LMFDB - Embedded newform 729.2.a.b.1.1

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,-3,0,9] 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:	$0$
Dimension:	$6$
Coefficient field:	6.6.7459857.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 $ x^6 - 3x^5 - 6x^4 + 13x^3 + 12x^2 - 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.17298$ 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)

$f(q)$	$=$	\(q-2.70506 q^{2} +5.31738 q^{4} +1.67238 q^{5} +0.500591 q^{7} -8.97372 q^{8} -4.52391 q^{10} +1.91772 q^{11} +3.11244 q^{13} -1.35413 q^{14} +13.6397 q^{16} -2.66467 q^{17} +5.79664 q^{19} +8.89270 q^{20} -5.18755 q^{22} +4.64587 q^{23} -2.20313 q^{25} -8.41934 q^{26} +2.66183 q^{28} +2.61507 q^{29} -4.61460 q^{31} -18.9489 q^{32} +7.20811 q^{34} +0.837181 q^{35} -4.85867 q^{37} -15.6803 q^{38} -15.0075 q^{40} +11.5482 q^{41} -9.00434 q^{43} +10.1972 q^{44} -12.5674 q^{46} -6.83224 q^{47} -6.74941 q^{49} +5.95961 q^{50} +16.5500 q^{52} +5.43322 q^{53} +3.20716 q^{55} -4.49216 q^{56} -7.07393 q^{58} +2.19131 q^{59} +6.84034 q^{61} +12.4828 q^{62} +23.9786 q^{64} +5.20519 q^{65} +12.4836 q^{67} -14.1691 q^{68} -2.26463 q^{70} +2.83568 q^{71} +9.93497 q^{73} +13.1430 q^{74} +30.8229 q^{76} +0.959993 q^{77} +5.31121 q^{79} +22.8109 q^{80} -31.2385 q^{82} +2.72815 q^{83} -4.45636 q^{85} +24.3573 q^{86} -17.2091 q^{88} -11.2189 q^{89} +1.55806 q^{91} +24.7038 q^{92} +18.4816 q^{94} +9.69421 q^{95} -6.88914 q^{97} +18.2576 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * q^98

Coefficient data

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

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

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

$2$	−2.70506	−1.91277	−0.956385	−	0.292110i	$-0.905643\pi$
$2$	−2.70506	−1.91277	−0.956385	+	0.292110i	$0.905643\pi$
$3$	0	0
$3$	0	0
$4$	5.31738	2.65869
$5$	1.67238	0.747913	0.373957	−	0.927446i	$-0.378001\pi$
$5$	1.67238	0.747913	0.373957	+	0.927446i	$0.378001\pi$
$6$	0	0
$7$	0.500591	0.189206	0.0946028	−	0.995515i	$-0.469842\pi$
$7$	0.500591	0.189206	0.0946028	+	0.995515i	$0.469842\pi$
$8$	−8.97372	−3.17269
$9$	0	0
$10$	−4.52391	−1.43059
$11$	1.91772	0.578214	0.289107	−	0.957297i	$-0.406642\pi$
$11$	1.91772	0.578214	0.289107	+	0.957297i	$0.406642\pi$
$12$	0	0
$13$	3.11244	0.863234	0.431617	−	0.902057i	$-0.357943\pi$
$13$	3.11244	0.863234	0.431617	+	0.902057i	$0.357943\pi$
$14$	−1.35413	−0.361907
$15$	0	0
$16$	13.6397	3.40993
$17$	−2.66467	−0.646278	−0.323139	−	0.946352i	$-0.604738\pi$
$17$	−2.66467	−0.646278	−0.323139	+	0.946352i	$0.604738\pi$
$18$	0	0
$19$	5.79664	1.32984	0.664920	−	0.746915i	$-0.268466\pi$
$19$	5.79664	1.32984	0.664920	+	0.746915i	$0.268466\pi$
$20$	8.89270	1.98847
$21$	0	0
$22$	−5.18755	−1.10599
$23$	4.64587	0.968731	0.484365	−	0.874866i	$-0.339051\pi$
$23$	4.64587	0.968731	0.484365	+	0.874866i	$0.339051\pi$
$24$	0	0
$25$	−2.20313	−0.440626
$26$	−8.41934	−1.65117
$27$	0	0
$28$	2.66183	0.503039
$29$	2.61507	0.485606	0.242803	−	0.970076i	$-0.421933\pi$
$29$	2.61507	0.485606	0.242803	+	0.970076i	$0.421933\pi$
$30$	0	0
