LMFDB - Embedded newform 729.2.a.c.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,2,Mod(1,729)] 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("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.82109430735$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{36})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 9x^{2} - 3 $ x^6 - 6x^4 + 9x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.96962$ of defining polynomial
Character	$\chi$	$=$	729.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.96962 q^{2} +1.87939 q^{4} -3.70167 q^{5} -2.34730 q^{7} -0.237565 q^{8} -7.29086 q^{10} +2.17853 q^{11} -4.71688 q^{13} -4.62327 q^{14} -4.22668 q^{16} +2.93512 q^{17} -6.22668 q^{19} -6.95686 q^{20} +4.29086 q^{22} -0.519030 q^{23} +8.70233 q^{25} -9.29044 q^{26} -4.41147 q^{28} +3.49276 q^{29} -4.30541 q^{31} -7.84981 q^{32} +5.78106 q^{34} +8.68891 q^{35} +2.41147 q^{37} -12.2642 q^{38} +0.879385 q^{40} -2.49860 q^{41} +1.06418 q^{43} +4.09429 q^{44} -1.02229 q^{46} +0.237565 q^{47} -1.49020 q^{49} +17.1403 q^{50} -8.86484 q^{52} -4.66717 q^{53} -8.06418 q^{55} +0.557635 q^{56} +6.87939 q^{58} +13.3122 q^{59} -3.67499 q^{61} -8.48000 q^{62} -7.00774 q^{64} +17.4603 q^{65} +14.2986 q^{67} +5.51622 q^{68} +17.1138 q^{70} -1.20307 q^{71} -4.68004 q^{73} +4.74968 q^{74} -11.7023 q^{76} -5.11365 q^{77} -12.8007 q^{79} +15.6458 q^{80} -4.92127 q^{82} +11.3040 q^{83} -10.8648 q^{85} +2.09602 q^{86} -0.517541 q^{88} -0.699287 q^{89} +11.0719 q^{91} -0.975457 q^{92} +0.467911 q^{94} +23.0491 q^{95} -7.08647 q^{97} -2.93512 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) $ 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

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 $

$2$	1.96962	1.39273	0.696364	−	0.717689i	$-0.254800\pi$
$2$	1.96962	1.39273	0.696364	+	0.717689i	$0.254800\pi$
$3$	0	0
$3$	0	0
$4$	1.87939	0.939693
$5$	−3.70167	−1.65544	−0.827718	−	0.561145i	$-0.810361\pi$
$5$	−3.70167	−1.65544	−0.827718	+	0.561145i	$0.810361\pi$
$6$	0	0
$7$	−2.34730	−0.887195	−0.443597	−	0.896226i	$-0.646298\pi$
$7$	−2.34730	−0.887195	−0.443597	+	0.896226i	$0.646298\pi$
$8$	−0.237565	−0.0839918
$9$	0	0
$10$	−7.29086	−2.30557
$11$	2.17853	0.656850	0.328425	−	0.944530i	$-0.393482\pi$
$11$	2.17853	0.656850	0.328425	+	0.944530i	$0.393482\pi$
$12$	0	0
$13$	−4.71688	−1.30823	−0.654114	−	0.756396i	$-0.726958\pi$
$13$	−4.71688	−1.30823	−0.654114	+	0.756396i	$0.726958\pi$
$14$	−4.62327	−1.23562
$15$	0	0
$16$	−4.22668	−1.05667
$17$	2.93512	0.711871	0.355936	−	0.934510i	$-0.384162\pi$
$17$	2.93512	0.711871	0.355936	+	0.934510i	$0.384162\pi$
$18$	0	0
$19$	−6.22668	−1.42850	−0.714249	−	0.699891i	$-0.753232\pi$
$19$	−6.22668	−1.42850	−0.714249	+	0.699891i	$0.753232\pi$
$20$	−6.95686	−1.55560
$21$	0	0
$22$	4.29086	0.914814
$23$	−0.519030	−0.108225	−0.0541126	−	0.998535i	$-0.517233\pi$
$23$	−0.519030	−0.108225	−0.0541126	+	0.998535i	$0.517233\pi$
$24$	0	0
$25$	8.70233	1.74047
$26$	−9.29044	−1.82201
$27$	0	0
$28$	−4.41147	−0.833690
$29$	3.49276	0.648588	0.324294	−	0.945956i	$-0.394873\pi$
$29$	3.49276	0.648588	0.324294	+	0.945956i	$0.394873\pi$
$30$	0	0
$31$	−4.30541	−0.773274	−0.386637	−	0.922232i	$-0.626363\pi$
$31$	−4.30541	−0.773274	−0.386637	+	0.922232i	$0.626363\pi$
$32$	−7.84981	−1.38766
$33$	0	0
$34$	5.78106	0.991443
$35$	8.68891	1.46869
$36$	0	0
$37$	2.41147	0.396444	0.198222	−	0.980157i	$-0.436483\pi$
$37$	2.41147	0.396444	0.198222	+	0.980157i	$0.436483\pi$
$38$	−12.2642	−1.98951
$39$	0	0
$40$	0.879385	0.139043
$41$	−2.49860	−0.390215	−0.195108	−	0.980782i	$-0.562506\pi$
$41$	−2.49860	−0.390215	−0.195108	+	0.980782i	$0.562506\pi$
$42$	0	0
$43$	1.06418	0.162286	0.0811428	−	0.996702i	$-0.474143\pi$
$43$	1.06418	0.162286	0.0811428	+	0.996702i	$0.474143\pi$
$44$	4.09429	0.617237
$45$	0	0
$46$	−1.02229	−0.150728
$47$	0.237565	0.0346524	0.0173262	−	0.999850i	$-0.494485\pi$
$47$	0.237565	0.0346524	0.0173262	+	0.999850i	$0.494485\pi$
$48$	0	0
$49$	−1.49020	−0.212886
$50$	17.1403	2.42400
$51$	0	0
$52$	−8.86484	−1.22933
$53$	−4.66717	−0.641085	−0.320543	−	0.947234i	$-0.603865\pi$
$53$	−4.66717	−0.641085	−0.320543	+	0.947234i	$0.603865\pi$
$54$	0	0
$55$	−8.06418	−1.08737
$56$	0.557635	0.0745171
$57$	0	0
$58$	6.87939	0.903308
$59$	13.3122	1.73310	0.866549	−	0.499092i	$-0.166333\pi$
$59$	13.3122	1.73310	0.866549	+	0.499092i	$0.166333\pi$
$60$	0	0
$61$	−3.67499	−0.470535	−0.235267	−	0.971931i	$-0.575597\pi$
$61$	−3.67499	−0.470535	−0.235267	+	0.971931i	$0.575597\pi$
$62$	−8.48000	−1.07696
$63$	0	0
$64$	−7.00774	−0.875968
$65$	17.4603	2.16569
$66$	0	0
$67$	14.2986	1.74685	0.873426	−	0.486957i	$-0.161893\pi$
$67$	14.2986	1.74685	0.873426	+	0.486957i	$0.161893\pi$
$68$	5.51622	0.668940
$69$	0	0
$70$	17.1138	2.04549
$71$	−1.20307	−0.142778	−0.0713891	−	0.997449i	$-0.522743\pi$
$71$	−1.20307	−0.142778	−0.0713891	+	0.997449i	$0.522743\pi$
$72$	0	0
$73$	−4.68004	−0.547758	−0.273879	−	0.961764i	$-0.588307\pi$
