969184i	$-0.420773\pi$
$313$	8.71629	0.492674	0.246337	+	0.969184i	$0.420773\pi$
$314$	0	0
$315$	10.9825	0.618794
$316$	0	0
$317$	13.1268	0.737273	0.368636	−	0.929574i	$-0.379825\pi$
$317$	13.1268	0.737273	0.368636	+	0.929574i	$0.379825\pi$
$318$	0	0
$319$	−0.952222	−0.0533142
$320$	0	0
$321$	11.1209	0.620707
$322$	0	0
$323$	−39.2193	−2.18222
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.20285	−0.0665178
$328$	0	0
$329$	4.09167	0.225581
$330$	0	0
$331$	9.23993	0.507872	0.253936	−	0.967221i	$-0.418275\pi$
$331$	9.23993	0.507872	0.253936	+	0.967221i	$0.418275\pi$
$332$	0	0
$333$	6.81185	0.373287
$334$	0	0
$335$	−9.52255	−0.520273
$336$	0	0
$337$	19.1626	1.04386	0.521928	−	0.852990i	$-0.325213\pi$
$337$	19.1626	1.04386	0.521928	+	0.852990i	$0.325213\pi$
$338$	0	0
$339$	−5.11207	−0.277650
$340$	0	0
$341$	−2.73979	−0.148368
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	16.2388	0.874270
$346$	0	0
$347$	10.5380	0.565711	0.282855	−	0.959163i	$-0.408718\pi$
$347$	10.5380	0.565711	0.282855	+	0.959163i	$0.408718\pi$
$348$	0	0
$349$	2.55764	0.136908	0.0684538	−	0.997654i	$-0.478193\pi$
$349$	2.55764	0.136908	0.0684538	+	0.997654i	$0.478193\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.14437	0.273807	0.136904	−	0.990584i	$-0.456285\pi$
$353$	5.14437	0.273807	0.136904	+	0.990584i	$0.456285\pi$
$354$	0	0
$355$	−23.9327	−1.27022
$356$	0	0
$357$	4.16574	0.220475
$358$	0	0
$359$	2.88203	0.152108	0.0760540	−	0.997104i	$-0.475768\pi$
$359$	2.88203	0.152108	0.0760540	+	0.997104i	$0.475768\pi$
$360$	0	0
$361$	20.7418	1.09167
$362$	0	0
$363$	7.06538	0.370836
$364$	0	0
$365$	−51.0284	−2.67095
$366$	0	0
$367$	35.3245	1.84392	0.921962	−	0.387280i	$-0.126585\pi$
$367$	35.3245	1.84392	0.921962	+	0.387280i	$0.126585\pi$
$368$	0	0
$369$	18.0058	0.937345
$370$	0	0
$371$	−5.42207	−0.281500
$372$	0	0
$373$	−6.04297	−0.312893	−0.156447	−	0.987686i	$-0.550004\pi$
$373$	−6.04297	−0.312893	−0.156447	+	0.987686i	$0.550004\pi$
$374$	0	0
$375$	−24.5703	−1.26881
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−10.2738	−0.527731	−0.263865	−	0.964560i	$-0.584997\pi$
$379$	−10.2738	−0.527731	−0.263865	+	0.964560i	$0.584997\pi$
$380$	0	0
$381$	−13.6218	−0.697867
$382$	0	0
$383$	−14.8030	−0.756400	−0.378200	−	0.925724i	$-0.623457\pi$
$383$	−14.8030	−0.756400	−0.378200	+	0.925724i	$0.623457\pi$
$384$	0	0
$385$	−2.88203	−0.146882
$386$	0	0
$387$	26.8568	1.36521
$388$	0	0
$389$	8.05258	0.408282	0.204141	−	0.978941i	$-0.434560\pi$
$389$	8.05258	0.408282	0.204141	+	0.978941i	$0.434560\pi$
$390$	0	0
$391$	−35.0536	−1.77274
$392$	0	0
$393$	−14.5028	−0.731570
$394$	0	0
$395$	1.34680	0.0677648
$396$	0	0
$397$	−29.6863	−1.48991	−0.744956	−	0.667114i	$-0.767530\pi$
$397$	−29.6863	−1.48991	−0.744956	+	0.667114i	$0.767530\pi$
$398$	0	0
$399$	−4.22124	−0.211326
$400$	0	0
$401$	21.1638	1.05687	0.528436	−	0.848973i	$-0.322779\pi$
$401$	21.1638	1.05687	0.528436	+	0.848973i	$0.322779\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	22.2339	1.10481
$406$	0	0