$31$	−4.61460	−0.828807	−0.414404	−	0.910093i	$-0.636010\pi$
$31$	−4.61460	−0.828807	−0.414404	+	0.910093i	$0.636010\pi$
$32$	−18.9489	−3.34973
$33$	0	0
$34$	7.20811	1.23618
$35$	0.837181	0.141509
$36$	0	0
$37$	−4.85867	−0.798761	−0.399381	−	0.916785i	$-0.630775\pi$
$37$	−4.85867	−0.798761	−0.399381	+	0.916785i	$0.630775\pi$
$38$	−15.6803	−2.54368
$39$	0	0
$40$	−15.0075	−2.37290
$41$	11.5482	1.80352	0.901760	−	0.432237i	$-0.142276\pi$
$41$	11.5482	1.80352	0.901760	+	0.432237i	$0.142276\pi$
$42$	0	0
$43$	−9.00434	−1.37315	−0.686574	−	0.727060i	$-0.740886\pi$
$43$	−9.00434	−1.37315	−0.686574	+	0.727060i	$0.740886\pi$
$44$	10.1972	1.53729
$45$	0	0
$46$	−12.5674	−1.85296
$47$	−6.83224	−0.996584	−0.498292	−	0.867009i	$-0.666039\pi$
$47$	−6.83224	−0.996584	−0.498292	+	0.867009i	$0.666039\pi$
$48$	0	0
$49$	−6.74941	−0.964201
$50$	5.95961	0.842816
$51$	0	0
$52$	16.5500	2.29507
$53$	5.43322	0.746309	0.373155	−	0.927769i	$-0.378276\pi$
$53$	5.43322	0.746309	0.373155	+	0.927769i	$0.378276\pi$
$54$	0	0
$55$	3.20716	0.432454
$56$	−4.49216	−0.600291
$57$	0	0
$58$	−7.07393	−0.928853
$59$	2.19131	0.285285	0.142642	−	0.989774i	$-0.454440\pi$
$59$	2.19131	0.285285	0.142642	+	0.989774i	$0.454440\pi$
$60$	0	0
$61$	6.84034	0.875815	0.437908	−	0.899020i	$-0.355720\pi$
$61$	6.84034	0.875815	0.437908	+	0.899020i	$0.355720\pi$
$62$	12.4828	1.58532
$63$	0	0
$64$	23.9786	2.99733
$65$	5.20519	0.645624
$66$	0	0
$67$	12.4836	1.52511	0.762557	−	0.646921i	$-0.223944\pi$
$67$	12.4836	1.52511	0.762557	+	0.646921i	$0.223944\pi$
$68$	−14.1691	−1.71825
$69$	0	0
$70$	−2.26463	−0.270675
$71$	2.83568	0.336534	0.168267	−	0.985741i	$-0.446183\pi$
$71$	2.83568	0.336534	0.168267	+	0.985741i	$0.446183\pi$
$72$	0	0
$73$	9.93497	1.16280	0.581400	−	0.813618i	$-0.302505\pi$
$73$	9.93497	1.16280	0.581400	+	0.813618i	$0.302505\pi$
$74$	13.1430	1.52785
$75$	0	0
$76$	30.8229	3.53563
$77$	0.959993	0.109401
$78$	0	0
$79$	5.31121	0.597558	0.298779	−	0.954322i	$-0.403421\pi$
$79$	5.31121	0.597558	0.298779	+	0.954322i	$0.403421\pi$
$80$	22.8109	2.55033
$81$	0	0
$82$	−31.2385	−3.44972
$83$	2.72815	0.299453	0.149727	−	0.988727i	$-0.452161\pi$
$83$	2.72815	0.299453	0.149727	+	0.988727i	$0.452161\pi$
$84$	0	0
$85$	−4.45636	−0.483360
$86$	24.3573	2.62652
$87$	0	0
$88$	−17.2091	−1.83449
$89$	−11.2189	−1.18920	−0.594600	−	0.804021i	$-0.702690\pi$
$89$	−11.2189	−1.18920	−0.594600	+	0.804021i	$0.702690\pi$
$90$	0	0
$91$	1.55806	0.163329
$92$	24.7038	2.57555
$93$	0	0
$94$	18.4816	1.90624
$95$	9.69421	0.994605
$96$	0	0
$97$	−6.88914	−0.699486	−0.349743	−	0.936846i	$-0.613731\pi$
$97$	−6.88914	−0.699486	−0.349743	+	0.936846i	$0.613731\pi$
$98$	18.2576	1.84429
$99$	0	0
$100$	−11.7149	−1.17149
$101$	−3.72965	−0.371114	−0.185557	−	0.982634i	$-0.559409\pi$
$101$	−3.72965	−0.371114	−0.185557	+	0.982634i	$0.559409\pi$
$102$	0	0
$103$	7.68080	0.756811	0.378406	−	0.925640i	$-0.376472\pi$
$103$	7.68080	0.756811	0.378406	+	0.925640i	$0.376472\pi$
$104$	−27.9301	−2.73877
$105$	0	0
$106$	−14.6972	−1.42752
$107$	10.7658	1.04077	0.520383	−	0.853933i	$-0.325789\pi$
$107$	10.7658	1.04077	0.520383	+	0.853933i	$0.325789\pi$