$73$	−4.68004	−0.547758	−0.273879	+	0.961764i	$0.588307\pi$
$74$	4.74968	0.552139
$75$	0	0
$76$	−11.7023	−1.34235
$77$	−5.11365	−0.582754
$78$	0	0
$79$	−12.8007	−1.44019	−0.720093	−	0.693877i	$-0.755901\pi$
$79$	−12.8007	−1.44019	−0.720093	+	0.693877i	$0.755901\pi$
$80$	15.6458	1.74925
$81$	0	0
$82$	−4.92127	−0.543464
$83$	11.3040	1.24077	0.620385	−	0.784297i	$-0.286976\pi$
$83$	11.3040	1.24077	0.620385	+	0.784297i	$0.286976\pi$
$84$	0	0
$85$	−10.8648	−1.17846
$86$	2.09602	0.226020
$87$	0	0
$88$	−0.517541	−0.0551701
$89$	−0.699287	−0.0741242	−0.0370621	−	0.999313i	$-0.511800\pi$
$89$	−0.699287	−0.0741242	−0.0370621	+	0.999313i	$0.511800\pi$
$90$	0	0
$91$	11.0719	1.16065
$92$	−0.975457	−0.101698
$93$	0	0
$94$	0.467911	0.0482613
$95$	23.0491	2.36479
$96$	0	0
$97$	−7.08647	−0.719522	−0.359761	−	0.933045i	$-0.617142\pi$
$97$	−7.08647	−0.719522	−0.359761	+	0.933045i	$0.617142\pi$
$98$	−2.93512	−0.296492
$99$	0	0
$100$	16.3550	1.63550
$101$	−4.67712	−0.465391	−0.232696	−	0.972550i	$-0.574755\pi$
$101$	−4.67712	−0.465391	−0.232696	+	0.972550i	$0.574755\pi$
$102$	0	0
$103$	−13.6236	−1.34237	−0.671187	−	0.741288i	$-0.734215\pi$
$103$	−13.6236	−1.34237	−0.671187	+	0.741288i	$0.734215\pi$
$104$	1.12056	0.109880
$105$	0	0
$106$	−9.19253	−0.892858
$107$	−11.6340	−1.12470	−0.562350	−	0.826900i	$-0.690102\pi$
$107$	−11.6340	−1.12470	−0.562350	+	0.826900i	$0.690102\pi$
$108$	0	0
$109$	14.6040	1.39881	0.699405	−	0.714725i	$-0.253448\pi$
$109$	14.6040	1.39881	0.699405	+	0.714725i	$0.253448\pi$
$110$	−15.8833	−1.51442
$111$	0	0
$112$	9.92127	0.937472
$113$	−4.69583	−0.441746	−0.220873	−	0.975303i	$-0.570891\pi$
$113$	−4.69583	−0.441746	−0.220873	+	0.975303i	$0.570891\pi$
$114$	0	0
$115$	1.92127	0.179160
$116$	6.56423	0.609474
$117$	0	0
$118$	26.2199	2.41374
$119$	−6.88960	−0.631568
$120$	0	0
$121$	−6.25402	−0.568548
$122$	−7.23832	−0.655327
$123$	0	0
$124$	−8.09152	−0.726640
$125$	−13.7048	−1.22579
$126$	0	0
$127$	6.09152	0.540535	0.270267	−	0.962785i	$-0.412888\pi$
$127$	6.09152	0.540535	0.270267	+	0.962785i	$0.412888\pi$
$128$	1.89706	0.167678
$129$	0	0
$130$	34.3901	3.01621
$131$	−10.3484	−0.904144	−0.452072	−	0.891981i	$-0.649315\pi$
$131$	−10.3484	−0.904144	−0.452072	+	0.891981i	$0.649315\pi$
$132$	0	0
$133$	14.6159	1.26736
$134$	28.1627	2.43289
$135$	0	0
$136$	−0.697281	−0.0597914
$137$	−19.0847	−1.63051	−0.815257	−	0.579100i	$-0.803404\pi$
$137$	−19.0847	−1.63051	−0.815257	+	0.579100i	$0.803404\pi$
$138$	0	0
$139$	−23.3037	−1.97659	−0.988295	−	0.152555i	$-0.951250\pi$
$139$	−23.3037	−1.97659	−0.988295	+	0.152555i	$0.951250\pi$
$140$	16.3298	1.38012
$141$	0	0
$142$	−2.36959	−0.198851
$143$	−10.2759	−0.859310
$144$	0	0
$145$	−12.9290	−1.07370
$146$	−9.21789	−0.762878
$147$	0	0
$148$	4.53209	0.372535
$149$	−15.5060	−1.27030	−0.635149	−	0.772390i	$-0.719061\pi$
$149$	−15.5060	−1.27030	−0.635149	+	0.772390i	$0.719061\pi$
$150$	0	0
$151$	−5.90167	−0.480271	−0.240136	−	0.970739i	$-0.577192\pi$
$151$	−5.90167	−0.480271	−0.240136	+	0.970739i	$0.577192\pi$
$152$	1.47924	0.119982
$153$	0	0
$154$	−10.0719	−0.811618
$155$	15.9372	1.28011
$156$	0	0
$157$	−1.21213	−0.0967388	−0.0483694	−	0.998830i	$-0.515402\pi$
$157$	−1.21213	−0.0967388	−0.0483694	+	0.998830i	$0.515402\pi$
$158$	−25.2124	−2.00579
$159$	0	0
$160$	29.0574	2.29719
$161$	1.21832	0.0960168
$162$	0	0
$163$	2.77332	0.217223	0.108612	−	0.994084i	$-0.465360\pi$
$163$	2.77332	0.217223	0.108612	+	0.994084i	$0.465360\pi$
$164$	−4.69583	−0.366682
$165$	0	0
$166$	22.2645	1.72806
$167$	3.82807	0.296225	0.148113	−	0.988971i	$-0.452680\pi$
$167$	3.82807	0.296225	0.148113	+	0.988971i	$0.452680\pi$
$168$	0	0
$169$	9.24897	0.711459
$170$	−21.3996	−1.64127
$171$	0	0
$172$	2.00000	0.152499
$173$	7.02941	0.534436	0.267218	−	0.963636i	$-0.413896\pi$
$173$	7.02941	0.534436	0.267218	+	0.963636i	$0.413896\pi$
$174$	0	0
$175$	−20.4270	−1.54413
$176$	−9.20794	−0.694074
$177$	0	0
$178$	−1.37733	−0.103235
$179$	14.3854	1.07521	0.537607	−	0.843195i	$-0.319328\pi$
$179$	14.3854	1.07521	0.537607	+	0.843195i	$0.319328\pi$
$180$	0	0
$181$	13.2003	0.981169	0.490584	−	0.871394i	$-0.336783\pi$
$181$	13.2003	0.981169	0.490584	+	0.871394i	$0.336783\pi$
$182$	21.8074	1.61647
$183$	0	0
$184$	0.123303	0.00909003
$185$	−8.92647	−0.656287
$186$	0	0
$187$	6.39424	0.467593
$188$	0.446476	0.0325626
$189$	0	0
$190$	45.3979	3.29351
$191$	−13.4233	−0.971279	−0.485639	−	0.874159i	$-0.661413\pi$
$191$	−13.4233	−0.971279	−0.485639	+	0.874159i	$0.661413\pi$
$192$	0	0
$193$	15.0051	1.08009	0.540044	−	0.841637i	$-0.318408\pi$
$193$	15.0051	1.08009	0.540044	+	0.841637i	$0.318408\pi$
$194$	−13.9576	−1.00210
$195$	0	0
$196$	−2.80066	−0.200047
$197$	22.3212	1.59032	0.795158	−	0.606402i	$-0.207388\pi$
$197$	22.3212	1.59032	0.795158	+	0.606402i	$0.207388\pi$
$198$	0	0
$199$	−9.10101	−0.645154	−0.322577	−	0.946543i	$-0.604549\pi$
$199$	−9.10101	−0.645154	−0.322577	+	0.946543i	$0.604549\pi$
$200$	−2.06737	−0.146185
$201$	0	0
$202$	−9.21213	−0.648163