$407$	−1.78757	−0.0886064
$408$	0	0
$409$	19.3022	0.954433	0.477216	−	0.878786i	$-0.341646\pi$
$409$	19.3022	0.954433	0.477216	+	0.878786i	$0.341646\pi$
$410$	0	0
$411$	5.83643	0.287890
$412$	0	0
$413$	−7.26901	−0.357685
$414$	0	0
$415$	13.0634	0.641256
$416$	0	0
$417$	−8.46784	−0.414672
$418$	0	0
$419$	28.7478	1.40442	0.702211	−	0.711969i	$-0.252197\pi$
$419$	28.7478	1.40442	0.702211	+	0.711969i	$0.252197\pi$
$420$	0	0
$421$	−23.9036	−1.16499	−0.582496	−	0.812834i	$-0.697924\pi$
$421$	−23.9036	−1.16499	−0.582496	+	0.812834i	$0.697924\pi$
$422$	0	0
$423$	10.4405	0.507632
$424$	0	0
$425$	84.1443	4.08160
$426$	0	0
$427$	−2.28262	−0.110464
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.05258	0.0988694	0.0494347	−	0.998777i	$-0.484258\pi$
$431$	2.05258	0.0988694	0.0494347	+	0.998777i	$0.484258\pi$
$432$	0	0
$433$	12.3723	0.594574	0.297287	−	0.954788i	$-0.403918\pi$
$433$	12.3723	0.594574	0.297287	+	0.954788i	$0.403918\pi$
$434$	0	0
$435$	4.09846	0.196506
$436$	0	0
$437$	35.5205	1.69918
$438$	0	0
$439$	−39.6063	−1.89031	−0.945153	−	0.326627i	$-0.894088\pi$
$439$	−39.6063	−1.89031	−0.945153	+	0.326627i	$0.894088\pi$
$440$	0	0
$441$	−2.55163	−0.121506
$442$	0	0
$443$	−31.9075	−1.51597	−0.757986	−	0.652271i	$-0.773816\pi$
$443$	−31.9075	−1.51597	−0.757986	+	0.652271i	$0.773816\pi$
$444$	0	0
$445$	−17.7362	−0.840776
$446$	0	0
$447$	10.5263	0.497878
$448$	0	0
$449$	42.0312	1.98357	0.991787	−	0.127899i	$-0.0408235\pi$
$449$	42.0312	1.98357	0.991787	+	0.127899i	$0.0408235\pi$
$450$	0	0
$451$	−4.72509	−0.222496
$452$	0	0
$453$	1.92981	0.0906705
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.5001	0.771844	0.385922	−	0.922531i	$-0.373883\pi$
$457$	16.5001	0.771844	0.385922	+	0.922531i	$0.373883\pi$
$458$	0	0
$459$	23.1267	1.07946
$460$	0	0
$461$	−2.34909	−0.109408	−0.0547041	−	0.998503i	$-0.517422\pi$
$461$	−2.34909	−0.109408	−0.0547041	+	0.998503i	$0.517422\pi$
$462$	0	0
$463$	21.6210	1.00481	0.502407	−	0.864631i	$-0.332448\pi$
$463$	21.6210	1.00481	0.502407	+	0.864631i	$0.332448\pi$
$464$	0	0
$465$	11.7923	0.546857
$466$	0	0
$467$	−26.3344	−1.21861	−0.609305	−	0.792936i	$-0.708552\pi$
$467$	−26.3344	−1.21861	−0.609305	+	0.792936i	$0.708552\pi$
$468$	0	0
$469$	2.21243	0.102161
$470$	0	0
$471$	11.4113	0.525803
$472$	0	0
$473$	−7.04778	−0.324057
$474$	0	0
$475$	−85.2652	−3.91224
$476$	0	0
$477$	−13.8352	−0.633468
$478$	0	0
$479$	−10.5632	−0.482646	−0.241323	−	0.970445i	$-0.577581\pi$
$479$	−10.5632	−0.482646	−0.241323	+	0.970445i	$0.577581\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−3.77287	−0.171672
$484$	0	0
$485$	−31.9833	−1.45229
$486$	0	0
$487$	−22.3547	−1.01299	−0.506494	−	0.862244i	$-0.669059\pi$
$487$	−22.3547	−1.01299	−0.506494	+	0.862244i	$0.669059\pi$
$488$	0	0
$489$	10.3469	0.467904
$490$	0	0
$491$	−10.4774	−0.472841	−0.236420	−	0.971651i	$-0.575974\pi$
$491$	−10.4774	−0.472841	−0.236420	+	0.971651i	$0.575974\pi$
$492$	0	0
$493$	−8.84706	−0.398451
$494$	0	0
$495$	−7.35390	−0.330533
$496$	0	0