$108$	0	0
$109$	12.2298	1.17141	0.585703	−	0.810526i	$-0.300819\pi$
$109$	12.2298	1.17141	0.585703	+	0.810526i	$0.300819\pi$
$110$	−8.67558	−0.827184
$111$	0	0
$112$	6.82793	0.645179
$113$	1.94395	0.182871	0.0914357	−	0.995811i	$-0.470854\pi$
$113$	1.94395	0.182871	0.0914357	+	0.995811i	$0.470854\pi$
$114$	0	0
$115$	7.76968	0.724526
$116$	13.9053	1.29108
$117$	0	0
$118$	−5.92764	−0.545684
$119$	−1.33391	−0.122279
$120$	0	0
$121$	−7.32236	−0.665669
$122$	−18.5036	−1.67523
$123$	0	0
$124$	−24.5376	−2.20354
$125$	−12.0464	−1.07746
$126$	0	0
$127$	2.34433	0.208026	0.104013	−	0.994576i	$-0.466832\pi$
$127$	2.34433	0.208026	0.104013	+	0.994576i	$0.466832\pi$
$128$	−26.9659	−2.38347
$129$	0	0
$130$	−14.0804	−1.23493
$131$	17.1024	1.49424	0.747121	−	0.664688i	$-0.231435\pi$
$131$	17.1024	1.49424	0.747121	+	0.664688i	$0.231435\pi$
$132$	0	0
$133$	2.90174	0.251613
$134$	−33.7689	−2.91719
$135$	0	0
$136$	23.9120	2.05044
$137$	−13.9272	−1.18988	−0.594939	−	0.803771i	$-0.702824\pi$
$137$	−13.9272	−1.18988	−0.594939	+	0.803771i	$0.702824\pi$
$138$	0	0
$139$	7.91664	0.671480	0.335740	−	0.941955i	$-0.391014\pi$
$139$	7.91664	0.671480	0.335740	+	0.941955i	$0.391014\pi$
$140$	4.45161	0.376229
$141$	0	0
$142$	−7.67071	−0.643712
$143$	5.96877	0.499134
$144$	0	0
$145$	4.37340	0.363191
$146$	−26.8747	−2.22417
$147$	0	0
$148$	−25.8354	−2.12366
$149$	−0.728134	−0.0596511	−0.0298255	−	0.999555i	$-0.509495\pi$
$149$	−0.728134	−0.0596511	−0.0298255	+	0.999555i	$0.509495\pi$
$150$	0	0
$151$	4.34209	0.353354	0.176677	−	0.984269i	$-0.443465\pi$
$151$	4.34209	0.353354	0.176677	+	0.984269i	$0.443465\pi$
$152$	−52.0174	−4.21917
$153$	0	0
$154$	−2.59684	−0.209260
$155$	−7.71739	−0.619876
$156$	0	0
$157$	−15.5233	−1.23889	−0.619446	−	0.785040i	$-0.712643\pi$
$157$	−15.5233	−1.23889	−0.619446	+	0.785040i	$0.712643\pi$
$158$	−14.3672	−1.14299
$159$	0	0
$160$	−31.6899	−2.50531
$161$	2.32568	0.183289
$162$	0	0
$163$	8.49738	0.665566	0.332783	−	0.943003i	$-0.392012\pi$
$163$	8.49738	0.665566	0.332783	+	0.943003i	$0.392012\pi$
$164$	61.4059	4.79500
$165$	0	0
$166$	−7.37982	−0.572785
$167$	−23.1419	−1.79078	−0.895389	−	0.445285i	$-0.853102\pi$
$167$	−23.1419	−1.79078	−0.895389	+	0.445285i	$0.853102\pi$
$168$	0	0
$169$	−3.31275	−0.254827
$170$	12.0547	0.924556
$171$	0	0
$172$	−47.8794	−3.65077
$173$	2.57848	0.196038	0.0980190	−	0.995185i	$-0.468749\pi$
$173$	2.57848	0.196038	0.0980190	+	0.995185i	$0.468749\pi$
$174$	0	0
$175$	−1.10287	−0.0833689
$176$	26.1572	1.97167
$177$	0	0
$178$	30.3478	2.27467
$179$	8.89613	0.664928	0.332464	−	0.943116i	$-0.392120\pi$
$179$	8.89613	0.664928	0.332464	+	0.943116i	$0.392120\pi$
$180$	0	0
$181$	7.91183	0.588082	0.294041	−	0.955793i	$-0.405000\pi$
$181$	7.91183	0.588082	0.294041	+	0.955793i	$0.405000\pi$
$182$	−4.21465	−0.312410
$183$	0	0
$184$	−41.6907	−3.07348
$185$	−8.12557	−0.597404
$186$	0	0
$187$	−5.11009	−0.373687
$188$	−36.3296	−2.64961
$189$	0	0
$190$	−26.2235	−1.90245
$191$	15.8973	1.15029	0.575146	−	0.818051i	$-0.304945\pi$
$191$	15.8973	1.15029	0.575146	+	0.818051i	$0.304945\pi$
$192$	0	0