$203$	−8.19853	−0.575424
$204$	0	0
$205$	9.24897	0.645976
$206$	−26.8333	−1.86956
$207$	0	0
$208$	19.9368	1.38237
$209$	−13.5650	−0.938310
$210$	0	0
$211$	−5.97771	−0.411523	−0.205761	−	0.978602i	$-0.565967\pi$
$211$	−5.97771	−0.411523	−0.205761	+	0.978602i	$0.565967\pi$
$212$	−8.77141	−0.602423
$213$	0	0
$214$	−22.9145	−1.56640
$215$	−3.93923	−0.268653
$216$	0	0
$217$	10.1061	0.686045
$218$	28.7643	1.94816
$219$	0	0
$220$	−15.1557	−1.02180
$221$	−13.8446	−0.931290
$222$	0	0
$223$	−8.90167	−0.596100	−0.298050	−	0.954550i	$-0.596336\pi$
$223$	−8.90167	−0.596100	−0.298050	+	0.954550i	$0.596336\pi$
$224$	18.4258	1.23113
$225$	0	0
$226$	−9.24897	−0.615232
$227$	10.6685	0.708092	0.354046	−	0.935228i	$-0.384806\pi$
$227$	10.6685	0.708092	0.354046	+	0.935228i	$0.384806\pi$
$228$	0	0
$229$	−8.27126	−0.546580	−0.273290	−	0.961932i	$-0.588112\pi$
$229$	−8.27126	−0.546580	−0.273290	+	0.961932i	$0.588112\pi$
$230$	3.78417	0.249521
$231$	0	0
$232$	−0.829755	−0.0544761
$233$	−12.7393	−0.834579	−0.417290	−	0.908774i	$-0.637020\pi$
$233$	−12.7393	−0.834579	−0.417290	+	0.908774i	$0.637020\pi$
$234$	0	0
$235$	−0.879385	−0.0573648
$236$	25.0187	1.62858
$237$	0	0
$238$	−13.5699	−0.879603
$239$	−15.0156	−0.971277	−0.485638	−	0.874160i	$-0.661413\pi$
$239$	−15.0156	−0.971277	−0.485638	+	0.874160i	$0.661413\pi$
$240$	0	0
$241$	0.795607	0.0512496	0.0256248	−	0.999672i	$-0.491842\pi$
$241$	0.795607	0.0512496	0.0256248	+	0.999672i	$0.491842\pi$
$242$	−12.3180	−0.791832
$243$	0	0
$244$	−6.90673	−0.442158
$245$	5.51622	0.352419
$246$	0	0
$247$	29.3705	1.86880
$248$	1.02281	0.0649487
$249$	0	0
$250$	−26.9932	−1.70720
$251$	−8.31499	−0.524837	−0.262419	−	0.964954i	$-0.584520\pi$
$251$	−8.31499	−0.524837	−0.262419	+	0.964954i	$0.584520\pi$
$252$	0	0
$253$	−1.13072	−0.0710877
$254$	11.9980	0.752818
$255$	0	0
$256$	17.7520	1.10950
$257$	25.6202	1.59815	0.799074	−	0.601233i	$-0.205324\pi$
$257$	25.6202	1.59815	0.799074	+	0.601233i	$0.205324\pi$
$258$	0	0
$259$	−5.66044	−0.351723
$260$	32.8147	2.03508
$261$	0	0
$262$	−20.3824	−1.25923
$263$	−28.0650	−1.73056	−0.865281	−	0.501287i	$-0.832860\pi$
$263$	−28.0650	−1.73056	−0.865281	+	0.501287i	$0.832860\pi$
$264$	0	0
$265$	17.2763	1.06128
$266$	28.7876	1.76508
$267$	0	0
$268$	26.8726	1.64150
$269$	30.1710	1.83956	0.919778	−	0.392439i	$-0.128369\pi$
$269$	30.1710	1.83956	0.919778	+	0.392439i	$0.128369\pi$
$270$	0	0
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	−12.4058	−0.752213
$273$	0	0
$274$	−37.5895	−2.27086
$275$	18.9583	1.14323
$276$	0	0
$277$	21.1215	1.26907	0.634535	−	0.772894i	$-0.281191\pi$
$277$	21.1215	1.26907	0.634535	+	0.772894i	$0.281191\pi$
$278$	−45.8992	−2.75285
$279$	0	0
$280$	−2.06418	−0.123358
$281$	1.74730	0.104235	0.0521175	−	0.998641i	$-0.483403\pi$
$281$	1.74730	0.104235	0.0521175	+	0.998641i	$0.483403\pi$
$282$	0	0
$283$	−7.28817	−0.433237	−0.216618	−	0.976256i	$-0.569503\pi$
$283$	−7.28817	−0.433237	−0.216618	+	0.976256i	$0.569503\pi$
$284$	−2.26103	−0.134168
$285$	0	0
$286$	−20.2395	−1.19679
$287$	5.86495	0.346197
$288$	0	0
$289$	−8.38507	−0.493239
$290$	−25.4652	−1.49537
$291$	0	0
$292$	−8.79561	−0.514724
$293$	15.3509	0.896809	0.448404	−	0.893831i	$-0.351992\pi$
$293$	15.3509	0.896809	0.448404	+	0.893831i	$0.351992\pi$
$294$	0	0
$295$	−49.2772	−2.86903
$296$	−0.572881	−0.0332980
$297$	0	0
$298$	−30.5408	−1.76918
$299$	2.44820	0.141583
$300$	0	0
$301$	−2.49794	−0.143979
$302$	−11.6240	−0.668888
$303$	0	0
$304$	26.3182	1.50945
$305$	13.6036	0.778940
$306$	0	0
$307$	16.7638	0.956762	0.478381	−	0.878152i	$-0.341224\pi$
$307$	16.7638	0.956762	0.478381	+	0.878152i	$0.341224\pi$
$308$	−9.61051	−0.547610
$309$	0	0
$310$	31.3901	1.78284
$311$	−16.0231	−0.908589	−0.454295	−	0.890852i	$-0.650109\pi$
$311$	−16.0231	−0.908589	−0.454295	+	0.890852i	$0.650109\pi$
$312$	0	0
$313$	34.4056	1.94472	0.972360	−	0.233488i	$-0.0750138\pi$
$313$	34.4056	1.94472	0.972360	+	0.233488i	$0.0750138\pi$
$314$	−2.38744	−0.134731
$315$	0	0
$316$	−24.0574	−1.35333
$317$	−16.3111	−0.916123	−0.458061	−	0.888921i	$-0.651456\pi$
$317$	−16.3111	−0.916123	−0.458061	+	0.888921i	$0.651456\pi$
$318$	0	0
$319$	7.60906	0.426026
$320$	25.9403	1.45011
$321$	0	0
$322$	2.39961	0.133725
$323$	−18.2761	−1.01691
$324$	0	0
$325$	−41.0479	−2.27693
$326$	5.46237	0.302533
$327$	0	0
$328$	0.593578	0.0327749
$329$	−0.557635	−0.0307434
$330$	0	0
$331$	−31.5800	−1.73579	−0.867896	−	0.496746i	$-0.834528\pi$
$331$	−31.5800	−1.73579	−0.867896	+	0.496746i	$0.834528\pi$
$332$	21.2445	1.16594
$333$	0	0
$334$	7.53983	0.412561
$335$	−52.9286	−2.89180
$336$	0	0
$337$	6.00505	0.327116	0.163558	−	0.986534i	$-0.447703\pi$
$337$	6.00505	0.327116	0.163558	+	0.986534i	$0.447703\pi$
$338$	18.2169	0.990870
$339$	0	0
$340$	−20.4192	−1.10739
$341$	−9.37944	−0.507925
$342$	0	0
$343$	19.9290	1.07607
$344$	−0.252811	−0.0136307
$345$	0	0
$346$	13.8452	0.744325
$347$	23.0304	1.23634	0.618168	−	0.786046i	$-0.287875\pi$