$497$	5.56044	0.249420
$498$	0	0
$499$	34.9191	1.56319	0.781597	−	0.623784i	$-0.214405\pi$
$499$	34.9191	1.56319	0.781597	+	0.623784i	$0.214405\pi$
$500$	0	0
$501$	−14.7417	−0.658609
$502$	0	0
$503$	35.2572	1.57204	0.786021	−	0.618200i	$-0.212138\pi$
$503$	35.2572	1.57204	0.786021	+	0.618200i	$0.212138\pi$
$504$	0	0
$505$	−47.7518	−2.12493
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.4832	−0.686279	−0.343139	−	0.939285i	$-0.611490\pi$
$509$	−15.4832	−0.686279	−0.343139	+	0.939285i	$0.611490\pi$
$510$	0	0
$511$	11.8557	0.524467
$512$	0	0
$513$	−23.4348	−1.03467
$514$	0	0
$515$	−19.0451	−0.839227
$516$	0	0
$517$	−2.73979	−0.120496
$518$	0	0
$519$	−8.25912	−0.362535
$520$	0	0
$521$	−14.0606	−0.616005	−0.308003	−	0.951386i	$-0.599661\pi$
$521$	−14.0606	−0.616005	−0.308003	+	0.951386i	$0.599661\pi$
$522$	0	0
$523$	11.0216	0.481941	0.240971	−	0.970532i	$-0.422534\pi$
$523$	11.0216	0.481941	0.240971	+	0.970532i	$0.422534\pi$
$524$	0	0
$525$	9.05658	0.395262
$526$	0	0
$527$	−25.4553	−1.10885
$528$	0	0
$529$	8.74767	0.380333
$530$	0	0
$531$	−18.5479	−0.804909
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−71.4835	−3.09050
$536$	0	0
$537$	0.497962	0.0214887
$538$	0	0
$539$	0.669601	0.0288417
$540$	0	0
$541$	34.3400	1.47639	0.738196	−	0.674587i	$-0.235678\pi$
$541$	34.3400	1.47639	0.738196	+	0.674587i	$0.235678\pi$
$542$	0	0
$543$	9.22232	0.395768
$544$	0	0
$545$	7.73177	0.331192
$546$	0	0
$547$	−25.8373	−1.10472	−0.552362	−	0.833604i	$-0.686273\pi$
$547$	−25.8373	−1.10472	−0.552362	+	0.833604i	$0.686273\pi$
$548$	0	0
$549$	−5.82441	−0.248580
$550$	0	0
$551$	8.96491	0.381918
$552$	0	0
$553$	−0.312910	−0.0133063
$554$	0	0
$555$	7.69388	0.326587
$556$	0	0
$557$	−21.5516	−0.913172	−0.456586	−	0.889679i	$-0.650928\pi$
$557$	−21.5516	−0.913172	−0.456586	+	0.889679i	$0.650928\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.78939	−0.117768
$562$	0	0
$563$	−29.0742	−1.22533	−0.612665	−	0.790342i	$-0.709903\pi$
$563$	−29.0742	−1.22533	−0.612665	+	0.790342i	$0.709903\pi$
$564$	0	0
$565$	32.8597	1.38242
$566$	0	0
$567$	−5.16574	−0.216941
$568$	0	0
$569$	−26.3559	−1.10490	−0.552448	−	0.833547i	$-0.686306\pi$
$569$	−26.3559	−1.10490	−0.552448	+	0.833547i	$0.686306\pi$
$570$	0	0
$571$	−34.3420	−1.43717	−0.718584	−	0.695440i	$-0.755209\pi$
$571$	−34.3420	−1.43717	−0.718584	+	0.695440i	$0.755209\pi$
$572$	0	0
$573$	11.7839	0.492277
$574$	0	0
$575$	−76.2086	−3.17812
$576$	0	0
$577$	−3.21643	−0.133902	−0.0669509	−	0.997756i	$-0.521327\pi$
$577$	−3.21643	−0.133902	−0.0669509	+	0.997756i	$0.521327\pi$
$578$	0	0
$579$	5.79928	0.241010
$580$	0	0
$581$	−3.03509	−0.125917
$582$	0	0
$583$	3.63062	0.150365
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.3041	−0.590394	−0.295197	−	0.955436i	$-0.595385\pi$
$587$	−14.3041	−0.590394	−0.295197	+	0.955436i	$0.595385\pi$
$588$	0	0
$589$	25.7944	1.06284
$590$	0	0
$591$	14.4716	0.595280
$592$	0	0
$593$	−6.72339	−0.276096	−0.138048	−	0.990426i	$-0.544083\pi$
$593$	−6.72339	−0.276096	−0.138048	+	0.990426i	$0.544083\pi$
