LMFDB - Embedded newform 3072.2.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3072,2,Mod(1,3072)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3072.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$3072 = 2^{10} \cdot 3$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3072.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,-4,0,-4,0,4,0,0,0,-8,0,-4,0,0] 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:	$24.5300435009$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 6x^{2} - 4x + 2$ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 48)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.334904$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.473626 q^{5} -4.55765 q^{7} +1.00000 q^{9} +3.49824 q^{11} +0.0840215 q^{13} +0.473626 q^{15} -3.61706 q^{17} +3.61706 q^{19} -4.55765 q^{21} -2.82843 q^{23} -4.77568 q^{25} +1.00000 q^{27} -7.30205 q^{29} +0.557647 q^{31} +3.49824 q^{33} -2.15862 q^{35} -6.20285 q^{37} +0.0840215 q^{39} -9.27391 q^{41} +2.27744 q^{43} +0.473626 q^{45} +2.82843 q^{47} +13.7721 q^{49} -3.61706 q^{51} -0.697947 q^{53} +1.65685 q^{55} +3.61706 q^{57} +5.65685 q^{59} +3.85970 q^{61} -4.55765 q^{63} +0.0397948 q^{65} -5.33962 q^{67} -2.82843 q^{69} +9.11529 q^{71} -0.541560 q^{73} -4.77568 q^{75} -15.9437 q^{77} -10.9937 q^{79} +1.00000 q^{81} -15.0496 q^{83} -1.71313 q^{85} -7.30205 q^{87} -14.6533 q^{89} -0.382941 q^{91} +0.557647 q^{93} +1.71313 q^{95} +4.31724 q^{97} +3.49824 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9} - 8 q^{13} - 4 q^{15} - 4 q^{21} + 4 q^{25} + 4 q^{27} - 12 q^{29} - 12 q^{31} - 16 q^{37} - 8 q^{39} - 4 q^{45} + 4 q^{49} - 20 q^{53} - 16 q^{55} - 16 q^{61}+ \cdots + 24 q^{95}+O(q^{100})$ 4 * q + 4 * q^3 - 4 * q^5 - 4 * q^7 + 4 * q^9 - 8 * q^13 - 4 * q^15 - 4 * q^21 + 4 * q^25 + 4 * q^27 - 12 * q^29 - 12 * q^31 - 16 * q^37 - 8 * q^39 - 4 * q^45 + 4 * q^49 - 20 * q^53 - 16 * q^55 - 16 * q^61 - 4 * q^63 - 8 * q^65 - 16 * q^67 + 8 * q^71 - 8 * q^73 + 4 * q^75 - 24 * q^77 - 12 * q^79 + 4 * q^81 - 24 * q^85 - 12 * q^87 - 8 * q^89 - 16 * q^91 - 12 * q^93 + 24 * q^95

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.473626	0.211812	0.105906	−	0.994376i	$-0.466226\pi$
$5$	0.473626	0.211812	0.105906	+	0.994376i	$0.466226\pi$
$6$	0	0
$7$	−4.55765	−1.72263	−0.861314	−	0.508072i	$-0.830358\pi$
$7$	−4.55765	−1.72263	−0.861314	+	0.508072i	$0.830358\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.49824	1.05476	0.527379	−	0.849630i	$-0.323175\pi$
$11$	3.49824	1.05476	0.527379	+	0.849630i	$0.323175\pi$
$12$	0	0
$13$	0.0840215	0.0233034	0.0116517	−	0.999932i	$-0.496291\pi$
$13$	0.0840215	0.0233034	0.0116517	+	0.999932i	$0.496291\pi$
$14$	0	0
$15$	0.473626	0.122290
$16$	0	0
$17$	−3.61706	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$17$	−3.61706	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$18$	0	0
$19$	3.61706	0.829810	0.414905	−	0.909865i	$-0.363815\pi$
$19$	3.61706	0.829810	0.414905	+	0.909865i	$0.363815\pi$
$20$	0	0
$21$	−4.55765	−0.994560
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	−4.77568	−0.955136
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.30205	−1.35596	−0.677979	−	0.735082i	$-0.737144\pi$
$29$	−7.30205	−1.35596	−0.677979	+	0.735082i	$0.737144\pi$
$30$	0	0
$31$	0.557647	0.100156	0.0500782	−	0.998745i	$-0.484053\pi$
$31$	0.557647	0.100156	0.0500782	+	0.998745i	$0.484053\pi$
$32$	0	0
$33$	3.49824	0.608965
$34$	0	0
$35$	−2.15862	−0.364873
$36$	0	0
$37$	−6.20285	−1.01974	−0.509871	−	0.860251i	$-0.670307\pi$
$37$	−6.20285	−1.01974	−0.509871	+	0.860251i	$0.670307\pi$
$38$	0	0
$39$	0.0840215	0.0134542
$40$	0	0
$41$	−9.27391	−1.44834	−0.724171	−	0.689620i	$-0.757777\pi$
$41$	−9.27391	−1.44834	−0.724171	+	0.689620i	$0.757777\pi$
$42$	0	0
$43$	2.27744	0.347307	0.173653	−	0.984807i	$-0.444443\pi$
$43$	2.27744	0.347307	0.173653	+	0.984807i	$0.444443\pi$
$44$	0	0
$45$	0.473626	0.0706040
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	13.7721	1.96745
$50$	0	0
$51$	−3.61706	−0.506490
$52$	0	0
$53$	−0.697947	−0.0958704	−0.0479352	−	0.998850i	$-0.515264\pi$
$53$	−0.697947	−0.0958704	−0.0479352	+	0.998850i	$0.515264\pi$
$54$	0	0
$55$	1.65685	0.223410
$56$	0	0
$57$	3.61706	0.479091
$58$	0	0
$59$	5.65685	0.736460	0.368230	−	0.929735i	$-0.379964\pi$
$59$	5.65685	0.736460	0.368230	+	0.929735i	$0.379964\pi$
$60$	0	0
$61$	3.85970	0.494184	0.247092	−	0.968992i	$-0.420525\pi$
$61$	3.85970	0.494184	0.247092	+	0.968992i	$0.420525\pi$
$62$	0	0
$63$	−4.55765	−0.574210
$64$	0	0
$65$	0.0397948	0.00493593
$66$	0	0
$67$	−5.33962	−0.652338	−0.326169	−	0.945311i	$-0.605758\pi$
$67$	−5.33962	−0.652338	−0.326169	+	0.945311i	$0.605758\pi$