$347$	23.0304	1.23634	0.618168	+	0.786046i	$0.287875\pi$
$348$	0	0
$349$	−11.5662	−0.619126	−0.309563	−	0.950879i	$-0.600183\pi$
$349$	−11.5662	−0.619126	−0.309563	+	0.950879i	$0.600183\pi$
$350$	−40.2332	−2.15056
$351$	0	0
$352$	−17.1010	−0.911487
$353$	−2.13463	−0.113615	−0.0568073	−	0.998385i	$-0.518092\pi$
$353$	−2.13463	−0.113615	−0.0568073	+	0.998385i	$0.518092\pi$
$354$	0	0
$355$	4.45336	0.236360
$356$	−1.31423	−0.0696540
$357$	0	0
$358$	28.3337	1.49748
$359$	24.2235	1.27847	0.639234	−	0.769012i	$-0.279251\pi$
$359$	24.2235	1.27847	0.639234	+	0.769012i	$0.279251\pi$
$360$	0	0
$361$	19.7716	1.04061
$362$	25.9995	1.36650
$363$	0	0
$364$	20.8084	1.09066
$365$	17.3240	0.906778
$366$	0	0
$367$	−2.83481	−0.147976	−0.0739879	−	0.997259i	$-0.523573\pi$
$367$	−2.83481	−0.147976	−0.0739879	+	0.997259i	$0.523573\pi$
$368$	2.19377	0.114358
$369$	0	0
$370$	−17.5817	−0.914030
$371$	10.9552	0.568767
$372$	0	0
$373$	−28.8334	−1.49294	−0.746468	−	0.665421i	$-0.768252\pi$
$373$	−28.8334	−1.49294	−0.746468	+	0.665421i	$0.768252\pi$
$374$	12.5942	0.651230
$375$	0	0
$376$	−0.0564370	−0.00291052
$377$	−16.4749	−0.848501
$378$	0	0
$379$	32.1985	1.65393	0.826963	−	0.562256i	$-0.190066\pi$
$379$	32.1985	1.65393	0.826963	+	0.562256i	$0.190066\pi$
$380$	43.3181	2.22217
$381$	0	0
$382$	−26.4388	−1.35273
$383$	0.672250	0.0343504	0.0171752	−	0.999852i	$-0.494533\pi$
$383$	0.672250	0.0343504	0.0171752	+	0.999852i	$0.494533\pi$
$384$	0	0
$385$	18.9290	0.964712
$386$	29.5542	1.50427
$387$	0	0
$388$	−13.3182	−0.676129
$389$	−4.25930	−0.215955	−0.107978	−	0.994153i	$-0.534437\pi$
$389$	−4.25930	−0.215955	−0.107978	+	0.994153i	$0.534437\pi$
$390$	0	0
$391$	−1.52341	−0.0770424
$392$	0.354019	0.0178807
$393$	0	0
$394$	43.9641	2.21488
$395$	47.3838	2.38414
$396$	0	0
$397$	−8.86484	−0.444913	−0.222457	−	0.974943i	$-0.571408\pi$
$397$	−8.86484	−0.444913	−0.222457	+	0.974943i	$0.571408\pi$
$398$	−17.9255	−0.898524
$399$	0	0
$400$	−36.7820	−1.83910
$401$	37.0839	1.85188	0.925941	−	0.377667i	$-0.123274\pi$
$401$	37.0839	1.85188	0.925941	+	0.377667i	$0.123274\pi$
$402$	0	0
$403$	20.3081	1.01162
$404$	−8.79012	−0.437325
$405$	0	0
$406$	−16.1480	−0.801410
$407$	5.25346	0.260404
$408$	0	0
$409$	−3.38919	−0.167584	−0.0837922	−	0.996483i	$-0.526703\pi$
$409$	−3.38919	−0.167584	−0.0837922	+	0.996483i	$0.526703\pi$
$410$	18.2169	0.899669
$411$	0	0
$412$	−25.6040	−1.26142
$413$	−31.2476	−1.53760
$414$	0	0
$415$	−41.8435	−2.05402
$416$	37.0266	1.81538
$417$	0	0
$418$	−26.7178	−1.30681
$419$	−20.6564	−1.00913	−0.504565	−	0.863374i	$-0.668347\pi$
$419$	−20.6564	−1.00913	−0.504565	+	0.863374i	$0.668347\pi$
$420$	0	0
$421$	27.5330	1.34188	0.670939	−	0.741513i	$-0.265891\pi$
$421$	27.5330	1.34188	0.670939	+	0.741513i	$0.265891\pi$
$422$	−11.7738	−0.573139
$423$	0	0
$424$	1.10876	0.0538459
$425$	25.5424	1.23899
$426$	0	0
$427$	8.62630	0.417456
$428$	−21.8647	−1.05687
$429$	0	0
$430$	−7.75877	−0.374161
$431$	−2.58110	−0.124327	−0.0621636	−	0.998066i	$-0.519800\pi$
$431$	−2.58110	−0.124327	−0.0621636	+	0.998066i	$0.519800\pi$
$432$	0	0
$433$	−27.0137	−1.29820	−0.649098	−	0.760704i	$-0.724854\pi$
$433$	−27.0137	−1.29820	−0.649098	+	0.760704i	$0.724854\pi$
$434$	19.9051	0.955474
$435$	0	0
$436$	27.4466	1.31445
$437$	3.23183	0.154599
$438$	0	0
$439$	11.7101	0.558891	0.279446	−	0.960162i	$-0.409849\pi$
$439$	11.7101	0.558891	0.279446	+	0.960162i	$0.409849\pi$
$440$	1.91576	0.0913305
$441$	0	0
$442$	−27.2686	−1.29703
$443$	2.08077	0.0988606	0.0494303	−	0.998778i	$-0.484259\pi$
$443$	2.08077	0.0988606	0.0494303	+	0.998778i	$0.484259\pi$
$444$	0	0
$445$	2.58853	0.122708
$446$	−17.5329	−0.830206
$447$	0	0
$448$	16.4492	0.777154
$449$	10.5508	0.497924	0.248962	−	0.968513i	$-0.419911\pi$
$449$	10.5508	0.497924	0.248962	+	0.968513i	$0.419911\pi$
$450$	0	0
$451$	−5.44326	−0.256313
$452$	−8.82526	−0.415106
$453$	0	0
$454$	21.0128	0.986179
$455$	−40.9845	−1.92139
$456$	0	0
$457$	−7.90941	−0.369987	−0.184993	−	0.982740i	$-0.559226\pi$
$457$	−7.90941	−0.369987	−0.184993	+	0.982740i	$0.559226\pi$
$458$	−16.2912	−0.761238
$459$	0	0
$460$	3.61081	0.168355
$461$	−39.0928	−1.82073	−0.910366	−	0.413804i	$-0.864200\pi$
$461$	−39.0928	−1.82073	−0.910366	+	0.413804i	$0.864200\pi$
$462$	0	0
$463$	23.6928	1.10110	0.550550	−	0.834802i	$-0.314418\pi$
$463$	23.6928	1.10110	0.550550	+	0.834802i	$0.314418\pi$
$464$	−14.7628	−0.685344
$465$	0	0
$466$	−25.0915	−1.16234
$467$	−34.7152	−1.60643	−0.803214	−	0.595691i	$-0.796878\pi$
$467$	−34.7152	−1.60643	−0.803214	+	0.595691i	$0.796878\pi$
$468$	0	0
$469$	−33.5631	−1.54980
$470$	−1.73205	−0.0798935
$471$	0	0
$472$	−3.16250	−0.145566
$473$	2.31834	0.106597
$474$	0	0
$475$	−54.1867	−2.48625
$476$	−12.9482	−0.593480
$477$	0	0
$478$	−29.5749	−1.35272
$479$	−6.48173	−0.296158	−0.148079	−	0.988976i	$-0.547309\pi$
$479$	−6.48173	−0.296158	−0.148079	+	0.988976i	$0.547309\pi$
$480$	0	0
$481$	−11.3746	−0.518639
$482$	1.56704	0.0713767
$483$	0	0
$484$	−11.7537	−0.534260