$193$	−4.58478	−0.330020	−0.165010	−	0.986292i	$-0.552766\pi$
$193$	−4.58478	−0.330020	−0.165010	+	0.986292i	$0.552766\pi$
$194$	18.6356	1.33796
$195$	0	0
$196$	−35.8891	−2.56351
$197$	−2.99417	−0.213326	−0.106663	−	0.994295i	$-0.534017\pi$
$197$	−2.99417	−0.213326	−0.106663	+	0.994295i	$0.534017\pi$
$198$	0	0
$199$	14.8885	1.05542	0.527709	−	0.849425i	$-0.323051\pi$
$199$	14.8885	1.05542	0.527709	+	0.849425i	$0.323051\pi$
$200$	19.7703	1.39797
$201$	0	0
$202$	10.0889	0.709855
$203$	1.30908	0.0918795
$204$	0	0
$205$	19.3130	1.34888
$206$	−20.7771	−1.44761
$207$	0	0
$208$	42.4528	2.94357
$209$	11.1163	0.768932
$210$	0	0
$211$	13.8790	0.955467	0.477733	−	0.878505i	$-0.341458\pi$
$211$	13.8790	0.955467	0.477733	+	0.878505i	$0.341458\pi$
$212$	28.8904	1.98420
$213$	0	0
$214$	−29.1221	−1.99074
$215$	−15.0587	−1.02700
$216$	0	0
$217$	−2.31003	−0.156815
$218$	−33.0825	−2.24063
$219$	0	0
$220$	17.0537	1.14976
$221$	−8.29362	−0.557889
$222$	0	0
$223$	−6.81198	−0.456164	−0.228082	−	0.973642i	$-0.573245\pi$
$223$	−6.81198	−0.456164	−0.228082	+	0.973642i	$0.573245\pi$
$224$	−9.48567	−0.633788
$225$	0	0
$226$	−5.25851	−0.349791
$227$	9.74084	0.646522	0.323261	−	0.946310i	$-0.395221\pi$
$227$	9.74084	0.646522	0.323261	+	0.946310i	$0.395221\pi$
$228$	0	0
$229$	−14.0448	−0.928109	−0.464055	−	0.885807i	$-0.653606\pi$
$229$	−14.0448	−0.928109	−0.464055	+	0.885807i	$0.653606\pi$
$230$	−21.0175	−1.38585
$231$	0	0
$232$	−23.4669	−1.54068
$233$	−5.32333	−0.348743	−0.174372	−	0.984680i	$-0.555789\pi$
$233$	−5.32333	−0.348743	−0.174372	+	0.984680i	$0.555789\pi$
$234$	0	0
$235$	−11.4261	−0.745359
$236$	11.6520	0.758483
$237$	0	0
$238$	3.60832	0.233892
$239$	−17.7735	−1.14967	−0.574836	−	0.818268i	$-0.694934\pi$
$239$	−17.7735	−1.14967	−0.574836	+	0.818268i	$0.694934\pi$
$240$	0	0
$241$	−2.00679	−0.129269	−0.0646344	−	0.997909i	$-0.520588\pi$
$241$	−2.00679	−0.129269	−0.0646344	+	0.997909i	$0.520588\pi$
$242$	19.8075	1.27327
$243$	0	0
$244$	36.3726	2.32852
$245$	−11.2876	−0.721139
$246$	0	0
$247$	18.0417	1.14796
$248$	41.4101	2.62955
$249$	0	0
$250$	32.5863	2.06094
$251$	−23.5643	−1.48737	−0.743683	−	0.668533i	$-0.766923\pi$
$251$	−23.5643	−1.48737	−0.743683	+	0.668533i	$0.766923\pi$
$252$	0	0
$253$	8.90947	0.560133
$254$	−6.34157	−0.397906
$255$	0	0
$256$	24.9871	1.56170
$257$	−5.87457	−0.366445	−0.183223	−	0.983071i	$-0.558653\pi$
$257$	−5.87457	−0.366445	−0.183223	+	0.983071i	$0.558653\pi$
$258$	0	0
$259$	−2.43221	−0.151130
$260$	27.6779	1.71651
$261$	0	0
$262$	−46.2631	−2.85814
$263$	−21.9781	−1.35523	−0.677615	−	0.735417i	$-0.736986\pi$
$263$	−21.9781	−1.35523	−0.677615	+	0.735417i	$0.736986\pi$
$264$	0	0
$265$	9.08643	0.558175
$266$	−7.84941	−0.481278
$267$	0	0
$268$	66.3800	4.05480
$269$	30.6026	1.86587	0.932937	−	0.360041i	$-0.117237\pi$
$269$	30.6026	1.86587	0.932937	+	0.360041i	$0.117237\pi$
$270$	0	0
$271$	−16.0823	−0.976928	−0.488464	−	0.872584i	$-0.662443\pi$
$271$	−16.0823	−0.976928	−0.488464	+	0.872584i	$0.662443\pi$
$272$	−36.3454	−2.20376
$273$	0	0
$274$	37.6739	2.27596
$275$	−4.22498	−0.254776
$276$	0	0