$68$	0	0
$69$	−2.82843	−0.340503
$70$	0	0
$71$	9.11529	1.08179	0.540893	−	0.841091i	$-0.318086\pi$
$71$	9.11529	1.08179	0.540893	+	0.841091i	$0.318086\pi$
$72$	0	0
$73$	−0.541560	−0.0633848	−0.0316924	−	0.999498i	$-0.510090\pi$
$73$	−0.541560	−0.0633848	−0.0316924	+	0.999498i	$0.510090\pi$
$74$	0	0
$75$	−4.77568	−0.551448
$76$	0	0
$77$	−15.9437	−1.81696
$78$	0	0
$79$	−10.9937	−1.23689	−0.618445	−	0.785828i	$-0.712237\pi$
$79$	−10.9937	−1.23689	−0.618445	+	0.785828i	$0.712237\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.0496	−1.65191	−0.825954	−	0.563738i	$-0.809363\pi$
$83$	−15.0496	−1.65191	−0.825954	+	0.563738i	$0.809363\pi$
$84$	0	0
$85$	−1.71313	−0.185815
$86$	0	0
$87$	−7.30205	−0.782862
$88$	0	0
$89$	−14.6533	−1.55325	−0.776625	−	0.629964i	$-0.783070\pi$
$89$	−14.6533	−1.55325	−0.776625	+	0.629964i	$0.783070\pi$
$90$	0	0
$91$	−0.382941	−0.0401431
$92$	0	0
$93$	0.557647	0.0578253
$94$	0	0
$95$	1.71313	0.175764
$96$	0	0
$97$	4.31724	0.438349	0.219175	−	0.975686i	$-0.429664\pi$
$97$	4.31724	0.438349	0.219175	+	0.975686i	$0.429664\pi$
$98$	0	0
$99$	3.49824	0.351586
$100$	0	0
$101$	0.641669	0.0638484	0.0319242	−	0.999490i	$-0.489836\pi$
$101$	0.641669	0.0638484	0.0319242	+	0.999490i	$0.489836\pi$
$102$	0	0
$103$	1.33686	0.131724	0.0658622	−	0.997829i	$-0.479020\pi$
$103$	1.33686	0.131724	0.0658622	+	0.997829i	$0.479020\pi$
$104$	0	0
$105$	−2.15862	−0.210660
$106$	0	0
$107$	−8.57373	−0.828854	−0.414427	−	0.910083i	$-0.636018\pi$
$107$	−8.57373	−0.828854	−0.414427	+	0.910083i	$0.636018\pi$
$108$	0	0
$109$	−8.08402	−0.774309	−0.387154	−	0.922015i	$-0.626542\pi$
$109$	−8.08402	−0.774309	−0.387154	+	0.922015i	$0.626542\pi$
$110$	0	0
$111$	−6.20285	−0.588748
$112$	0	0
$113$	9.55136	0.898516	0.449258	−	0.893402i	$-0.351688\pi$
$113$	9.55136	0.898516	0.449258	+	0.893402i	$0.351688\pi$
$114$	0	0
$115$	−1.33962	−0.124920
$116$	0	0
$117$	0.0840215	0.00776779
$118$	0	0
$119$	16.4853	1.51120
$120$	0	0
$121$	1.23765	0.112514
$122$	0	0
$123$	−9.27391	−0.836201
$124$	0	0
$125$	−4.63001	−0.414121
$126$	0	0
$127$	−5.09921	−0.452481	−0.226241	−	0.974071i	$-0.572644\pi$
$127$	−5.09921	−0.452481	−0.226241	+	0.974071i	$0.572644\pi$
$128$	0	0
$129$	2.27744	0.200518
$130$	0	0
$131$	−2.99647	−0.261803	−0.130901	−	0.991395i	$-0.541787\pi$
$131$	−2.99647	−0.261803	−0.130901	+	0.991395i	$0.541787\pi$
$132$	0	0
$133$	−16.4853	−1.42946
$134$	0	0
$135$	0.473626	0.0407632
$136$	0	0
$137$	−3.37941	−0.288723	−0.144361	−	0.989525i	$-0.546113\pi$
$137$	−3.37941	−0.288723	−0.144361	+	0.989525i	$0.546113\pi$
$138$	0	0
$139$	−8.31724	−0.705459	−0.352729	−	0.935725i	$-0.614746\pi$
$139$	−8.31724	−0.705459	−0.352729	+	0.935725i	$0.614746\pi$
$140$	0	0
$141$	2.82843	0.238197
$142$	0	0
$143$	0.293927	0.0245794
$144$	0	0
$145$	−3.45844	−0.287208
$146$	0	0
$147$	13.7721	1.13591
$148$	0	0
$149$	−14.1305	−1.15761	−0.578807	−	0.815465i	$-0.696482\pi$
$149$	−14.1305	−1.15761	−0.578807	+	0.815465i	$0.696482\pi$
$150$	0	0
$151$	−9.97685	−0.811905	−0.405952	−	0.913894i	$-0.633060\pi$
$151$	−9.97685	−0.811905	−0.405952	+	0.913894i	$0.633060\pi$
$152$	0	0
$153$	−3.61706	−0.292422
$154$	0	0
$155$	0.264116	0.0212143
$156$	0	0
$157$	−22.8562	−1.82412	−0.912060	−	0.410056i	$-0.865509\pi$
$157$	−22.8562	−1.82412	−0.912060	+	0.410056i	$0.865509\pi$
$158$	0	0
$159$	−0.697947	−0.0553508
$160$	0	0
$161$	12.8910	1.01595
$162$	0	0
$163$	10.6135	0.831316	0.415658	−	0.909521i	$-0.363551\pi$
$163$	10.6135	0.831316	0.415658	+	0.909521i	$0.363551\pi$
$164$	0	0
$165$	1.65685	0.128986
$166$	0	0
$167$	−5.83822	−0.451775	−0.225888	−	0.974153i	$-0.572528\pi$
$167$	−5.83822	−0.451775	−0.225888	+	0.974153i	$0.572528\pi$
$168$	0	0
$169$	−12.9929	−0.999457
$170$	0	0
$171$	3.61706	0.276603
$172$	0	0
$173$	5.12695	0.389795	0.194897	−	0.980824i	$-0.437563\pi$
$173$	5.12695	0.389795	0.194897	+	0.980824i	$0.437563\pi$
$174$	0	0
$175$	21.7659	1.64534
$176$	0	0
$177$	5.65685	0.425195
$178$	0	0
$179$	13.1286	0.981279	0.490640	−	0.871363i	$-0.336763\pi$
$179$	13.1286	0.981279	0.490640	+	0.871363i	$0.336763\pi$
$180$	0	0
$181$	−15.3181	−1.13859	−0.569294	−	0.822134i	$-0.692783\pi$
$181$	−15.3181	−1.13859	−0.569294	+	0.822134i	$0.692783\pi$
$182$	0	0
$183$	3.85970	0.285317
$184$	0	0
$185$	−2.93783	−0.215993
$186$	0	0
$187$	−12.6533	−0.925303
$188$	0	0
$189$	−4.55765	−0.331520
$190$	0	0
$191$	8.63001	0.624446	0.312223	−	0.950009i	$-0.398926\pi$
$191$	8.63001	0.624446	0.312223	+	0.950009i	$0.398926\pi$
$192$	0	0
$193$	11.4514	0.824288	0.412144	−	0.911119i	$-0.364780\pi$
$193$	11.4514	0.824288	0.412144	+	0.911119i	$0.364780\pi$
$194$	0	0
$195$	0.0397948	0.00284976
$196$	0	0