$485$	26.2317	1.19112
$486$	0	0
$487$	19.9828	0.905505	0.452753	−	0.891636i	$-0.350442\pi$
$487$	19.9828	0.905505	0.452753	+	0.891636i	$0.350442\pi$
$488$	0.873048	0.0395210
$489$	0	0
$490$	10.8648	0.490823
$491$	−26.2271	−1.18361	−0.591806	−	0.806081i	$-0.701585\pi$
$491$	−26.2271	−1.18361	−0.591806	+	0.806081i	$0.701585\pi$
$492$	0	0
$493$	10.2517	0.461711
$494$	57.8486	2.60273
$495$	0	0
$496$	18.1976	0.817096
$497$	2.82396	0.126672
$498$	0	0
$499$	6.31727	0.282800	0.141400	−	0.989953i	$-0.454840\pi$
$499$	6.31727	0.282800	0.141400	+	0.989953i	$0.454840\pi$
$500$	−25.7566	−1.15187
$501$	0	0
$502$	−16.3773	−0.730956
$503$	−21.8261	−0.973179	−0.486589	−	0.873631i	$-0.661759\pi$
$503$	−21.8261	−0.973179	−0.486589	+	0.873631i	$0.661759\pi$
$504$	0	0
$505$	17.3131	0.770425
$506$	−2.22708	−0.0990059
$507$	0	0
$508$	11.4483	0.507937
$509$	29.0931	1.28953	0.644765	−	0.764381i	$-0.276955\pi$
$509$	29.0931	1.28953	0.644765	+	0.764381i	$0.276955\pi$
$510$	0	0
$511$	10.9855	0.485968
$512$	31.1704	1.37755
$513$	0	0
$514$	50.4620	2.22579
$515$	50.4301	2.22221
$516$	0	0
$517$	0.517541	0.0227614
$518$	−11.1489	−0.489855
$519$	0	0
$520$	−4.14796	−0.181900
$521$	−13.6949	−0.599982	−0.299991	−	0.953942i	$-0.596984\pi$
$521$	−13.6949	−0.599982	−0.299991	+	0.953942i	$0.596984\pi$
$522$	0	0
$523$	13.1506	0.575038	0.287519	−	0.957775i	$-0.407170\pi$
$523$	13.1506	0.575038	0.287519	+	0.957775i	$0.407170\pi$
$524$	−19.4486	−0.849618
$525$	0	0
$526$	−55.2772	−2.41020
$527$	−12.6369	−0.550472
$528$	0	0
$529$	−22.7306	−0.988287
$530$	34.0277	1.47807
$531$	0	0
$532$	27.4688	1.19093
$533$	11.7856	0.510490
$534$	0	0
$535$	43.0651	1.86187
$536$	−3.39684	−0.146721
$537$	0	0
$538$	59.4252	2.56200
$539$	−3.24644	−0.139834
$540$	0	0
$541$	6.26083	0.269174	0.134587	−	0.990902i	$-0.457029\pi$
$541$	6.26083	0.269174	0.134587	+	0.990902i	$0.457029\pi$
$542$	−37.4227	−1.60744
$543$	0	0
$544$	−23.0401	−0.987838
$545$	−54.0592	−2.31564
$546$	0	0
$547$	−31.3783	−1.34164	−0.670819	−	0.741621i	$-0.734057\pi$
$547$	−31.3783	−1.34164	−0.670819	+	0.741621i	$0.734057\pi$
$548$	−35.8674	−1.53218
$549$	0	0
$550$	37.3405	1.59220
$551$	−21.7483	−0.926508
$552$	0	0
$553$	30.0469	1.27773
$554$	41.6013	1.76747
$555$	0	0
$556$	−43.7965	−1.85739
$557$	−43.4392	−1.84058	−0.920290	−	0.391237i	$-0.872047\pi$
$557$	−43.4392	−1.84058	−0.920290	+	0.391237i	$0.872047\pi$
$558$	0	0
$559$	−5.01960	−0.212306
$560$	−36.7252	−1.55192
$561$	0	0
$562$	3.44150	0.145171
$563$	−32.0176	−1.34938	−0.674691	−	0.738100i	$-0.735723\pi$
$563$	−32.0176	−1.34938	−0.674691	+	0.738100i	$0.735723\pi$
$564$	0	0
$565$	17.3824	0.731282
$566$	−14.3549	−0.603381
$567$	0	0
$568$	0.285807	0.0119922
$569$	−6.77194	−0.283895	−0.141947	−	0.989874i	$-0.545336\pi$
$569$	−6.77194	−0.283895	−0.141947	+	0.989874i	$0.545336\pi$
$570$	0	0
$571$	−21.1530	−0.885226	−0.442613	−	0.896713i	$-0.645948\pi$
$571$	−21.1530	−0.885226	−0.442613	+	0.896713i	$0.645948\pi$
$572$	−19.3123	−0.807487
$573$	0	0
$574$	11.5517	0.482158
$575$	−4.51677	−0.188362
$576$	0	0
$577$	11.9162	0.496079	0.248039	−	0.968750i	$-0.420214\pi$
$577$	11.9162	0.496079	0.248039	+	0.968750i	$0.420214\pi$
$578$	−16.5154	−0.686948
$579$	0	0
$580$	−24.2986	−1.00894
$581$	−26.5337	−1.10081
$582$	0	0
$583$	−10.1676	−0.421097
$584$	1.11181	0.0460072
$585$	0	0
$586$	30.2354	1.24901
$587$	−0.129862	−0.00535996	−0.00267998	−	0.999996i	$-0.500853\pi$
$587$	−0.129862	−0.00535996	−0.00267998	+	0.999996i	$0.500853\pi$
$588$	0	0
$589$	26.8084	1.10462
$590$	−97.0572	−3.99578
$591$	0	0
$592$	−10.1925	−0.418911
$593$	26.2622	1.07846	0.539230	−	0.842158i	$-0.318715\pi$
$593$	26.2622	1.07846	0.539230	+	0.842158i	$0.318715\pi$
$594$	0	0
$595$	25.5030	1.04552
$596$	−29.1417	−1.19369
$597$	0	0
$598$	4.82201	0.197187
$599$	27.5952	1.12751	0.563754	−	0.825943i	$-0.309357\pi$
$599$	27.5952	1.12751	0.563754	+	0.825943i	$0.309357\pi$
$600$	0	0
$601$	1.33275	0.0543639	0.0271820	−	0.999631i	$-0.491347\pi$
$601$	1.33275	0.0543639	0.0271820	+	0.999631i	$0.491347\pi$
$602$	−4.91998	−0.200524
$603$	0	0
$604$	−11.0915	−0.451308
$605$	23.1503	0.941194
$606$	0	0
$607$	−12.5645	−0.509977	−0.254988	−	0.966944i	$-0.582072\pi$
$607$	−12.5645	−0.509977	−0.254988	+	0.966944i	$0.582072\pi$
$608$	48.8783	1.98228
$609$	0	0
$610$	26.7939	1.08485
$611$	−1.12056	−0.0453332
$612$	0	0
$613$	−13.9982	−0.565384	−0.282692	−	0.959211i	$-0.591227\pi$
$613$	−13.9982	−0.565384	−0.282692	+	0.959211i	$0.591227\pi$
$614$	33.0183	1.33251
$615$	0	0
$616$	1.21482	0.0489466
$617$	24.0467	0.968084	0.484042	−	0.875045i	$-0.339168\pi$
$617$	24.0467	0.968084	0.484042	+	0.875045i	$0.339168\pi$
$618$	0	0
$619$	6.87164	0.276195	0.138097	−	0.990419i	$-0.455901\pi$
$619$	6.87164	0.276195	0.138097	+	0.990419i	$0.455901\pi$
$620$	29.9521	1.20291
$621$	0	0
$622$	−31.5594	−1.26542
$623$	1.64143	0.0657626
$624$	0	0
$625$	7.21894	0.288758
$626$	67.7658	2.70847
$627$	0	0
$628$	−2.27807	−0.0909047
$629$	7.07797	0.282217