$594$	0	0
$595$	−26.7769	−1.09774
$596$	0	0
$597$	2.94931	0.120707
$598$	0	0
$599$	−19.2152	−0.785113	−0.392556	−	0.919728i	$-0.628409\pi$
$599$	−19.2152	−0.785113	−0.392556	+	0.919728i	$0.628409\pi$
$600$	0	0
$601$	39.4428	1.60890	0.804452	−	0.594018i	$-0.202459\pi$
$601$	39.4428	1.60890	0.804452	+	0.594018i	$0.202459\pi$
$602$	0	0
$603$	5.64532	0.229895
$604$	0	0
$605$	−45.4154	−1.84640
$606$	0	0
$607$	−4.99520	−0.202749	−0.101374	−	0.994848i	$-0.532324\pi$
$607$	−4.99520	−0.202749	−0.101374	+	0.994848i	$0.532324\pi$
$608$	0	0
$609$	−0.952222	−0.0385860
$610$	0	0
$611$	0	0
$612$	0	0
$613$	31.7582	1.28270	0.641350	−	0.767248i	$-0.278375\pi$
$613$	31.7582	1.28270	0.641350	+	0.767248i	$0.278375\pi$
$614$	0	0
$615$	20.3373	0.820079
$616$	0	0
$617$	−35.2164	−1.41776	−0.708880	−	0.705329i	$-0.750799\pi$
$617$	−35.2164	−1.41776	−0.708880	+	0.705329i	$0.750799\pi$
$618$	0	0
$619$	27.9882	1.12494	0.562471	−	0.826817i	$-0.309851\pi$
$619$	27.9882	1.12494	0.562471	+	0.826817i	$0.309851\pi$
$620$	0	0
$621$	−20.9456	−0.840518
$622$	0	0
$623$	4.12076	0.165095
$624$	0	0
$625$	90.3082	3.61233
$626$	0	0
$627$	2.82654	0.112881
$628$	0	0
$629$	−16.6082	−0.662213
$630$	0	0
$631$	15.1638	0.603663	0.301832	−	0.953361i	$-0.402402\pi$
$631$	15.1638	0.603663	0.301832	+	0.953361i	$0.402402\pi$
$632$	0	0
$633$	9.23671	0.367126
$634$	0	0
$635$	87.5593	3.47469
$636$	0	0
$637$	0	0
$638$	0	0
$639$	14.1882	0.561277
$640$	0	0
$641$	1.35560	0.0535430	0.0267715	−	0.999642i	$-0.491477\pi$
$641$	1.35560	0.0535430	0.0267715	+	0.999642i	$0.491477\pi$
$642$	0	0
$643$	10.1308	0.399518	0.199759	−	0.979845i	$-0.435984\pi$
$643$	10.1308	0.399518	0.199759	+	0.979845i	$0.435984\pi$
$644$	0	0
$645$	30.3344	1.19442
$646$	0	0
$647$	10.4396	0.410422	0.205211	−	0.978718i	$-0.434212\pi$
$647$	10.4396	0.410422	0.205211	+	0.978718i	$0.434212\pi$
$648$	0	0
$649$	4.86734	0.191060
$650$	0	0
$651$	−2.73979	−0.107381
$652$	0	0
$653$	6.56524	0.256918	0.128459	−	0.991715i	$-0.458997\pi$
$653$	6.56524	0.256918	0.128459	+	0.991715i	$0.458997\pi$
$654$	0	0
$655$	93.2222	3.64249
$656$	0	0
$657$	30.2515	1.18022
$658$	0	0
$659$	5.42766	0.211432	0.105716	−	0.994396i	$-0.466287\pi$
$659$	5.42766	0.211432	0.105716	+	0.994396i	$0.466287\pi$
$660$	0	0
$661$	−4.88695	−0.190080	−0.0950402	−	0.995473i	$-0.530298\pi$
$661$	−4.88695	−0.190080	−0.0950402	+	0.995473i	$0.530298\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	27.1336	1.05219
$666$	0	0
$667$	8.01268	0.310252
$668$	0	0
$669$	−18.4238	−0.712306
$670$	0	0
$671$	1.52844	0.0590050
$672$	0	0
$673$	−33.8157	−1.30350	−0.651749	−	0.758434i	$-0.725965\pi$
$673$	−33.8157	−1.30350	−0.651749	+	0.758434i	$0.725965\pi$
$674$	0	0
$675$	50.2788	1.93523
$676$	0	0
$677$	5.27223	0.202628	0.101314	−	0.994854i	$-0.467695\pi$
$677$	5.27223	0.202628	0.101314	+	0.994854i	$0.467695\pi$
$678$	0	0
$679$	7.43088	0.285171
$680$	0	0
$681$	13.6883	0.524537
$682$	0	0
$683$	49.4115	1.89068	0.945339	−	0.326089i	$-0.105731\pi$