$197$	−10.5925	−0.754681	−0.377340	−	0.926075i	$-0.623161\pi$
$197$	−10.5925	−0.754681	−0.377340	+	0.926075i	$0.623161\pi$
$198$	0	0
$199$	−3.68000	−0.260868	−0.130434	−	0.991457i	$-0.541637\pi$
$199$	−3.68000	−0.260868	−0.130434	+	0.991457i	$0.541637\pi$
$200$	0	0
$201$	−5.33962	−0.376627
$202$	0	0
$203$	33.2802	2.33581
$204$	0	0
$205$	−4.39236	−0.306776
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	12.6533	0.875249
$210$	0	0
$211$	14.3102	0.985153	0.492577	−	0.870269i	$-0.336055\pi$
$211$	14.3102	0.985153	0.492577	+	0.870269i	$0.336055\pi$
$212$	0	0
$213$	9.11529	0.624570
$214$	0	0
$215$	1.07866	0.0735637
$216$	0	0
$217$	−2.54156	−0.172532
$218$	0	0
$219$	−0.541560	−0.0365952
$220$	0	0
$221$	−0.303911	−0.0204433
$222$	0	0
$223$	4.86156	0.325554	0.162777	−	0.986663i	$-0.447955\pi$
$223$	4.86156	0.325554	0.162777	+	0.986663i	$0.447955\pi$
$224$	0	0
$225$	−4.77568	−0.318379
$226$	0	0
$227$	−15.0496	−0.998877	−0.499438	−	0.866349i	$-0.666460\pi$
$227$	−15.0496	−0.998877	−0.499438	+	0.866349i	$0.666460\pi$
$228$	0	0
$229$	28.5264	1.88507	0.942537	−	0.334101i	$-0.108433\pi$
$229$	28.5264	1.88507	0.942537	+	0.334101i	$0.108433\pi$
$230$	0	0
$231$	−15.9437	−1.04902
$232$	0	0
$233$	13.5702	0.889014	0.444507	−	0.895775i	$-0.353379\pi$
$233$	13.5702	0.889014	0.444507	+	0.895775i	$0.353379\pi$
$234$	0	0
$235$	1.33962	0.0873869
$236$	0	0
$237$	−10.9937	−0.714118
$238$	0	0
$239$	29.3629	1.89933	0.949665	−	0.313267i	$-0.101424\pi$
$239$	29.3629	1.89933	0.949665	+	0.313267i	$0.101424\pi$
$240$	0	0
$241$	−24.0063	−1.54638	−0.773190	−	0.634175i	$-0.781340\pi$
$241$	−24.0063	−1.54638	−0.773190	+	0.634175i	$0.781340\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.52284	0.416729
$246$	0	0
$247$	0.303911	0.0193374
$248$	0	0
$249$	−15.0496	−0.953729
$250$	0	0
$251$	22.2837	1.40654	0.703268	−	0.710925i	$-0.251724\pi$
$251$	22.2837	1.40654	0.703268	+	0.710925i	$0.251724\pi$
$252$	0	0
$253$	−9.89450	−0.622062
$254$	0	0
$255$	−1.71313	−0.107281
$256$	0	0
$257$	8.66038	0.540220	0.270110	−	0.962829i	$-0.412940\pi$
$257$	8.66038	0.540220	0.270110	+	0.962829i	$0.412940\pi$
$258$	0	0
$259$	28.2704	1.75664
$260$	0	0
$261$	−7.30205	−0.451986
$262$	0	0
$263$	13.3208	0.821394	0.410697	−	0.911772i	$-0.365285\pi$
$263$	13.3208	0.821394	0.410697	+	0.911772i	$0.365285\pi$
$264$	0	0
$265$	−0.330566	−0.0203065
$266$	0	0
$267$	−14.6533	−0.896769
$268$	0	0
$269$	16.5058	1.00638	0.503188	−	0.864177i	$-0.332160\pi$
$269$	16.5058	1.00638	0.503188	+	0.864177i	$0.332160\pi$
$270$	0	0
$271$	−21.9769	−1.33500	−0.667499	−	0.744610i	$-0.732635\pi$
$271$	−21.9769	−1.33500	−0.667499	+	0.744610i	$0.732635\pi$
$272$	0	0
$273$	−0.382941	−0.0231766
$274$	0	0
$275$	−16.7064	−1.00744
$276$	0	0
$277$	−15.4862	−0.930475	−0.465237	−	0.885186i	$-0.654031\pi$
$277$	−15.4862	−0.930475	−0.465237	+	0.885186i	$0.654031\pi$
$278$	0	0
$279$	0.557647	0.0333855
$280$	0	0
$281$	22.8910	1.36556	0.682780	−	0.730624i	$-0.260771\pi$
$281$	22.8910	1.36556	0.682780	+	0.730624i	$0.260771\pi$
$282$	0	0
$283$	6.34315	0.377061	0.188530	−	0.982067i	$-0.439628\pi$
$283$	6.34315	0.377061	0.188530	+	0.982067i	$0.439628\pi$
$284$	0	0
$285$	1.71313	0.101477
$286$	0	0
$287$	42.2672	2.49496
$288$	0	0
$289$	−3.91688	−0.230405
$290$	0	0
$291$	4.31724	0.253081
$292$	0	0
$293$	−30.5783	−1.78641	−0.893203	−	0.449654i	$-0.851547\pi$
$293$	−30.5783	−1.78641	−0.893203	+	0.449654i	$0.851547\pi$
$294$	0	0
$295$	2.67923	0.155991
$296$	0	0
$297$	3.49824	0.202988
$298$	0	0
$299$	−0.237649	−0.0137436
$300$	0	0
$301$	−10.3798	−0.598281
$302$	0	0
$303$	0.641669	0.0368629
$304$	0	0
$305$	1.82805	0.104674
$306$	0	0
$307$	−17.1286	−0.977582	−0.488791	−	0.872401i	$-0.662562\pi$
$307$	−17.1286	−0.977582	−0.488791	+	0.872401i	$0.662562\pi$
$308$	0	0
$309$	1.33686	0.0760511
$310$	0	0
$311$	26.8651	1.52338	0.761689	−	0.647943i	$-0.224370\pi$
$311$	26.8651	1.52338	0.761689	+	0.647943i	$0.224370\pi$
$312$	0	0
$313$	−19.6890	−1.11289	−0.556445	−	0.830885i	$-0.687835\pi$
$313$	−19.6890	−1.11289	−0.556445	+	0.830885i	$0.687835\pi$
$314$	0	0
$315$	−2.15862	−0.121624
$316$	0	0
$317$	30.1860	1.69541	0.847706	−	0.530466i	$-0.177983\pi$
$317$	30.1860	1.69541	0.847706	+	0.530466i	$0.177983\pi$
$318$	0	0
$319$	−25.5443	−1.43021
$320$	0	0
$321$	−8.57373	−0.478539
$322$	0	0
$323$	−13.0831	−0.727964
$324$	0	0
$325$	−0.401260	−0.0222579
$326$	0	0
$327$	−8.08402	−0.447047
$328$	0	0
$329$	−12.8910	−0.710702
$330$	0	0
$331$	−20.7784	−1.14209	−0.571043	−	0.820920i	$-0.693461\pi$
$331$	−20.7784	−1.14209	−0.571043	+	0.820920i	$0.693461\pi$
$332$	0	0
$333$	−6.20285	−0.339914
$334$	0	0
$335$	−2.52898	−0.138173
$336$	0	0