$630$	0	0
$631$	−35.3773	−1.40835	−0.704175	−	0.710027i	$-0.748683\pi$
$631$	−35.3773	−1.40835	−0.704175	+	0.710027i	$0.748683\pi$
$632$	3.04098	0.120964
$633$	0	0
$634$	−32.1266	−1.27591
$635$	−22.5488	−0.894821
$636$	0	0
$637$	7.02910	0.278503
$638$	14.9869	0.593338
$639$	0	0
$640$	−7.02229	−0.277580
$641$	−19.1737	−0.757314	−0.378657	−	0.925537i	$-0.623614\pi$
$641$	−19.1737	−0.757314	−0.378657	+	0.925537i	$0.623614\pi$
$642$	0	0
$643$	19.3764	0.764130	0.382065	−	0.924135i	$-0.375213\pi$
$643$	19.3764	0.764130	0.382065	+	0.924135i	$0.375213\pi$
$644$	2.28969	0.0902263
$645$	0	0
$646$	−35.9968	−1.41628
$647$	8.77141	0.344840	0.172420	−	0.985024i	$-0.444841\pi$
$647$	8.77141	0.344840	0.172420	+	0.985024i	$0.444841\pi$
$648$	0	0
$649$	29.0009	1.13839
$650$	−80.8485	−3.17114
$651$	0	0
$652$	5.21213	0.204123
$653$	32.8094	1.28393	0.641965	−	0.766734i	$-0.278119\pi$
$653$	32.8094	1.28393	0.641965	+	0.766734i	$0.278119\pi$
$654$	0	0
$655$	38.3063	1.49675
$656$	10.5608	0.412329
$657$	0	0
$658$	−1.09833	−0.0428172
$659$	18.6516	0.726563	0.363282	−	0.931679i	$-0.381656\pi$
$659$	18.6516	0.726563	0.363282	+	0.931679i	$0.381656\pi$
$660$	0	0
$661$	36.4074	1.41608	0.708041	−	0.706171i	$-0.249579\pi$
$661$	36.4074	1.41608	0.708041	+	0.706171i	$0.249579\pi$
$662$	−62.2004	−2.41749
$663$	0	0
$664$	−2.68542	−0.104215
$665$	−54.1031	−2.09803
$666$	0	0
$667$	−1.81284	−0.0701936
$668$	7.19442	0.278361
$669$	0	0
$670$	−104.249	−4.02749
$671$	−8.00607	−0.309071
$672$	0	0
$673$	39.3901	1.51838	0.759189	−	0.650871i	$-0.225596\pi$
$673$	39.3901	1.51838	0.759189	+	0.650871i	$0.225596\pi$
$674$	11.8276	0.455584
$675$	0	0
$676$	17.3824	0.668553
$677$	31.6754	1.21738	0.608692	−	0.793406i	$-0.291694\pi$
$677$	31.6754	1.21738	0.608692	+	0.793406i	$0.291694\pi$
$678$	0	0
$679$	16.6340	0.638356
$680$	2.58110	0.0989807
$681$	0	0
$682$	−18.4739	−0.707402
$683$	−29.0656	−1.11217	−0.556083	−	0.831127i	$-0.687696\pi$
$683$	−29.0656	−1.11217	−0.556083	+	0.831127i	$0.687696\pi$
$684$	0	0
$685$	70.6451	2.69921
$686$	39.2525	1.49867
$687$	0	0
$688$	−4.49794	−0.171482
$689$	22.0145	0.838685
$690$	0	0
$691$	5.35740	0.203805	0.101903	−	0.994794i	$-0.467507\pi$
$691$	5.35740	0.203805	0.101903	+	0.994794i	$0.467507\pi$
$692$	13.2110	0.502206
$693$	0	0
$694$	45.3610	1.72188
$695$	86.2623	3.27212
$696$	0	0
$697$	−7.33368	−0.277783
$698$	−22.7810	−0.862275
$699$	0	0
$700$	−38.3901	−1.45101
$701$	−25.6536	−0.968922	−0.484461	−	0.874813i	$-0.660984\pi$
$701$	−25.6536	−0.968922	−0.484461	+	0.874813i	$0.660984\pi$
$702$	0	0
$703$	−15.0155	−0.566320
$704$	−15.2665	−0.575380
$705$	0	0
$706$	−4.20439	−0.158234
$707$	10.9786	0.412893
$708$	0	0
$709$	−4.69047	−0.176154	−0.0880772	−	0.996114i	$-0.528072\pi$
$709$	−4.69047	−0.176154	−0.0880772	+	0.996114i	$0.528072\pi$
$710$	8.77141	0.329185
$711$	0	0
$712$	0.166126	0.00622583
$713$	2.23463	0.0836877
$714$	0	0
$715$	38.0378	1.42253
$716$	27.0357	1.01037
$717$	0	0
$718$	47.7110	1.78056
$719$	39.0669	1.45695	0.728476	−	0.685072i	$-0.240229\pi$
$719$	39.0669	1.45695	0.728476	+	0.685072i	$0.240229\pi$
$720$	0	0
$721$	31.9786	1.19095
$722$	38.9424	1.44929
$723$	0	0
$724$	24.8084	0.921997
$725$	30.3951	1.12885
$726$	0	0
$727$	−10.8253	−0.401489	−0.200744	−	0.979644i	$-0.564336\pi$
$727$	−10.8253	−0.401489	−0.200744	+	0.979644i	$0.564336\pi$
$728$	−2.63030	−0.0974853
$729$	0	0
$730$	34.1215	1.26290
$731$	3.12349	0.115526
$732$	0	0
$733$	−3.41241	−0.126040	−0.0630201	−	0.998012i	$-0.520073\pi$
$733$	−3.41241	−0.126040	−0.0630201	+	0.998012i	$0.520073\pi$
$734$	−5.58348	−0.206090
$735$	0	0
$736$	4.07428	0.150180
$737$	31.1499	1.14742
$738$	0	0
$739$	26.3010	0.967497	0.483748	−	0.875207i	$-0.339275\pi$
$739$	26.3010	0.967497	0.483748	+	0.875207i	$0.339275\pi$
$740$	−16.7763	−0.616708
$741$	0	0
$742$	21.5776	0.792139
$743$	28.9439	1.06185	0.530924	−	0.847419i	$-0.321845\pi$
$743$	28.9439	1.06185	0.530924	+	0.847419i	$0.321845\pi$
$744$	0	0
$745$	57.3979	2.10289
$746$	−56.7907	−2.07925
$747$	0	0
$748$	12.0172	0.439394
$749$	27.3084	0.997827
$750$	0	0
$751$	−19.4483	−0.709679	−0.354839	−	0.934927i	$-0.615464\pi$
$751$	−19.4483	−0.709679	−0.354839	+	0.934927i	$0.615464\pi$
$752$	−1.00411	−0.0366161
$753$	0	0
$754$	−32.4492	−1.18173
$755$	21.8460	0.795058
$756$	0	0
$757$	6.59627	0.239745	0.119873	−	0.992789i	$-0.461751\pi$
$757$	6.59627	0.239745	0.119873	+	0.992789i	$0.461751\pi$
$758$	63.4187	2.30347
$759$	0	0
$760$	−5.47565	−0.198623
$761$	10.3297	0.374451	0.187226	−	0.982317i	$-0.440050\pi$
$761$	10.3297	0.374451	0.187226	+	0.982317i	$0.440050\pi$
$762$	0	0
$763$	−34.2799	−1.24102
$764$	−25.2276	−0.912703
$765$	0	0
$766$	1.32407	0.0478407
$767$	−62.7920	−2.26729
$768$	0	0
$769$	44.8590	1.61766	0.808828	−	0.588046i	$-0.200102\pi$
$769$	44.8590	1.61766	0.808828	+	0.588046i	$0.200102\pi$
$770$	37.2829	1.34358
$771$	0	0
$772$	28.2003	1.01495
$773$	42.9355	1.54428	0.772141	−	0.635452i	$-0.219186\pi$
$773$	42.9355	1.54428	0.772141	+	0.635452i	$0.219186\pi$