$277$	−20.8136	−1.25057	−0.625283	−	0.780398i	$-0.715017\pi$
$277$	−20.8136	−1.25057	−0.625283	+	0.780398i	$0.715017\pi$
$278$	−21.4150	−1.28439
$279$	0	0
$280$	−7.51263	−0.448965
$281$	−12.2603	−0.731387	−0.365693	−	0.930735i	$-0.619168\pi$
$281$	−12.2603	−0.731387	−0.365693	+	0.930735i	$0.619168\pi$
$282$	0	0
$283$	4.57258	0.271812	0.135906	−	0.990722i	$-0.456606\pi$
$283$	4.57258	0.271812	0.135906	+	0.990722i	$0.456606\pi$
$284$	15.0784	0.894738
$285$	0	0
$286$	−16.1459	−0.954728
$287$	5.78091	0.341236
$288$	0	0
$289$	−9.89952	−0.582325
$290$	−11.8303	−0.694702
$291$	0	0
$292$	52.8280	3.09152
$293$	−26.1234	−1.52615	−0.763073	−	0.646312i	$-0.776310\pi$
$293$	−26.1234	−1.52615	−0.763073	+	0.646312i	$0.776310\pi$
$294$	0	0
$295$	3.66472	0.213368
$296$	43.6004	2.53422
$297$	0	0
$298$	1.96965	0.114099
$299$	14.4600	0.836241
$300$	0	0
$301$	−4.50749	−0.259807
$302$	−11.7456	−0.675885
$303$	0	0
$304$	79.0646	4.53466
$305$	11.4397	0.655034
$306$	0	0
$307$	3.29277	0.187928	0.0939641	−	0.995576i	$-0.470046\pi$
$307$	3.29277	0.187928	0.0939641	+	0.995576i	$0.470046\pi$
$308$	5.10464	0.290864
$309$	0	0
$310$	20.8760	1.18568
$311$	−34.7926	−1.97291	−0.986455	−	0.164034i	$-0.947549\pi$
$311$	−34.7926	−1.97291	−0.986455	+	0.164034i	$0.947549\pi$
$312$	0	0
$313$	10.6278	0.600721	0.300360	−	0.953826i	$-0.402893\pi$
$313$	10.6278	0.600721	0.300360	+	0.953826i	$0.402893\pi$
$314$	41.9914	2.36971
$315$	0	0
$316$	28.2417	1.58872
$317$	15.5002	0.870579	0.435290	−	0.900290i	$-0.356646\pi$
$317$	15.5002	0.870579	0.435290	+	0.900290i	$0.356646\pi$
$318$	0	0
$319$	5.01497	0.280784
$320$	40.1015	2.24174
$321$	0	0
$322$	−6.29112	−0.350590
$323$	−15.4461	−0.859446
$324$	0	0
$325$	−6.85710	−0.380363
$326$	−22.9860	−1.27308
$327$	0	0
$328$	−103.630	−5.72201
$329$	−3.42016	−0.188559
$330$	0	0
$331$	−14.5703	−0.800855	−0.400427	−	0.916328i	$-0.631138\pi$
$331$	−14.5703	−0.800855	−0.400427	+	0.916328i	$0.631138\pi$
$332$	14.5066	0.796153
$333$	0	0
$334$	62.6005	3.42534
$335$	20.8774	1.14065
$336$	0	0
$337$	−20.6148	−1.12296	−0.561479	−	0.827491i	$-0.689768\pi$
$337$	−20.6148	−1.12296	−0.561479	+	0.827491i	$0.689768\pi$
$338$	8.96120	0.487425
$339$	0	0
$340$	−23.6961	−1.28510
$341$	−8.84951	−0.479228
$342$	0	0
$343$	−6.88283	−0.371638
$344$	80.8024	4.35657
$345$	0	0
$346$	−6.97495	−0.374976
$347$	21.1835	1.13719	0.568596	−	0.822617i	$-0.307487\pi$
$347$	21.1835	1.13719	0.568596	+	0.822617i	$0.307487\pi$
$348$	0	0
$349$	−4.62597	−0.247623	−0.123811	−	0.992306i	$-0.539512\pi$
$349$	−4.62597	−0.247623	−0.123811	+	0.992306i	$0.539512\pi$
$350$	2.98333	0.159466
$351$	0	0
$352$	−36.3387	−1.93686
$353$	13.6434	0.726165	0.363083	−	0.931757i	$-0.381724\pi$
$353$	13.6434	0.726165	0.363083	+	0.931757i	$0.381724\pi$
$354$	0	0
$355$	4.74236	0.251698
$356$	−59.6551	−3.16171
$357$	0	0
$358$	−24.0646	−1.27185
$359$	28.2447	1.49070	0.745349	−	0.666675i	$-0.232283\pi$
$359$	28.2447	1.49070	0.745349	+	0.666675i	$0.232283\pi$
$360$	0	0
$361$	14.6010	0.768473
$362$	−21.4020	−1.12486
$363$	0	0
$364$	8.28478	0.434240
$365$	16.6151	0.869674
$366$	0	0
$367$	34.8802	1.82073	0.910366	−	0.413804i	$-0.135800\pi$