$683$	49.4115	1.89068	0.945339	+	0.326089i	$0.105731\pi$
$684$	0	0
$685$	−37.5158	−1.43341
$686$	0	0
$687$	−9.18122	−0.350285
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−6.04499	−0.229962	−0.114981	−	0.993368i	$-0.536681\pi$
$691$	−6.04499	−0.229962	−0.114981	+	0.993368i	$0.536681\pi$
$692$	0	0
$693$	1.70858	0.0649035
$694$	0	0
$695$	54.4302	2.06465
$696$	0	0
$697$	−43.9007	−1.66286
$698$	0	0
$699$	−0.355717	−0.0134545
$700$	0	0
$701$	−30.8395	−1.16479	−0.582395	−	0.812906i	$-0.697884\pi$
$701$	−30.8395	−1.16479	−0.582395	+	0.812906i	$0.697884\pi$
$702$	0	0
$703$	16.8295	0.634735
$704$	0	0
$705$	11.7923	0.444125
$706$	0	0
$707$	11.0945	0.417250
$708$	0	0
$709$	6.69497	0.251435	0.125717	−	0.992066i	$-0.459877\pi$
$709$	6.69497	0.251435	0.125717	+	0.992066i	$0.459877\pi$
$710$	0	0
$711$	−0.798432	−0.0299436
$712$	0	0
$713$	23.0546	0.863400
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.71125	0.0639079
$718$	0	0
$719$	−3.44728	−0.128562	−0.0642808	−	0.997932i	$-0.520475\pi$
$719$	−3.44728	−0.128562	−0.0642808	+	0.997932i	$0.520475\pi$
$720$	0	0
$721$	4.42487	0.164791
$722$	0	0
$723$	−5.68321	−0.211361
$724$	0	0
$725$	−19.2340	−0.714334
$726$	0	0
$727$	−29.9826	−1.11199	−0.555997	−	0.831184i	$-0.687663\pi$
$727$	−29.9826	−1.11199	−0.555997	+	0.831184i	$0.687663\pi$
$728$	0	0
$729$	−5.71361	−0.211615
$730$	0	0
$731$	−65.4806	−2.42189
$732$	0	0
$733$	−18.5400	−0.684792	−0.342396	−	0.939556i	$-0.611238\pi$
$733$	−18.5400	−0.684792	−0.342396	+	0.939556i	$0.611238\pi$
$734$	0	0
$735$	−2.88203	−0.106305
$736$	0	0
$737$	−1.48145	−0.0545698
$738$	0	0
$739$	−17.2543	−0.634710	−0.317355	−	0.948307i	$-0.602795\pi$
$739$	−17.2543	−0.634710	−0.317355	+	0.948307i	$0.602795\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.7644	−1.27538	−0.637690	−	0.770293i	$-0.720110\pi$
$743$	−34.7644	−1.27538	−0.637690	+	0.770293i	$0.720110\pi$
$744$	0	0
$745$	−67.6618	−2.47894
$746$	0	0
$747$	−7.74445	−0.283355
$748$	0	0
$749$	16.6082	0.606851
$750$	0	0
$751$	−33.8355	−1.23467	−0.617337	−	0.786699i	$-0.711788\pi$
$751$	−33.8355	−1.23467	−0.617337	+	0.786699i	$0.711788\pi$
$752$	0	0
$753$	4.51297	0.164462
$754$	0	0
$755$	−12.4046	−0.451449
$756$	0	0
$757$	−27.2291	−0.989659	−0.494830	−	0.868990i	$-0.664769\pi$
$757$	−27.2291	−0.989659	−0.494830	+	0.868990i	$0.664769\pi$
$758$	0	0
$759$	2.52632	0.0916995
$760$	0	0
$761$	15.6724	0.568124	0.284062	−	0.958806i	$-0.408318\pi$
$761$	15.6724	0.568124	0.284062	+	0.958806i	$0.408318\pi$
$762$	0	0
$763$	−1.79637	−0.0650329
$764$	0	0
$765$	−68.3248	−2.47029
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0.481563	0.0173656	0.00868280	−	0.999962i	$-0.497236\pi$
$769$	0.481563	0.0173656	0.00868280	+	0.999962i	$0.497236\pi$
$770$	0	0
$771$	2.26130	0.0814388
$772$	0	0
$773$	10.7740	0.387512	0.193756	−	0.981050i	$-0.437933\pi$
$773$	10.7740	0.387512	0.193756	+	0.981050i	$0.437933\pi$
$774$	0	0
$775$	−55.3413	−1.98792
$776$	0	0
$777$	−1.78757	−0.0641286
$778$	0	0
$779$	44.4854	1.59386
$780$	0	0
$781$	−3.72327	−0.133229