$337$	23.0098	1.25342	0.626712	−	0.779251i	$-0.284400\pi$
$337$	23.0098	1.25342	0.626712	+	0.779251i	$0.284400\pi$
$338$	0	0
$339$	9.55136	0.518759
$340$	0	0
$341$	1.95078	0.105641
$342$	0	0
$343$	−30.8651	−1.66656
$344$	0	0
$345$	−1.33962	−0.0721225
$346$	0	0
$347$	−15.4186	−0.827716	−0.413858	−	0.910341i	$-0.635819\pi$
$347$	−15.4186	−0.827716	−0.413858	+	0.910341i	$0.635819\pi$
$348$	0	0
$349$	28.3638	1.51828	0.759140	−	0.650927i	$-0.225620\pi$
$349$	28.3638	1.51828	0.759140	+	0.650927i	$0.225620\pi$
$350$	0	0
$351$	0.0840215	0.00448474
$352$	0	0
$353$	−12.2117	−0.649965	−0.324983	−	0.945720i	$-0.605358\pi$
$353$	−12.2117	−0.649965	−0.324983	+	0.945720i	$0.605358\pi$
$354$	0	0
$355$	4.31724	0.229135
$356$	0	0
$357$	16.4853	0.872494
$358$	0	0
$359$	33.4780	1.76690	0.883452	−	0.468522i	$-0.155214\pi$
$359$	33.4780	1.76690	0.883452	+	0.468522i	$0.155214\pi$
$360$	0	0
$361$	−5.91688	−0.311415
$362$	0	0
$363$	1.23765	0.0649597
$364$	0	0
$365$	−0.256497	−0.0134256
$366$	0	0
$367$	−0.702379	−0.0366639	−0.0183319	−	0.999832i	$-0.505836\pi$
$367$	−0.702379	−0.0366639	−0.0183319	+	0.999832i	$0.505836\pi$
$368$	0	0
$369$	−9.27391	−0.482781
$370$	0	0
$371$	3.18100	0.165149
$372$	0	0
$373$	26.8132	1.38834	0.694168	−	0.719813i	$-0.255773\pi$
$373$	26.8132	1.38834	0.694168	+	0.719813i	$0.255773\pi$
$374$	0	0
$375$	−4.63001	−0.239093
$376$	0	0
$377$	−0.613530	−0.0315984
$378$	0	0
$379$	−2.51509	−0.129192	−0.0645958	−	0.997912i	$-0.520576\pi$
$379$	−2.51509	−0.129192	−0.0645958	+	0.997912i	$0.520576\pi$
$380$	0	0
$381$	−5.09921	−0.261240
$382$	0	0
$383$	−25.4880	−1.30238	−0.651188	−	0.758916i	$-0.725729\pi$
$383$	−25.4880	−1.30238	−0.651188	+	0.758916i	$0.725729\pi$
$384$	0	0
$385$	−7.55136	−0.384853
$386$	0	0
$387$	2.27744	0.115769
$388$	0	0
$389$	16.5532	0.839281	0.419641	−	0.907690i	$-0.362156\pi$
$389$	16.5532	0.839281	0.419641	+	0.907690i	$0.362156\pi$
$390$	0	0
$391$	10.2306	0.517383
$392$	0	0
$393$	−2.99647	−0.151152
$394$	0	0
$395$	−5.20690	−0.261988
$396$	0	0
$397$	12.7936	0.642094	0.321047	−	0.947063i	$-0.395965\pi$
$397$	12.7936	0.642094	0.321047	+	0.947063i	$0.395965\pi$
$398$	0	0
$399$	−16.4853	−0.825296
$400$	0	0
$401$	18.0853	0.903137	0.451568	−	0.892237i	$-0.350865\pi$
$401$	18.0853	0.903137	0.451568	+	0.892237i	$0.350865\pi$
$402$	0	0
$403$	0.0468544	0.00233398
$404$	0	0
$405$	0.473626	0.0235347
$406$	0	0
$407$	−21.6990	−1.07558
$408$	0	0
$409$	25.2271	1.24740	0.623699	−	0.781665i	$-0.285629\pi$
$409$	25.2271	1.24740	0.623699	+	0.781665i	$0.285629\pi$
$410$	0	0
$411$	−3.37941	−0.166694
$412$	0	0
$413$	−25.7819	−1.26865
$414$	0	0
$415$	−7.12787	−0.349894
$416$	0	0
$417$	−8.31724	−0.407297
$418$	0	0
$419$	−10.2571	−0.501090	−0.250545	−	0.968105i	$-0.580610\pi$
$419$	−10.2571	−0.501090	−0.250545	+	0.968105i	$0.580610\pi$
$420$	0	0
$421$	−3.38775	−0.165109	−0.0825543	−	0.996587i	$-0.526308\pi$
$421$	−3.38775	−0.165109	−0.0825543	+	0.996587i	$0.526308\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	17.2739	0.837908
$426$	0	0
$427$	−17.5912	−0.851296
$428$	0	0
$429$	0.293927	0.0141909
$430$	0	0
$431$	−4.42454	−0.213123	−0.106561	−	0.994306i	$-0.533984\pi$
$431$	−4.42454	−0.213123	−0.106561	+	0.994306i	$0.533984\pi$
$432$	0	0
$433$	−7.31371	−0.351474	−0.175737	−	0.984437i	$-0.556231\pi$
$433$	−7.31371	−0.351474	−0.175737	+	0.984437i	$0.556231\pi$
$434$	0	0
$435$	−3.45844	−0.165820
$436$	0	0
$437$	−10.2306	−0.489395
$438$	0	0
$439$	29.6533	1.41527	0.707637	−	0.706576i	$-0.249761\pi$
$439$	29.6533	1.41527	0.707637	+	0.706576i	$0.249761\pi$
$440$	0	0
$441$	13.7721	0.655817
$442$	0	0
$443$	14.5743	0.692446	0.346223	−	0.938152i	$-0.387464\pi$
$443$	14.5743	0.692446	0.346223	+	0.938152i	$0.387464\pi$
$444$	0	0
$445$	−6.94019	−0.328997
$446$	0	0
$447$	−14.1305	−0.668349
$448$	0	0
$449$	−6.48844	−0.306208	−0.153104	−	0.988210i	$-0.548927\pi$
$449$	−6.48844	−0.306208	−0.153104	+	0.988210i	$0.548927\pi$
$450$	0	0
$451$	−32.4423	−1.52765
$452$	0	0
$453$	−9.97685	−0.468753
$454$	0	0
$455$	−0.181370	−0.00850278
$456$	0	0
$457$	−9.00353	−0.421167	−0.210584	−	0.977576i	$-0.567536\pi$
$457$	−9.00353	−0.421167	−0.210584	+	0.977576i	$0.567536\pi$
$458$	0	0
$459$	−3.61706	−0.168830
$460$	0	0
$461$	−20.6783	−0.963085	−0.481542	−	0.876423i	$-0.659923\pi$
$461$	−20.6783	−0.963085	−0.481542	+	0.876423i	$0.659923\pi$
$462$	0	0
$463$	−18.6435	−0.866437	−0.433219	−	0.901289i	$-0.642622\pi$
$463$	−18.6435	−0.866437	−0.433219	+	0.901289i	$0.642622\pi$
$464$	0	0
$465$	0.264116	0.0122481
$466$	0	0
$467$	−33.2535	−1.53879	−0.769395	−	0.638773i	$-0.779442\pi$
$467$	−33.2535	−1.53879	−0.769395	+	0.638773i	$0.779442\pi$
$468$	0	0
$469$	24.3361	1.12374