$367$	34.8802	1.82073	0.910366	+	0.413804i	$0.135800\pi$
$368$	63.3684	3.30331
$369$	0	0
$370$	21.9802	1.14270
$371$	2.71982	0.141206
$372$	0	0
$373$	3.05944	0.158412	0.0792059	−	0.996858i	$-0.474762\pi$
$373$	3.05944	0.158412	0.0792059	+	0.996858i	$0.474762\pi$
$374$	13.8231	0.714777
$375$	0	0
$376$	61.3106	3.16185
$377$	8.13924	0.419192
$378$	0	0
$379$	−7.67705	−0.394344	−0.197172	−	0.980369i	$-0.563176\pi$
$379$	−7.67705	−0.394344	−0.197172	+	0.980369i	$0.563176\pi$
$380$	51.5477	2.64434
$381$	0	0
$382$	−43.0034	−2.20024
$383$	−10.3619	−0.529470	−0.264735	−	0.964321i	$-0.585285\pi$
$383$	−10.3619	−0.529470	−0.264735	+	0.964321i	$0.585285\pi$
$384$	0	0
$385$	1.60548	0.0818227
$386$	12.4021	0.631252
$387$	0	0
$388$	−36.6322	−1.85972
$389$	0.426951	0.0216473	0.0108236	−	0.999941i	$-0.496555\pi$
$389$	0.426951	0.0216473	0.0108236	+	0.999941i	$0.496555\pi$
$390$	0	0
$391$	−12.3797	−0.626069
$392$	60.5673	3.05911
$393$	0	0
$394$	8.09942	0.408043
$395$	8.88239	0.446922
$396$	0	0
$397$	−27.0891	−1.35956	−0.679781	−	0.733416i	$-0.737925\pi$
$397$	−27.0891	−1.35956	−0.679781	+	0.733416i	$0.737925\pi$
$398$	−40.2743	−2.01877
$399$	0	0
$400$	−30.0501	−1.50250
$401$	−29.3605	−1.46619	−0.733096	−	0.680125i	$-0.761925\pi$
$401$	−29.3605	−1.46619	−0.733096	+	0.680125i	$0.761925\pi$
$402$	0	0
$403$	−14.3627	−0.715455
$404$	−19.8319	−0.986675
$405$	0	0
$406$	−3.54115	−0.175744
$407$	−9.31757	−0.461855
$408$	0	0
$409$	−20.1839	−0.998031	−0.499015	−	0.866593i	$-0.666305\pi$
$409$	−20.1839	−0.998031	−0.499015	+	0.866593i	$0.666305\pi$
$410$	−52.2428	−2.58009
$411$	0	0
$412$	40.8417	2.01213
$413$	1.09695	0.0539775
$414$	0	0
$415$	4.56252	0.223965
$416$	−58.9773	−2.89160
$417$	0	0
$418$	−30.0704	−1.47079
$419$	−24.3703	−1.19057	−0.595284	−	0.803515i	$-0.702960\pi$
$419$	−24.3703	−1.19057	−0.595284	+	0.803515i	$0.702960\pi$
$420$	0	0
$421$	−26.1745	−1.27567	−0.637835	−	0.770173i	$-0.720170\pi$
$421$	−26.1745	−1.27567	−0.637835	+	0.770173i	$0.720170\pi$
$422$	−37.5435	−1.82759
$423$	0	0
$424$	−48.7561	−2.36781
$425$	5.87062	0.284767
$426$	0	0
$427$	3.42421	0.165709
$428$	57.2456	2.76707
$429$	0	0
$430$	40.7348	1.96441
$431$	−31.9185	−1.53746	−0.768731	−	0.639572i	$-0.779111\pi$
$431$	−31.9185	−1.53746	−0.768731	+	0.639572i	$0.779111\pi$
$432$	0	0
$433$	0.0123080	0.000591484 0	0.000295742	−	1.00000i	$-0.499906\pi$
$433$	0.0123080	0.000591484 0	0.000295742		1.00000i	$0.499906\pi$
$434$	6.24878	0.299951
$435$	0	0
$436$	65.0306	3.11440
$437$	26.9304	1.28826
$438$	0	0
$439$	5.30480	0.253185	0.126592	−	0.991955i	$-0.459596\pi$
$439$	5.30480	0.253185	0.126592	+	0.991955i	$0.459596\pi$
$440$	−28.7802	−1.37204
$441$	0	0
$442$	22.4348	1.06711
$443$	−40.6963	−1.93354	−0.966771	−	0.255645i	$-0.917712\pi$
$443$	−40.6963	−1.93354	−0.966771	+	0.255645i	$0.917712\pi$
$444$	0	0
$445$	−18.7623	−0.889419
$446$	18.4268	0.872536
$447$	0	0
$448$	12.0035	0.567111
$449$	−15.4280	−0.728093	−0.364047	−	0.931381i	$-0.618605\pi$
$449$	−15.4280	−0.728093	−0.364047	+	0.931381i	$0.618605\pi$
$450$	0	0
$451$	22.1461	1.04282
$452$	10.3367	0.486198
$453$	0	0