$782$	0	0
$783$	−5.28639	−0.188920
$784$	0	0
$785$	−73.3501	−2.61798
$786$	0	0
$787$	19.4272	0.692503	0.346252	−	0.938142i	$-0.387454\pi$
$787$	19.4272	0.692503	0.346252	+	0.938142i	$0.387454\pi$
$788$	0	0
$789$	12.5391	0.446405
$790$	0	0
$791$	−7.63451	−0.271452
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−15.6266	−0.554218
$796$	0	0
$797$	−48.7507	−1.72684	−0.863419	−	0.504487i	$-0.831681\pi$
$797$	−48.7507	−1.72684	−0.863419	+	0.504487i	$0.831681\pi$
$798$	0	0
$799$	−25.4553	−0.900543
$800$	0	0
$801$	10.5147	0.371518
$802$	0	0
$803$	−7.93861	−0.280148
$804$	0	0
$805$	24.2515	0.854754
$806$	0	0
$807$	−19.7056	−0.693670
$808$	0	0
$809$	15.3089	0.538233	0.269116	−	0.963108i	$-0.413268\pi$
$809$	15.3089	0.538233	0.269116	+	0.963108i	$0.413268\pi$
$810$	0	0
$811$	41.0761	1.44238	0.721188	−	0.692740i	$-0.243597\pi$
$811$	41.0761	1.44238	0.721188	+	0.692740i	$0.243597\pi$
$812$	0	0
$813$	0.473734	0.0166146
$814$	0	0
$815$	−66.5086	−2.32970
$816$	0	0
$817$	66.3529	2.32139
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37.6787	−1.31500	−0.657498	−	0.753457i	$-0.728385\pi$
$821$	−37.6787	−1.31500	−0.657498	+	0.753457i	$0.728385\pi$
$822$	0	0
$823$	−23.9941	−0.836382	−0.418191	−	0.908359i	$-0.637336\pi$
$823$	−23.9941	−0.836382	−0.418191	+	0.908359i	$0.637336\pi$
$824$	0	0
$825$	−6.06429	−0.211132
$826$	0	0
$827$	−46.5086	−1.61726	−0.808632	−	0.588315i	$-0.799792\pi$
$827$	−46.5086	−1.61726	−0.808632	+	0.588315i	$0.799792\pi$
$828$	0	0
$829$	25.3851	0.881660	0.440830	−	0.897591i	$-0.354684\pi$
$829$	25.3851	0.881660	0.440830	+	0.897591i	$0.354684\pi$
$830$	0	0
$831$	−12.1959	−0.423070
$832$	0	0
$833$	6.22124	0.215553
$834$	0	0
$835$	94.7575	3.27922
$836$	0	0
$837$	−15.2103	−0.525745
$838$	0	0
$839$	−16.6317	−0.574191	−0.287095	−	0.957902i	$-0.592690\pi$
$839$	−16.6317	−0.574191	−0.287095	+	0.957902i	$0.592690\pi$
$840$	0	0
$841$	−26.9777	−0.930266
$842$	0	0
$843$	17.4873	0.602296
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.5516	0.362558
$848$	0	0
$849$	−20.9338	−0.718446
$850$	0	0
$851$	15.0419	0.515629
$852$	0	0
$853$	−51.5261	−1.76422	−0.882110	−	0.471043i	$-0.843878\pi$
$853$	−51.5261	−1.76422	−0.882110	+	0.471043i	$0.843878\pi$
$854$	0	0
$855$	69.2349	2.36778
$856$	0	0
$857$	−7.65890	−0.261623	−0.130812	−	0.991407i	$-0.541758\pi$
$857$	−7.65890	−0.261623	−0.130812	+	0.991407i	$0.541758\pi$
$858$	0	0
$859$	19.3413	0.659917	0.329958	−	0.943995i	$-0.392965\pi$
$859$	19.3413	0.659917	0.329958	+	0.943995i	$0.392965\pi$
$860$	0	0
$861$	−4.72509	−0.161031
$862$	0	0
$863$	4.99011	0.169865	0.0849326	−	0.996387i	$-0.472932\pi$
$863$	4.99011	0.169865	0.0849326	+	0.996387i	$0.472932\pi$
$864$	0	0
$865$	53.0886	1.80507
$866$	0	0
$867$	−14.5329	−0.493562
$868$	0	0
$869$	0.209525	0.00710764
$870$	0	0
$871$	0	0
$872$	0	0
$873$	18.9609	0.641729
$874$	0	0
$875$	−36.6940	−1.24048
$876$	0	0
$877$	−41.1201	−1.38853	−0.694263	−	0.719721i	$-0.744270\pi$
$877$	−41.1201	−1.38853	−0.694263	+	0.719721i	$0.744270\pi$
$878$	0	0