$774$	0	0
$775$	−37.4671	−1.34586
$776$	1.68349	0.0604339
$777$	0	0
$778$	−8.38919	−0.300767
$779$	15.5580	0.557422
$780$	0	0
$781$	−2.62092	−0.0937839
$782$	−3.00054	−0.107299
$783$	0	0
$784$	6.29860	0.224950
$785$	4.48691	0.160145
$786$	0	0
$787$	0.477407	0.0170177	0.00850885	−	0.999964i	$-0.497292\pi$
$787$	0.477407	0.0170177	0.00850885	+	0.999964i	$0.497292\pi$
$788$	41.9501	1.49441
$789$	0	0
$790$	93.3278	3.32045
$791$	11.0225	0.391915
$792$	0	0
$793$	17.3345	0.615566
$794$	−17.4603	−0.619644
$795$	0	0
$796$	−17.1043	−0.606246
$797$	11.4690	0.406252	0.203126	−	0.979153i	$-0.434890\pi$
$797$	11.4690	0.406252	0.203126	+	0.979153i	$0.434890\pi$
$798$	0	0
$799$	0.697281	0.0246680
$800$	−68.3116	−2.41518
$801$	0	0
$802$	73.0411	2.57917
$803$	−10.1956	−0.359795
$804$	0	0
$805$	−4.50980	−0.158950
$806$	39.9991	1.40891
$807$	0	0
$808$	1.11112	0.0390890
$809$	42.7873	1.50432	0.752161	−	0.658979i	$-0.229012\pi$
$809$	42.7873	1.50432	0.752161	+	0.658979i	$0.229012\pi$
$810$	0	0
$811$	−31.2098	−1.09592	−0.547962	−	0.836503i	$-0.684596\pi$
$811$	−31.2098	−1.09592	−0.547962	+	0.836503i	$0.684596\pi$
$812$	−15.4082	−0.540722
$813$	0	0
$814$	10.3473	0.362673
$815$	−10.2659	−0.359599
$816$	0	0
$817$	−6.62630	−0.231825
$818$	−6.67539	−0.233400
$819$	0	0
$820$	17.3824	0.607019
$821$	27.2511	0.951070	0.475535	−	0.879697i	$-0.342255\pi$
$821$	27.2511	0.951070	0.475535	+	0.879697i	$0.342255\pi$
$822$	0	0
$823$	−14.4406	−0.503367	−0.251683	−	0.967810i	$-0.580984\pi$
$823$	−14.4406	−0.503367	−0.251683	+	0.967810i	$0.580984\pi$
$824$	3.23649	0.112748
$825$	0	0
$826$	−61.5458	−2.14145
$827$	7.02757	0.244373	0.122186	−	0.992507i	$-0.461009\pi$
$827$	7.02757	0.244373	0.122186	+	0.992507i	$0.461009\pi$
$828$	0	0
$829$	−42.0806	−1.46152	−0.730760	−	0.682635i	$-0.760834\pi$
$829$	−42.0806	−1.46152	−0.730760	+	0.682635i	$0.760834\pi$
$830$	−82.4156	−2.86069
$831$	0	0
$832$	33.0547	1.14596
$833$	−4.37392	−0.151547
$834$	0	0
$835$	−14.1702	−0.490382
$836$	−25.4938	−0.881723
$837$	0	0
$838$	−40.6851	−1.40544
$839$	−39.9213	−1.37824	−0.689118	−	0.724649i	$-0.742002\pi$
$839$	−39.9213	−1.37824	−0.689118	+	0.724649i	$0.742002\pi$
$840$	0	0
$841$	−16.8007	−0.579333
$842$	54.2295	1.86887
$843$	0	0
$844$	−11.2344	−0.386705
$845$	−34.2366	−1.17777
$846$	0	0
$847$	14.6800	0.504412
$848$	19.7266	0.677416
$849$	0	0
$850$	50.3087	1.72557
$851$	−1.25163	−0.0429052
$852$	0	0
$853$	−23.4810	−0.803975	−0.401988	−	0.915645i	$-0.631680\pi$
$853$	−23.4810	−0.803975	−0.401988	+	0.915645i	$0.631680\pi$
$854$	16.9905	0.581402
$855$	0	0
$856$	2.76382	0.0944655
$857$	−8.03472	−0.274461	−0.137230	−	0.990539i	$-0.543820\pi$
$857$	−8.03472	−0.274461	−0.137230	+	0.990539i	$0.543820\pi$
$858$	0	0
$859$	−24.2959	−0.828966	−0.414483	−	0.910057i	$-0.636038\pi$
$859$	−24.2959	−0.828966	−0.414483	+	0.910057i	$0.636038\pi$
$860$	−7.40333	−0.252452
$861$	0	0
$862$	−5.08378	−0.173154
$863$	10.2828	0.350029	0.175015	−	0.984566i	$-0.444003\pi$
$863$	10.2828	0.350029	0.175015	+	0.984566i	$0.444003\pi$
$864$	0	0
$865$	−26.0205	−0.884725
$866$	−53.2067	−1.80804
$867$	0	0
$868$	18.9932	0.644671
$869$	−27.8866	−0.945987
$870$	0	0
$871$	−67.4448	−2.28528
$872$	−3.46940	−0.117489
$873$	0	0
$874$	6.36547	0.215315
$875$	32.1692	1.08752
$876$	0	0
$877$	−26.2276	−0.885644	−0.442822	−	0.896610i	$-0.646023\pi$
$877$	−26.2276	−0.885644	−0.442822	+	0.896610i	$0.646023\pi$
$878$	23.0643	0.778384
$879$	0	0
$880$	34.0847	1.14900
$881$	39.2326	1.32178	0.660890	−	0.750483i	$-0.270179\pi$
$881$	39.2326	1.32178	0.660890	+	0.750483i	$0.270179\pi$
$882$	0	0
$883$	7.79830	0.262434	0.131217	−	0.991354i	$-0.458112\pi$
$883$	7.79830	0.262434	0.131217	+	0.991354i	$0.458112\pi$
$884$	−26.0194	−0.875126
$885$	0	0
$886$	4.09833	0.137686
$887$	22.8571	0.767465	0.383732	−	0.923444i	$-0.374639\pi$
$887$	22.8571	0.767465	0.383732	+	0.923444i	$0.374639\pi$
$888$	0	0
$889$	−14.2986	−0.479560
$890$	5.09840	0.170899
$891$	0	0
$892$	−16.7297	−0.560151
$893$	−1.47924	−0.0495009
$894$	0	0
$895$	−53.2499	−1.77995
$896$	−4.45297	−0.148763
$897$	0	0
$898$	20.7811	0.693473
$899$	−15.0377	−0.501537
$900$	0	0
$901$	−13.6987	−0.456370
$902$	−10.7211	−0.356974
$903$	0	0
$904$	1.11556	0.0371031
$905$	−48.8630	−1.62426
$906$	0	0
$907$	−37.8485	−1.25674	−0.628370	−	0.777915i	$-0.716278\pi$
$907$	−37.8485	−1.25674	−0.628370	+	0.777915i	$0.716278\pi$
$908$	20.0502	0.665388
$909$	0	0
$910$	−80.7238	−2.67597
$911$	52.7045	1.74618	0.873089	−	0.487561i	$-0.162113\pi$
$911$	52.7045	1.74618	0.873089	+	0.487561i	$0.162113\pi$
$912$	0	0
$913$	24.6260	0.815001
$914$	−15.5785	−0.515291
$915$	0	0
$916$	−15.5449	−0.513617
$917$	24.2908	0.802152
$918$	0	0
$919$	−49.1052	−1.61983	−0.809916	−	0.586545i	$-0.800488\pi$
$919$	−49.1052	−1.61983	−0.809916	+	0.586545i	$0.800488\pi$
$920$	−0.456427	−0.0150480
$921$	0	0
$922$	−76.9977	−2.53579
$923$	5.67474	0.186786
$924$	0	0
$925$	20.9855	0.689997
$926$	46.6658	1.53353
$927$	0	0