$470$	0	0
$471$	−22.8562	−1.05316
$472$	0	0
$473$	7.96703	0.366325
$474$	0	0
$475$	−17.2739	−0.792582
$476$	0	0
$477$	−0.697947	−0.0319568
$478$	0	0
$479$	1.08864	0.0497412	0.0248706	−	0.999691i	$-0.492083\pi$
$479$	1.08864	0.0497412	0.0248706	+	0.999691i	$0.492083\pi$
$480$	0	0
$481$	−0.521173	−0.0237634
$482$	0	0
$483$	12.8910	0.586560
$484$	0	0
$485$	2.04476	0.0928476
$486$	0	0
$487$	−35.3298	−1.60095	−0.800473	−	0.599369i	$-0.795418\pi$
$487$	−35.3298	−1.60095	−0.800473	+	0.599369i	$0.795418\pi$
$488$	0	0
$489$	10.6135	0.479960
$490$	0	0
$491$	18.2306	0.822735	0.411367	−	0.911470i	$-0.365051\pi$
$491$	18.2306	0.822735	0.411367	+	0.911470i	$0.365051\pi$
$492$	0	0
$493$	26.4120	1.18953
$494$	0	0
$495$	1.65685	0.0744701
$496$	0	0
$497$	−41.5443	−1.86352
$498$	0	0
$499$	20.3361	0.910368	0.455184	−	0.890397i	$-0.349573\pi$
$499$	20.3361	0.910368	0.455184	+	0.890397i	$0.349573\pi$
$500$	0	0
$501$	−5.83822	−0.260833
$502$	0	0
$503$	30.2969	1.35087	0.675435	−	0.737420i	$-0.263956\pi$
$503$	30.2969	1.35087	0.675435	+	0.737420i	$0.263956\pi$
$504$	0	0
$505$	0.303911	0.0135239
$506$	0	0
$507$	−12.9929	−0.577037
$508$	0	0
$509$	14.9660	0.663355	0.331677	−	0.943393i	$-0.392385\pi$
$509$	14.9660	0.663355	0.331677	+	0.943393i	$0.392385\pi$
$510$	0	0
$511$	2.46824	0.109188
$512$	0	0
$513$	3.61706	0.159697
$514$	0	0
$515$	0.633169	0.0279008
$516$	0	0
$517$	9.89450	0.435160
$518$	0	0
$519$	5.12695	0.225048
$520$	0	0
$521$	24.9049	1.09110	0.545551	−	0.838078i	$-0.316320\pi$
$521$	24.9049	1.09110	0.545551	+	0.838078i	$0.316320\pi$
$522$	0	0
$523$	18.2445	0.797775	0.398888	−	0.917000i	$-0.369396\pi$
$523$	18.2445	0.797775	0.398888	+	0.917000i	$0.369396\pi$
$524$	0	0
$525$	21.7659	0.949940
$526$	0	0
$527$	−2.01704	−0.0878638
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	5.65685	0.245487
$532$	0	0
$533$	−0.779208	−0.0337513
$534$	0	0
$535$	−4.06074	−0.175561
$536$	0	0
$537$	13.1286	0.566542
$538$	0	0
$539$	48.1782	2.07518
$540$	0	0
$541$	25.8471	1.11125	0.555627	−	0.831432i	$-0.312478\pi$
$541$	25.8471	1.11125	0.555627	+	0.831432i	$0.312478\pi$
$542$	0	0
$543$	−15.3181	−0.657364
$544$	0	0
$545$	−3.82880	−0.164008
$546$	0	0
$547$	−19.4249	−0.830549	−0.415275	−	0.909696i	$-0.636315\pi$
$547$	−19.4249	−0.830549	−0.415275	+	0.909696i	$0.636315\pi$
$548$	0	0
$549$	3.85970	0.164728
$550$	0	0
$551$	−26.4120	−1.12519
$552$	0	0
$553$	50.1055	2.13070
$554$	0	0
$555$	−2.93783	−0.124704
$556$	0	0
$557$	38.9652	1.65101	0.825504	−	0.564397i	$-0.190891\pi$
$557$	38.9652	1.65101	0.825504	+	0.564397i	$0.190891\pi$
$558$	0	0
$559$	0.191354	0.00809342
$560$	0	0
$561$	−12.6533	−0.534224
$562$	0	0
$563$	−28.1327	−1.18565	−0.592826	−	0.805330i	$-0.701988\pi$
$563$	−28.1327	−1.18565	−0.592826	+	0.805330i	$0.701988\pi$
$564$	0	0
$565$	4.52377	0.190316
$566$	0	0
$567$	−4.55765	−0.191403
$568$	0	0
$569$	−13.4849	−0.565317	−0.282658	−	0.959221i	$-0.591216\pi$
$569$	−13.4849	−0.565317	−0.282658	+	0.959221i	$0.591216\pi$
$570$	0	0
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	0	0
$573$	8.63001	0.360524
$574$	0	0
$575$	13.5077	0.563308
$576$	0	0
$577$	−11.6176	−0.483648	−0.241824	−	0.970320i	$-0.577746\pi$
$577$	−11.6176	−0.483648	−0.241824	+	0.970320i	$0.577746\pi$
$578$	0	0
$579$	11.4514	0.475903
$580$	0	0
$581$	68.5907	2.84562
$582$	0	0
$583$	−2.44158	−0.101120
$584$	0	0
$585$	0.0397948	0.00164531
$586$	0	0
$587$	24.0796	0.993871	0.496936	−	0.867787i	$-0.334459\pi$
$587$	24.0796	0.993871	0.496936	+	0.867787i	$0.334459\pi$
$588$	0	0
$589$	2.01704	0.0831108
$590$	0	0
$591$	−10.5925	−0.435715
$592$	0	0
$593$	−41.5372	−1.70573	−0.852865	−	0.522132i	$-0.825137\pi$
$593$	−41.5372	−1.70573	−0.852865	+	0.522132i	$0.825137\pi$
$594$	0	0
$595$	7.80785	0.320091
$596$	0	0
$597$	−3.68000	−0.150612
$598$	0	0
$599$	6.43160	0.262788	0.131394	−	0.991330i	$-0.458055\pi$
$599$	6.43160	0.262788	0.131394	+	0.991330i	$0.458055\pi$
$600$	0	0
$601$	−3.45844	−0.141073	−0.0705364	−	0.997509i	$-0.522471\pi$
$601$	−3.45844	−0.141073	−0.0705364	+	0.997509i	$0.522471\pi$
$602$	0	0
$603$	−5.33962	−0.217446
$604$	0	0
$605$	0.586182	0.0238317
$606$	0	0
$607$	30.1019	1.22180	0.610900	−	0.791708i	$-0.290808\pi$
$607$	30.1019	1.22180	0.610900	+	0.791708i	$0.290808\pi$
$608$	0	0
$609$	33.2802	1.34858
$610$	0	0
$611$	0.237649	0.00961424
$612$	0	0
$613$	−3.54246	−0.143079	−0.0715393	−	0.997438i	$-0.522791\pi$
$613$	−3.54246	−0.143079	−0.0715393	+	0.997438i	$0.522791\pi$
$614$	0	0
$615$	−4.39236	−0.177117
$616$	0	0
$617$	−22.9098	−0.922315	−0.461157	−	0.887318i	$-0.652566\pi$
$617$	−22.9098	−0.922315	−0.461157	+	0.887318i	$0.652566\pi$
$618$	0	0
$619$	40.4612	1.62627	0.813136	−	0.582074i	$-0.197758\pi$