$879$	16.4881	0.556131
$880$	0	0
$881$	−56.8753	−1.91618	−0.958089	−	0.286470i	$-0.907518\pi$
$881$	−56.8753	−1.91618	−0.958089	+	0.286470i	$0.907518\pi$
$882$	0	0
$883$	−6.41928	−0.216026	−0.108013	−	0.994149i	$-0.534449\pi$
$883$	−6.41928	−0.216026	−0.108013	+	0.994149i	$0.534449\pi$
$884$	0	0
$885$	−20.9495	−0.704211
$886$	0	0
$887$	13.2340	0.444355	0.222178	−	0.975006i	$-0.428683\pi$
$887$	13.2340	0.444355	0.222178	+	0.975006i	$0.428683\pi$
$888$	0	0
$889$	−20.3432	−0.682289
$890$	0	0
$891$	3.45899	0.115880
$892$	0	0
$893$	25.7944	0.863175
$894$	0	0
$895$	−3.20084	−0.106992
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.81866	0.194063
$900$	0	0
$901$	33.7320	1.12378
$902$	0	0
$903$	−7.04778	−0.234535
$904$	0	0
$905$	−59.2799	−1.97053
$906$	0	0
$907$	−10.2662	−0.340884	−0.170442	−	0.985368i	$-0.554520\pi$
$907$	−10.2662	−0.340884	−0.170442	+	0.985368i	$0.554520\pi$
$908$	0	0
$909$	28.3090	0.938951
$910$	0	0
$911$	−21.0028	−0.695854	−0.347927	−	0.937522i	$-0.613114\pi$
$911$	−21.0028	−0.695854	−0.347927	+	0.937522i	$0.613114\pi$
$912$	0	0
$913$	2.03230	0.0672594
$914$	0	0
$915$	−6.57859	−0.217482
$916$	0	0
$917$	−21.6589	−0.715240
$918$	0	0
$919$	38.1855	1.25962	0.629811	−	0.776749i	$-0.283132\pi$
$919$	38.1855	1.25962	0.629811	+	0.776749i	$0.283132\pi$
$920$	0	0
$921$	21.1884	0.698181
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−36.1073	−1.18720
$926$	0	0
$927$	11.2906	0.370833
$928$	0	0
$929$	−38.9414	−1.27763	−0.638813	−	0.769362i	$-0.720574\pi$
$929$	−38.9414	−1.27763	−0.638813	+	0.769362i	$0.720574\pi$
$930$	0	0
$931$	−6.30411	−0.206609
$932$	0	0
$933$	12.2885	0.402308
$934$	0	0
$935$	17.9298	0.586368
$936$	0	0
$937$	51.7398	1.69026	0.845132	−	0.534557i	$-0.179522\pi$
$937$	51.7398	1.69026	0.845132	+	0.534557i	$0.179522\pi$
$938$	0	0
$939$	−5.83643	−0.190465
$940$	0	0
$941$	−13.1309	−0.428055	−0.214027	−	0.976828i	$-0.568658\pi$
$941$	−13.1309	−0.428055	−0.214027	+	0.976828i	$0.568658\pi$
$942$	0	0
$943$	39.7604	1.29478
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	11.0411	0.358787	0.179393	−	0.983777i	$-0.442587\pi$
$947$	11.0411	0.358787	0.179393	+	0.983777i	$0.442587\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−8.78969	−0.285025
$952$	0	0
$953$	−9.90752	−0.320936	−0.160468	−	0.987041i	$-0.551300\pi$
$953$	−9.90752	−0.320936	−0.160468	+	0.987041i	$0.551300\pi$
$954$	0	0
$955$	−75.7451	−2.45105
$956$	0	0
$957$	0.637608	0.0206110
$958$	0	0
$959$	8.71629	0.281464
$960$	0	0
$961$	−14.2582	−0.459942
$962$	0	0
$963$	42.3781	1.36562
$964$	0	0
$965$	−37.2770	−1.19999
$966$	0	0
$967$	−18.7339	−0.602441	−0.301221	−	0.953554i	$-0.597394\pi$
$967$	−18.7339	−0.602441	−0.301221	+	0.953554i	$0.597394\pi$
$968$	0	0
$969$	26.2613	0.843635
$970$	0	0
$971$	33.6824	1.08092	0.540460	−	0.841370i	$-0.318250\pi$
$971$	33.6824	1.08092	0.540460	+	0.841370i	$0.318250\pi$
$972$	0	0
$973$	−12.6461	−0.405415
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.0029	0.639950	0.319975	−	0.947426i	$-0.396326\pi$