$928$	−27.4175	−0.900022
$929$	17.9507	0.588943	0.294472	−	0.955660i	$-0.404856\pi$
$929$	17.9507	0.588943	0.294472	+	0.955660i	$0.404856\pi$
$930$	0	0
$931$	9.27900	0.304107
$932$	−23.9420	−0.784248
$933$	0	0
$934$	−68.3756	−2.23732
$935$	−23.6693	−0.774070
$936$	0	0
$937$	−29.7980	−0.973457	−0.486729	−	0.873553i	$-0.661810\pi$
$937$	−29.7980	−0.973457	−0.486729	+	0.873553i	$0.661810\pi$
$938$	−66.1063	−2.15845
$939$	0	0
$940$	−1.65270	−0.0539052
$941$	−30.0112	−0.978339	−0.489169	−	0.872189i	$-0.662700\pi$
$941$	−30.0112	−0.978339	−0.489169	+	0.872189i	$0.662700\pi$
$942$	0	0
$943$	1.29685	0.0422311
$944$	−56.2663	−1.83131
$945$	0	0
$946$	4.56624	0.148461
$947$	45.6013	1.48184	0.740922	−	0.671591i	$-0.234389\pi$
$947$	45.6013	1.48184	0.740922	+	0.671591i	$0.234389\pi$
$948$	0	0
$949$	22.0752	0.716592
$950$	−106.727	−3.46268
$951$	0	0
$952$	1.63673	0.0530466
$953$	−13.1957	−0.427451	−0.213726	−	0.976894i	$-0.568560\pi$
$953$	−13.1957	−0.427451	−0.213726	+	0.976894i	$0.568560\pi$
$954$	0	0
$955$	49.6887	1.60789
$956$	−28.2201	−0.912702
$957$	0	0
$958$	−12.7665	−0.412467
$959$	44.7974	1.44658
$960$	0	0
$961$	−12.4635	−0.402047
$962$	−22.4037	−0.722323
$963$	0	0
$964$	1.49525	0.0481588
$965$	−55.5437	−1.78801
$966$	0	0
$967$	−20.1548	−0.648133	−0.324067	−	0.946034i	$-0.605050\pi$
$967$	−20.1548	−0.648133	−0.324067	+	0.946034i	$0.605050\pi$
$968$	1.48574	0.0477533
$969$	0	0
$970$	51.6664	1.65891
$971$	−26.5839	−0.853118	−0.426559	−	0.904460i	$-0.640274\pi$
$971$	−26.5839	−0.853118	−0.426559	+	0.904460i	$0.640274\pi$
$972$	0	0
$973$	54.7006	1.75362
$974$	39.3584	1.26112
$975$	0	0
$976$	15.5330	0.497200
$977$	14.4041	0.460828	0.230414	−	0.973093i	$-0.425992\pi$
$977$	14.4041	0.460828	0.230414	+	0.973093i	$0.425992\pi$
$978$	0	0
$979$	−1.52341	−0.0486885
$980$	10.3671	0.331165
$981$	0	0
$982$	−51.6573	−1.64845
$983$	−42.9881	−1.37111	−0.685554	−	0.728022i	$-0.740440\pi$
$983$	−42.9881	−1.37111	−0.685554	+	0.728022i	$0.740440\pi$
$984$	0	0
$985$	−82.6255	−2.63267
$986$	20.1918	0.643039
$987$	0	0
$988$	55.1985	1.75610
$989$	−0.552340	−0.0175634
$990$	0	0
$991$	−15.5794	−0.494895	−0.247447	−	0.968901i	$-0.579592\pi$
$991$	−15.5794	−0.494895	−0.247447	+	0.968901i	$0.579592\pi$
$992$	33.7966	1.07304
$993$	0	0
$994$	5.56212	0.176420
$995$	33.6889	1.06801
$996$	0	0
$997$	40.6186	1.28640	0.643201	−	0.765697i	$-0.277606\pi$
$997$	40.6186	1.28640	0.643201	+	0.765697i	$0.277606\pi$
$998$	12.4426	0.393863
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.c.1.6	yes	6
3.2	odd	2	inner	729.2.a.c.1.1	✓	6
9.2	odd	6		729.2.c.c.244.6		12
9.4	even	3		729.2.c.c.487.1		12
9.5	odd	6		729.2.c.c.487.6		12
9.7	even	3		729.2.c.c.244.1		12
27.2	odd	18		729.2.e.r.568.2		12
27.4	even	9		729.2.e.q.406.2		12
27.5	odd	18		729.2.e.m.649.1		12
27.7	even	9		729.2.e.q.325.2		12
27.11	odd	18		729.2.e.m.82.1		12
27.13	even	9		729.2.e.r.163.1		12
27.14	odd	18		729.2.e.r.163.2		12
27.16	even	9		729.2.e.m.82.2		12
27.20	odd	18		729.2.e.q.325.1		12
27.22	even	9		729.2.e.m.649.2		12
27.23	odd	18		729.2.e.q.406.1		12
27.25	even	9		729.2.e.r.568.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.a.c.1.1	✓	6	3.2	odd	2	inner
729.2.a.c.1.6	yes	6	1.1	even	1	trivial
729.2.c.c.244.1		12	9.7	even	3
729.2.c.c.244.6		12	9.2	odd	6
729.2.c.c.487.1		12	9.4	even	3
729.2.c.c.487.6		12	9.5	odd	6
729.2.e.m.82.1		12	27.11	odd	18
729.2.e.m.82.2		12	27.16	even	9
729.2.e.m.649.1		12	27.5	odd	18
729.2.e.m.649.2		12	27.22	even	9
729.2.e.q.325.1		12	27.20	odd	18
729.2.e.q.325.2		12	27.7	even	9
729.2.e.q.406.1		12	27.23	odd	18
729.2.e.q.406.2		12	27.4	even	9
729.2.e.r.163.1		12	27.13	even	9
729.2.e.r.163.2		12	27.14	odd	18
729.2.e.r.568.1		12	27.25	even	9
729.2.e.r.568.2		12	27.2	odd	18

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q+1.96962 q^{2} +1.87939 q^{4} -3.70167 q^{5} -2.34730 q^{7} -0.237565 q^{8} -7.29086 q^{10} +2.17853 q^{11} -4.71688 q^{13} -4.62327 q^{14} -4.22668 q^{16} +2.93512 q^{17} -6.22668 q^{19} -6.95686 q^{20} +4.29086 q^{22} -0.519030 q^{23} +8.70233 q^{25} -9.29044 q^{26} -4.41147 q^{28} +3.49276 q^{29} -4.30541 q^{31} -7.84981 q^{32} +5.78106 q^{34} +8.68891 q^{35} +2.41147 q^{37} -12.2642 q^{38} +0.879385 q^{40} -2.49860 q^{41} +1.06418 q^{43} +4.09429 q^{44} -1.02229 q^{46} +0.237565 q^{47} -1.49020 q^{49} +17.1403 q^{50} -8.86484 q^{52} -4.66717 q^{53} -8.06418 q^{55} +0.557635 q^{56} +6.87939 q^{58} +13.3122 q^{59} -3.67499 q^{61} -8.48000 q^{62} -7.00774 q^{64} +17.4603 q^{65} +14.2986 q^{67} +5.51622 q^{68} +17.1138 q^{70} -1.20307 q^{71} -4.68004 q^{73} +4.74968 q^{74} -11.7023 q^{76} -5.11365 q^{77} -12.8007 q^{79} +15.6458 q^{80} -4.92127 q^{82} +11.3040 q^{83} -10.8648 q^{85} +2.09602 q^{86} -0.517541 q^{88} -0.699287 q^{89} +11.0719 q^{91} -0.975457 q^{92} +0.467911 q^{94} +23.0491 q^{95} -7.08647 q^{97} -2.93512 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) \) 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more