$619$	40.4612	1.62627	0.813136	+	0.582074i	$0.197758\pi$
$620$	0	0
$621$	−2.82843	−0.113501
$622$	0	0
$623$	66.7847	2.67567
$624$	0	0
$625$	21.6855	0.867420
$626$	0	0
$627$	12.6533	0.505325
$628$	0	0
$629$	22.4361	0.894584
$630$	0	0
$631$	11.1851	0.445270	0.222635	−	0.974902i	$-0.428534\pi$
$631$	11.1851	0.445270	0.222635	+	0.974902i	$0.428534\pi$
$632$	0	0
$633$	14.3102	0.568779
$634$	0	0
$635$	−2.41512	−0.0958409
$636$	0	0
$637$	1.15716	0.0458482
$638$	0	0
$639$	9.11529	0.360595
$640$	0	0
$641$	−6.69312	−0.264362	−0.132181	−	0.991226i	$-0.542198\pi$
$641$	−6.69312	−0.264362	−0.132181	+	0.991226i	$0.542198\pi$
$642$	0	0
$643$	25.3724	1.00059	0.500294	−	0.865856i	$-0.333225\pi$
$643$	25.3724	1.00059	0.500294	+	0.865856i	$0.333225\pi$
$644$	0	0
$645$	1.07866	0.0424720
$646$	0	0
$647$	−6.72999	−0.264583	−0.132292	−	0.991211i	$-0.542234\pi$
$647$	−6.72999	−0.264583	−0.132292	+	0.991211i	$0.542234\pi$
$648$	0	0
$649$	19.7890	0.776786
$650$	0	0
$651$	−2.54156	−0.0996116
$652$	0	0
$653$	−37.0144	−1.44849	−0.724243	−	0.689545i	$-0.757810\pi$
$653$	−37.0144	−1.44849	−0.724243	+	0.689545i	$0.757810\pi$
$654$	0	0
$655$	−1.41921	−0.0554529
$656$	0	0
$657$	−0.541560	−0.0211283
$658$	0	0
$659$	19.7624	0.769832	0.384916	−	0.922952i	$-0.374230\pi$
$659$	19.7624	0.769832	0.384916	+	0.922952i	$0.374230\pi$
$660$	0	0
$661$	16.8632	0.655904	0.327952	−	0.944694i	$-0.393642\pi$
$661$	16.8632	0.655904	0.327952	+	0.944694i	$0.393642\pi$
$662$	0	0
$663$	−0.303911	−0.0118029
$664$	0	0
$665$	−7.80785	−0.302776
$666$	0	0
$667$	20.6533	0.799700
$668$	0	0
$669$	4.86156	0.187959
$670$	0	0
$671$	13.5021	0.521244
$672$	0	0
$673$	−37.3066	−1.43807	−0.719033	−	0.694976i	$-0.755415\pi$
$673$	−37.3066	−1.43807	−0.719033	+	0.694976i	$0.755415\pi$
$674$	0	0
$675$	−4.77568	−0.183816
$676$	0	0
$677$	−0.632805	−0.0243207	−0.0121603	−	0.999926i	$-0.503871\pi$
$677$	−0.632805	−0.0243207	−0.0121603	+	0.999926i	$0.503871\pi$
$678$	0	0
$679$	−19.6764	−0.755113
$680$	0	0
$681$	−15.0496	−0.576702
$682$	0	0
$683$	6.04606	0.231346	0.115673	−	0.993287i	$-0.463098\pi$
$683$	6.04606	0.231346	0.115673	+	0.993287i	$0.463098\pi$
$684$	0	0
$685$	−1.60058	−0.0611549
$686$	0	0
$687$	28.5264	1.08835
$688$	0	0
$689$	−0.0586426	−0.00223410
$690$	0	0
$691$	28.3955	1.08021	0.540107	−	0.841596i	$-0.318384\pi$
$691$	28.3955	1.08021	0.540107	+	0.841596i	$0.318384\pi$
$692$	0	0
$693$	−15.9437	−0.605652
$694$	0	0
$695$	−3.93926	−0.149425
$696$	0	0
$697$	33.5443	1.27058
$698$	0	0
$699$	13.5702	0.513272
$700$	0	0
$701$	−14.7738	−0.558000	−0.279000	−	0.960291i	$-0.590003\pi$
$701$	−14.7738	−0.558000	−0.279000	+	0.960291i	$0.590003\pi$
$702$	0	0
$703$	−22.4361	−0.846192
$704$	0	0
$705$	1.33962	0.0504529
$706$	0	0
$707$	−2.92450	−0.109987
$708$	0	0
$709$	−22.7569	−0.854655	−0.427327	−	0.904097i	$-0.640545\pi$
$709$	−22.7569	−0.854655	−0.427327	+	0.904097i	$0.640545\pi$
$710$	0	0
$711$	−10.9937	−0.412296
$712$	0	0
$713$	−1.57726	−0.0590690
$714$	0	0
$715$	0.139211	0.00520621
$716$	0	0
$717$	29.3629	1.09658
$718$	0	0
$719$	−30.9957	−1.15594	−0.577972	−	0.816057i	$-0.696156\pi$
$719$	−30.9957	−1.15594	−0.577972	+	0.816057i	$0.696156\pi$
$720$	0	0
$721$	−6.09292	−0.226912
$722$	0	0
$723$	−24.0063	−0.892803
$724$	0	0
$725$	34.8723	1.29512
$726$	0	0
$727$	−41.1117	−1.52475	−0.762375	−	0.647135i	$-0.775967\pi$
$727$	−41.1117	−1.52475	−0.762375	+	0.647135i	$0.775967\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.23765	−0.304680
$732$	0	0
$733$	0.206562	0.00762954	0.00381477	−	0.999993i	$-0.498786\pi$
$733$	0.206562	0.00762954	0.00381477	+	0.999993i	$0.498786\pi$
$734$	0	0
$735$	6.52284	0.240599
$736$	0	0
$737$	−18.6792	−0.688058
$738$	0	0
$739$	2.13215	0.0784325	0.0392162	−	0.999231i	$-0.487514\pi$
$739$	2.13215	0.0784325	0.0392162	+	0.999231i	$0.487514\pi$
$740$	0	0
$741$	0.303911	0.0111644
$742$	0	0
$743$	40.5175	1.48644	0.743221	−	0.669046i	$-0.233297\pi$
$743$	40.5175	1.48644	0.743221	+	0.669046i	$0.233297\pi$
$744$	0	0
$745$	−6.69256	−0.245196
$746$	0	0
$747$	−15.0496	−0.550636
$748$	0	0
$749$	39.0761	1.42781
$750$	0	0
$751$	−12.5843	−0.459208	−0.229604	−	0.973284i	$-0.573743\pi$
$751$	−12.5843	−0.459208	−0.229604	+	0.973284i	$0.573743\pi$
$752$	0	0
$753$	22.2837	0.812064
$754$	0	0
$755$	−4.72529	−0.171971
$756$	0	0
$757$	−10.6052	−0.385452	−0.192726	−	0.981253i	$-0.561733\pi$
$757$	−10.6052	−0.385452	−0.192726	+	0.981253i	$0.561733\pi$
$758$	0	0
$759$	−9.89450	−0.359148
$760$	0	0
$761$	42.8182	1.55216	0.776079	−	0.630635i	$-0.217206\pi$
$761$	42.8182	1.55216	0.776079	+	0.630635i	$0.217206\pi$
$762$	0	0
$763$	36.8441	1.33385
$764$	0	0
$765$	−1.71313	−0.0619384