$977$	20.0029	0.639950	0.319975	+	0.947426i	$0.396326\pi$
$978$	0	0
$979$	−2.75926	−0.0881864
$980$	0	0
$981$	−4.58368	−0.146346
$982$	0	0
$983$	−16.6334	−0.530524	−0.265262	−	0.964176i	$-0.585458\pi$
$983$	−16.6334	−0.530524	−0.265262	+	0.964176i	$0.585458\pi$
$984$	0	0
$985$	−93.0213	−2.96390
$986$	0	0
$987$	−2.73979	−0.0872084
$988$	0	0
$989$	59.3051	1.88579
$990$	0	0
$991$	−3.16198	−0.100444	−0.0502218	−	0.998738i	$-0.515993\pi$
$991$	−3.16198	−0.100444	−0.0502218	+	0.998738i	$0.515993\pi$
$992$	0	0
$993$	−6.18706	−0.196341
$994$	0	0
$995$	−18.9578	−0.601003
$996$	0	0
$997$	−19.5420	−0.618902	−0.309451	−	0.950915i	$-0.600145\pi$
$997$	−19.5420	−0.618902	−0.309451	+	0.950915i	$0.600145\pi$
$998$	0	0
$999$	−9.92392	−0.313979

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.ba.1.2		4
13.12	even	2		728.2.a.h.1.2	✓	4
39.38	odd	2		6552.2.a.bt.1.4		4
52.51	odd	2		1456.2.a.u.1.3		4
91.90	odd	2		5096.2.a.t.1.3		4
104.51	odd	2		5824.2.a.cf.1.2		4
104.77	even	2		5824.2.a.cc.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.h.1.2	✓	4	13.12	even	2
1456.2.a.u.1.3		4	52.51	odd	2
5096.2.a.t.1.3		4	91.90	odd	2
5824.2.a.cc.1.3		4	104.77	even	2
5824.2.a.cf.1.2		4	104.51	odd	2
6552.2.a.bt.1.4		4	39.38	odd	2
9464.2.a.ba.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q-0.669601 q^{3} +4.30411 q^{5} -1.00000 q^{7} -2.55163 q^{9} +0.669601 q^{11} -2.88203 q^{15} +6.22124 q^{17} -6.30411 q^{19} +0.669601 q^{21} -5.63451 q^{23} +13.5253 q^{25} +3.71738 q^{27} -1.42207 q^{29} -4.09167 q^{31} -0.448365 q^{33} -4.30411 q^{35} -2.66960 q^{37} -7.05658 q^{41} -10.5253 q^{43} -10.9825 q^{45} -4.09167 q^{47} +1.00000 q^{49} -4.16574 q^{51} +5.42207 q^{53} +2.88203 q^{55} +4.22124 q^{57} +7.26901 q^{59} +2.28262 q^{61} +2.55163 q^{63} -2.21243 q^{67} +3.77287 q^{69} -5.56044 q^{71} -11.8557 q^{73} -9.05658 q^{75} -0.669601 q^{77} +0.312910 q^{79} +5.16574 q^{81} +3.03509 q^{83} +26.7769 q^{85} +0.952222 q^{87} -4.12076 q^{89} +2.73979 q^{93} -27.1336 q^{95} -7.43088 q^{97} -1.70858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 4 q^{7} + 9 q^{9} - q^{11} + 4 q^{15} + 2 q^{17} - 8 q^{19} - q^{21} - 9 q^{23} + 14 q^{25} + 7 q^{27} - 4 q^{29} - 11 q^{31} - 21 q^{33} - 7 q^{37} - 13 q^{41} - 2 q^{43} - 12 q^{45} - 11 q^{47}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q + q^3 - 4 * q^7 + 9 * q^9 - q^11 + 4 * q^15 + 2 * q^17 - 8 * q^19 - q^21 - 9 * q^23 + 14 * q^25 + 7 * q^27 - 4 * q^29 - 11 * q^31 - 21 * q^33 - 7 * q^37 - 13 * q^41 - 2 * q^43 - 12 * q^45 - 11 * q^47 + 4 * q^49 - 28 * q^51 + 20 * q^53 - 4 * q^55 - 6 * q^57 + 2 * q^59 + 17 * q^61 - 9 * q^63 + 3 * q^67 - 27 * q^69 + 8 * q^71 - 11 * q^73 - 21 * q^75 + q^77 - 27 * q^79 + 32 * q^81 + 22 * q^83 + 8 * q^85 + 8 * q^87 - 10 * q^89 + 27 * q^93 - 34 * q^95 - 17 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

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Fields

Representations

Groups

Database

Properties

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