$454$	−26.3496	−1.23665
$455$	2.60567	0.122156
$456$	0	0
$457$	2.38875	0.111741	0.0558704	−	0.998438i	$-0.482207\pi$
$457$	2.38875	0.111741	0.0558704	+	0.998438i	$0.482207\pi$
$458$	37.9922	1.77526
$459$	0	0
$460$	41.3143	1.92629
$461$	−32.4839	−1.51292	−0.756462	−	0.654037i	$-0.773074\pi$
$461$	−32.4839	−1.51292	−0.756462	+	0.654037i	$0.773074\pi$
$462$	0	0
$463$	33.8785	1.57447	0.787234	−	0.616654i	$-0.211512\pi$
$463$	33.8785	1.57447	0.787234	+	0.616654i	$0.211512\pi$
$464$	35.6689	1.65589
$465$	0	0
$466$	14.4000	0.667065
$467$	13.8027	0.638711	0.319356	−	0.947635i	$-0.396534\pi$
$467$	13.8027	0.638711	0.319356	+	0.947635i	$0.396534\pi$
$468$	0	0
$469$	6.24918	0.288560
$470$	30.9084	1.42570
$471$	0	0
$472$	−19.6642	−0.905119
$473$	−17.2678	−0.793973
$474$	0	0
$475$	−12.7707	−0.585962
$476$	−7.09291	−0.325103
$477$	0	0
$478$	48.0785	2.19906
$479$	−5.89112	−0.269172	−0.134586	−	0.990902i	$-0.542970\pi$
$479$	−5.89112	−0.269172	−0.134586	+	0.990902i	$0.542970\pi$
$480$	0	0
$481$	−15.1223	−0.689518
$482$	5.42850	0.247262
$483$	0	0
$484$	−38.9357	−1.76981
$485$	−11.5213	−0.523155
$486$	0	0
$487$	29.0299	1.31547	0.657736	−	0.753249i	$-0.271514\pi$
$487$	29.0299	1.31547	0.657736	+	0.753249i	$0.271514\pi$
$488$	−61.3832	−2.77869
$489$	0	0
$490$	30.5337	1.37937
$491$	−4.38351	−0.197825	−0.0989125	−	0.995096i	$-0.531536\pi$
$491$	−4.38351	−0.197825	−0.0989125	+	0.995096i	$0.531536\pi$
$492$	0	0
$493$	−6.96830	−0.313837
$494$	−48.8038	−2.19579
$495$	0	0
$496$	−62.9420	−2.82618
$497$	1.41952	0.0636741
$498$	0	0
$499$	35.3435	1.58219	0.791096	−	0.611691i	$-0.209511\pi$
$499$	35.3435	1.58219	0.791096	+	0.611691i	$0.209511\pi$
$500$	−64.0553	−2.86464
$501$	0	0
$502$	63.7430	2.84499
$503$	8.37659	0.373494	0.186747	−	0.982408i	$-0.440206\pi$
$503$	8.37659	0.373494	0.186747	+	0.982408i	$0.440206\pi$
$504$	0	0
$505$	−6.23740	−0.277561
$506$	−24.1007	−1.07141
$507$	0	0
$508$	12.4657	0.553076
$509$	3.84297	0.170337	0.0851683	−	0.996367i	$-0.472857\pi$
$509$	3.84297	0.170337	0.0851683	+	0.996367i	$0.472857\pi$
$510$	0	0
$511$	4.97336	0.220008
$512$	−13.6601	−0.603699
$513$	0	0
$514$	15.8911	0.700925
$515$	12.8452	0.566029
$516$	0	0
$517$	−13.1023	−0.576239
$518$	6.57928	0.289077
$519$	0	0
$520$	−46.7099	−2.04836
$521$	−19.6523	−0.860983	−0.430491	−	0.902595i	$-0.641660\pi$
$521$	−19.6523	−0.860983	−0.430491	+	0.902595i	$0.641660\pi$
$522$	0	0
$523$	−39.6103	−1.73204	−0.866018	−	0.500012i	$-0.833329\pi$
$523$	−39.6103	−1.73204	−0.866018	+	0.500012i	$0.833329\pi$
$524$	90.9398	3.97272
$525$	0	0
$526$	59.4523	2.59224
$527$	12.2964	0.535640
$528$	0	0
$529$	−1.41591	−0.0615611
$530$	−24.5794	−1.06766
$531$	0	0
$532$	15.4297	0.668961
$533$	35.9429	1.55686
$534$	0	0
$535$	18.0045	0.778402
$536$	−112.024	−4.83871
$537$	0	0
$538$	−82.7820	−3.56899
$539$	−12.9435	−0.557514
$540$	0	0
$541$	−41.8257	−1.79823	−0.899115	−	0.437713i	$-0.855788\pi$
$541$	−41.8257	−1.79823	−0.899115	+	0.437713i	$0.855788\pi$
$542$	43.5036	1.86864
$543$	0	0
$544$	50.4927	2.16486
$545$	20.4530	0.876110
$546$	0	0
$547$	−17.0343	−0.728335	−0.364168	−	0.931333i	$-0.618646\pi$