$766$	0	0
$767$	0.475298	0.0171620
$768$	0	0
$769$	12.7455	0.459614	0.229807	−	0.973236i	$-0.426190\pi$
$769$	12.7455	0.459614	0.229807	+	0.973236i	$0.426190\pi$
$770$	0	0
$771$	8.66038	0.311896
$772$	0	0
$773$	32.3522	1.16363	0.581814	−	0.813322i	$-0.302343\pi$
$773$	32.3522	1.16363	0.581814	+	0.813322i	$0.302343\pi$
$774$	0	0
$775$	−2.66314	−0.0956630
$776$	0	0
$777$	28.2704	1.01419
$778$	0	0
$779$	−33.5443	−1.20185
$780$	0	0
$781$	31.8874	1.14102
$782$	0	0
$783$	−7.30205	−0.260954
$784$	0	0
$785$	−10.8253	−0.386370
$786$	0	0
$787$	7.36056	0.262376	0.131188	−	0.991358i	$-0.458121\pi$
$787$	7.36056	0.262376	0.131188	+	0.991358i	$0.458121\pi$
$788$	0	0
$789$	13.3208	0.474232
$790$	0	0
$791$	−43.5317	−1.54781
$792$	0	0
$793$	0.324298	0.0115162
$794$	0	0
$795$	−0.330566	−0.0117240
$796$	0	0
$797$	24.0627	0.852344	0.426172	−	0.904642i	$-0.359862\pi$
$797$	24.0627	0.852344	0.426172	+	0.904642i	$0.359862\pi$
$798$	0	0
$799$	−10.2306	−0.361932
$800$	0	0
$801$	−14.6533	−0.517750
$802$	0	0
$803$	−1.89450	−0.0668556
$804$	0	0
$805$	6.10550	0.215190
$806$	0	0
$807$	16.5058	0.581032
$808$	0	0
$809$	−7.83586	−0.275494	−0.137747	−	0.990467i	$-0.543986\pi$
$809$	−7.83586	−0.275494	−0.137747	+	0.990467i	$0.543986\pi$
$810$	0	0
$811$	−45.7351	−1.60598	−0.802988	−	0.595995i	$-0.796758\pi$
$811$	−45.7351	−1.60598	−0.802988	+	0.595995i	$0.796758\pi$
$812$	0	0
$813$	−21.9769	−0.770762
$814$	0	0
$815$	5.02684	0.176083
$816$	0	0
$817$	8.23765	0.288199
$818$	0	0
$819$	−0.382941	−0.0133810
$820$	0	0
$821$	27.3709	0.955250	0.477625	−	0.878564i	$-0.341498\pi$
$821$	27.3709	0.955250	0.477625	+	0.878564i	$0.341498\pi$
$822$	0	0
$823$	28.8560	1.00586	0.502929	−	0.864328i	$-0.332256\pi$
$823$	28.8560	1.00586	0.502929	+	0.864328i	$0.332256\pi$
$824$	0	0
$825$	−16.7064	−0.581644
$826$	0	0
$827$	−14.4227	−0.501528	−0.250764	−	0.968048i	$-0.580682\pi$
$827$	−14.4227	−0.501528	−0.250764	+	0.968048i	$0.580682\pi$
$828$	0	0
$829$	21.7497	0.755400	0.377700	−	0.925928i	$-0.376715\pi$
$829$	21.7497	0.755400	0.377700	+	0.925928i	$0.376715\pi$
$830$	0	0
$831$	−15.4862	−0.537210
$832$	0	0
$833$	−49.8147	−1.72598
$834$	0	0
$835$	−2.76513	−0.0956914
$836$	0	0
$837$	0.557647	0.0192751
$838$	0	0
$839$	44.4557	1.53478	0.767390	−	0.641181i	$-0.221555\pi$
$839$	44.4557	1.53478	0.767390	+	0.641181i	$0.221555\pi$
$840$	0	0
$841$	24.3200	0.838620
$842$	0	0
$843$	22.8910	0.788407
$844$	0	0
$845$	−6.15379	−0.211697
$846$	0	0
$847$	−5.64077	−0.193819
$848$	0	0
$849$	6.34315	0.217696
$850$	0	0
$851$	17.5443	0.601411
$852$	0	0
$853$	−16.5648	−0.567169	−0.283585	−	0.958947i	$-0.591524\pi$
$853$	−16.5648	−0.567169	−0.283585	+	0.958947i	$0.591524\pi$
$854$	0	0
$855$	1.71313	0.0585879
$856$	0	0
$857$	−19.0888	−0.652062	−0.326031	−	0.945359i	$-0.605711\pi$
$857$	−19.0888	−0.652062	−0.326031	+	0.945359i	$0.605711\pi$
$858$	0	0
$859$	−53.9272	−1.83997	−0.919987	−	0.391949i	$-0.871801\pi$
$859$	−53.9272	−1.83997	−0.919987	+	0.391949i	$0.871801\pi$
$860$	0	0
$861$	42.2672	1.44046
$862$	0	0
$863$	−3.64533	−0.124089	−0.0620443	−	0.998073i	$-0.519762\pi$
$863$	−3.64533	−0.124089	−0.0620443	+	0.998073i	$0.519762\pi$
$864$	0	0
$865$	2.42826	0.0825632
$866$	0	0
$867$	−3.91688	−0.133024
$868$	0	0
$869$	−38.4586	−1.30462
$870$	0	0
$871$	−0.448643	−0.0152017
$872$	0	0
$873$	4.31724	0.146116
$874$	0	0
$875$	21.1020	0.713377
$876$	0	0
$877$	−56.6481	−1.91287	−0.956435	−	0.291945i	$-0.905698\pi$
$877$	−56.6481	−1.91287	−0.956435	+	0.291945i	$0.905698\pi$
$878$	0	0
$879$	−30.5783	−1.03138
$880$	0	0
$881$	20.0118	0.674214	0.337107	−	0.941466i	$-0.390552\pi$
$881$	20.0118	0.674214	0.337107	+	0.941466i	$0.390552\pi$
$882$	0	0
$883$	−15.0292	−0.505773	−0.252887	−	0.967496i	$-0.581380\pi$
$883$	−15.0292	−0.505773	−0.252887	+	0.967496i	$0.581380\pi$
$884$	0	0
$885$	2.67923	0.0900614
$886$	0	0
$887$	26.1180	0.876958	0.438479	−	0.898742i	$-0.355517\pi$
$887$	26.1180	0.876958	0.438479	+	0.898742i	$0.355517\pi$
$888$	0	0
$889$	23.2404	0.779458
$890$	0	0
$891$	3.49824	0.117195
$892$	0	0
$893$	10.2306	0.342354
$894$	0	0
$895$	6.21805	0.207847
$896$	0	0
$897$	−0.237649	−0.00793486
$898$	0	0
$899$	−4.07197	−0.135808
$900$	0	0
$901$	2.52452	0.0841038
$902$	0	0
$903$	−10.3798	−0.345418
$904$	0	0
$905$	−7.25507	−0.241167
$906$	0	0
$907$	−51.2480	−1.70166	−0.850831	−	0.525439i	$-0.823901\pi$
$907$	−51.2480	−1.70166	−0.850831	+	0.525439i	$0.823901\pi$
$908$	0	0
$909$	0.641669	0.0212828
$910$	0	0
$911$	−21.0535	−0.697533	−0.348767	−	0.937210i	$-0.613399\pi$
$911$	−21.0535	−0.697533	−0.348767	+	0.937210i	$0.613399\pi$
$912$	0	0
$913$	−52.6470	−1.74236
$914$	0	0
$915$	1.82805	0.0604336
$916$	0	0
$917$	13.6569	0.450989
$918$	0	0
$919$	−17.8839	−0.589937	−0.294968	−	0.955507i	$-0.595309\pi$
$919$	−17.8839	−0.589937	−0.294968	+	0.955507i	$0.595309\pi$
$920$	0	0
$921$	−17.1286	−0.564407
$922$	0	0
$923$	0.765881	0.0252093
$924$	0	0
$925$	29.6228	0.973992
$926$	0	0
$927$	1.33686	0.0439081
$928$	0	0
$929$	10.2774	0.337192	0.168596	−	0.985685i	$-0.446077\pi$
$929$	10.2774	0.337192	0.168596	+	0.985685i	$0.446077\pi$
$930$	0	0
$931$	49.8147	1.63261
$932$	0	0
$933$	26.8651	0.879523
$934$	0	0
$935$	−5.99294	−0.195990
$936$	0	0
$937$	−13.5780	−0.443574	−0.221787	−	0.975095i	$-0.571189\pi$
$937$	−13.5780	−0.443574	−0.221787	+	0.975095i	$0.571189\pi$
$938$	0	0
$939$	−19.6890	−0.642527
$940$	0	0
$941$	5.59890	0.182519	0.0912595	−	0.995827i	$-0.470911\pi$
$941$	5.59890	0.182519	0.0912595	+	0.995827i	$0.470911\pi$
$942$	0	0
$943$	26.2306	0.854186
$944$	0	0
$945$	−2.15862	−0.0702199
$946$	0	0
$947$	−46.9106	−1.52439	−0.762194	−	0.647348i	$-0.775878\pi$
$947$	−46.9106	−1.52439	−0.762194	+	0.647348i	$0.775878\pi$
$948$	0	0
$949$	−0.0455027	−0.00147708
$950$	0	0
$951$	30.1860	0.978847
$952$	0	0
$953$	5.59115	0.181115	0.0905576	−	0.995891i	$-0.471135\pi$
$953$	5.59115	0.181115	0.0905576	+	0.995891i	$0.471135\pi$
$954$	0	0
$955$	4.08740	0.132265
$956$	0	0
$957$	−25.5443	−0.825730
$958$	0	0
$959$	15.4022	0.497362
$960$	0	0
$961$	−30.6890	−0.989969
$962$	0	0
$963$	−8.57373	−0.276285
$964$	0	0
$965$	5.42367	0.174594
$966$	0	0
$967$	30.7561	0.989048	0.494524	−	0.869164i	$-0.335342\pi$
$967$	30.7561	0.989048	0.494524	+	0.869164i	$0.335342\pi$
$968$	0	0
$969$	−13.0831	−0.420290
$970$	0	0
$971$	−11.3668	−0.364779	−0.182389	−	0.983226i	$-0.558383\pi$
$971$	−11.3668	−0.364779	−0.182389	+	0.983226i	$0.558383\pi$
$972$	0	0
$973$	37.9070	1.21524
$974$	0	0
$975$	−0.401260	−0.0128506
$976$	0	0
$977$	−22.8323	−0.730471	−0.365235	−	0.930915i	$-0.619012\pi$
$977$	−22.8323	−0.730471	−0.365235	+	0.930915i	$0.619012\pi$
$978$	0	0
$979$	−51.2608	−1.63830
$980$	0	0
$981$	−8.08402	−0.258103
$982$	0	0
$983$	46.3557	1.47852	0.739258	−	0.673422i	$-0.235176\pi$
$983$	46.3557	1.47852	0.739258	+	0.673422i	$0.235176\pi$
$984$	0	0
$985$	−5.01686	−0.159850
$986$	0	0
$987$	−12.8910	−0.410324
$988$	0	0
$989$	−6.44158	−0.204830
$990$	0	0
$991$	−3.43683	−0.109175	−0.0545873	−	0.998509i	$-0.517384\pi$
$991$	−3.43683	−0.109175	−0.0545873	+	0.998509i	$0.517384\pi$
$992$	0	0
$993$	−20.7784	−0.659383
$994$	0	0
$995$	−1.74294	−0.0552550
$996$	0	0
$997$	31.0320	0.982794	0.491397	−	0.870936i	$-0.336486\pi$
$997$	31.0320	0.982794	0.491397	+	0.870936i	$0.336486\pi$
$998$	0	0
$999$	−6.20285	−0.196249

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.o.1.3		4
3.2	odd	2		9216.2.a.bn.1.2		4
4.3	odd	2		3072.2.a.i.1.3		4
8.3	odd	2		3072.2.a.t.1.2		4
8.5	even	2		3072.2.a.n.1.2		4
12.11	even	2		9216.2.a.bo.1.2		4
16.3	odd	4		3072.2.d.f.1537.2		8
16.5	even	4		3072.2.d.i.1537.3		8
16.11	odd	4		3072.2.d.f.1537.7		8
16.13	even	4		3072.2.d.i.1537.6		8
24.5	odd	2		9216.2.a.x.1.3		4
24.11	even	2		9216.2.a.y.1.3		4
32.3	odd	8		48.2.j.a.13.4	✓	8
32.5	even	8		384.2.j.a.97.2		8
32.11	odd	8		48.2.j.a.37.4	yes	8
32.13	even	8		384.2.j.a.289.2		8
32.19	odd	8		384.2.j.b.289.4		8
32.21	even	8		192.2.j.a.49.3		8
32.27	odd	8		384.2.j.b.97.4		8
32.29	even	8		192.2.j.a.145.3		8
96.5	odd	8		1152.2.k.f.865.2		8
96.11	even	8		144.2.k.b.37.1		8
96.29	odd	8		576.2.k.b.145.3		8
96.35	even	8		144.2.k.b.109.1		8
96.53	odd	8		576.2.k.b.433.3		8
96.59	even	8		1152.2.k.c.865.2		8
96.77	odd	8		1152.2.k.f.289.2		8
96.83	even	8		1152.2.k.c.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.4	✓	8	32.3	odd	8
48.2.j.a.37.4	yes	8	32.11	odd	8
144.2.k.b.37.1		8	96.11	even	8
144.2.k.b.109.1		8	96.35	even	8
192.2.j.a.49.3		8	32.21	even	8
192.2.j.a.145.3		8	32.29	even	8
384.2.j.a.97.2		8	32.5	even	8
384.2.j.a.289.2		8	32.13	even	8
384.2.j.b.97.4		8	32.27	odd	8
384.2.j.b.289.4		8	32.19	odd	8
576.2.k.b.145.3		8	96.29	odd	8
576.2.k.b.433.3		8	96.53	odd	8
1152.2.k.c.289.2		8	96.83	even	8
1152.2.k.c.865.2		8	96.59	even	8
1152.2.k.f.289.2		8	96.77	odd	8
1152.2.k.f.865.2		8	96.5	odd	8
3072.2.a.i.1.3		4	4.3	odd	2
3072.2.a.n.1.2		4	8.5	even	2
3072.2.a.o.1.3		4	1.1	even	1	trivial
3072.2.a.t.1.2		4	8.3	odd	2
3072.2.d.f.1537.2		8	16.3	odd	4
3072.2.d.f.1537.7		8	16.11	odd	4
3072.2.d.i.1537.3		8	16.5	even	4
3072.2.d.i.1537.6		8	16.13	even	4
9216.2.a.x.1.3		4	24.5	odd	2
9216.2.a.y.1.3		4	24.11	even	2
9216.2.a.bn.1.2		4	3.2	odd	2
9216.2.a.bo.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3072.2.a.o.1.3

Level	$3072$
Weight	$2$
Character	3072.1
Self dual	yes
Analytic conductor	$24.530$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$1$