$547$	−17.0343	−0.728335	−0.364168	+	0.931333i	$0.618646\pi$
$548$	−74.0559	−3.16351
$549$	0	0
$550$	11.4288	0.487328
$551$	15.1586	0.645779
$552$	0	0
$553$	2.65875	0.113061
$554$	56.3021	2.39205
$555$	0	0
$556$	42.0957	1.78526
$557$	33.7680	1.43080	0.715398	−	0.698717i	$-0.246245\pi$
$557$	33.7680	1.43080	0.715398	+	0.698717i	$0.246245\pi$
$558$	0	0
$559$	−28.0254	−1.18535
$560$	11.4189	0.482538
$561$	0	0
$562$	33.1648	1.39897
$563$	22.7113	0.957166	0.478583	−	0.878042i	$-0.341151\pi$
$563$	22.7113	0.957166	0.478583	+	0.878042i	$0.341151\pi$
$564$	0	0
$565$	3.25103	0.136772
$566$	−12.3691	−0.519913
$567$	0	0
$568$	−25.4466	−1.06772
$569$	−10.8714	−0.455755	−0.227877	−	0.973690i	$-0.573179\pi$
$569$	−10.8714	−0.455755	−0.227877	+	0.973690i	$0.573179\pi$
$570$	0	0
$571$	14.8837	0.622864	0.311432	−	0.950268i	$-0.399191\pi$
$571$	14.8837	0.622864	0.311432	+	0.950268i	$0.399191\pi$
$572$	31.7382	1.32704
$573$	0	0
$574$	−15.6377	−0.652706
$575$	−10.2354	−0.426848
$576$	0	0
$577$	−37.1163	−1.54517	−0.772586	−	0.634910i	$-0.781037\pi$
$577$	−37.1163	−1.54517	−0.772586	+	0.634910i	$0.781037\pi$
$578$	26.7789	1.11385
$579$	0	0
$580$	23.2550	0.965613
$581$	1.36569	0.0566583
$582$	0	0
$583$	10.4194	0.431526
$584$	−89.1536	−3.68920
$585$	0	0
$586$	70.6655	2.91917
$587$	−14.6432	−0.604390	−0.302195	−	0.953246i	$-0.597719\pi$
$587$	−14.6432	−0.604390	−0.302195	+	0.953246i	$0.597719\pi$
$588$	0	0
$589$	−26.7492	−1.10218
$590$	−9.91330	−0.408124
$591$	0	0
$592$	−66.2710	−2.72372
$593$	36.4392	1.49638	0.748189	−	0.663485i	$-0.230924\pi$
$593$	36.4392	1.49638	0.748189	+	0.663485i	$0.230924\pi$
$594$	0	0
$595$	−2.23081	−0.0914544
$596$	−3.87176	−0.158594
$597$	0	0
$598$	−39.1151	−1.59954
$599$	27.4949	1.12341	0.561705	−	0.827337i	$-0.310145\pi$
$599$	27.4949	1.12341	0.561705	+	0.827337i	$0.310145\pi$
$600$	0	0
$601$	45.1770	1.84281	0.921404	−	0.388606i	$-0.127043\pi$
$601$	45.1770	1.84281	0.921404	+	0.388606i	$0.127043\pi$
$602$	12.1931	0.496952
$603$	0	0
$604$	23.0885	0.939459
$605$	−12.2458	−0.497863
$606$	0	0
$607$	17.1372	0.695576	0.347788	−	0.937573i	$-0.386933\pi$
$607$	17.1372	0.695576	0.347788	+	0.937573i	$0.386933\pi$
$608$	−109.840	−4.45460
$609$	0	0
$610$	−30.9451	−1.25293
$611$	−21.2649	−0.860286
$612$	0	0
$613$	−0.468761	−0.0189331	−0.00946653	−	0.999955i	$-0.503013\pi$
$613$	−0.468761	−0.0189331	−0.00946653	+	0.999955i	$0.503013\pi$
$614$	−8.90715	−0.359463
$615$	0	0
$616$	−8.61470	−0.347096
$617$	−2.13485	−0.0859458	−0.0429729	−	0.999076i	$-0.513683\pi$
$617$	−2.13485	−0.0859458	−0.0429729	+	0.999076i	$0.513683\pi$
$618$	0	0
$619$	8.53760	0.343155	0.171578	−	0.985171i	$-0.445114\pi$
$619$	8.53760	0.343155	0.171578	+	0.985171i	$0.445114\pi$
$620$	−41.0363	−1.64806
$621$	0	0
$622$	94.1163	3.77372
$623$	−5.61608	−0.225004
$624$	0	0
$625$	−9.13058	−0.365223
$626$	−28.7490	−1.14904
$627$	0	0
$628$	−82.5430	−3.29383
$629$	12.9468	0.516222
$630$	0	0
$631$	−11.8708	−0.472569	−0.236284	−	0.971684i	$-0.575930\pi$
$631$	−11.8708	−0.472569	−0.236284	+	0.971684i	$0.575930\pi$
$632$	−47.6613	−1.89587
$633$	0	0
$634$	−41.9291	−1.66522
$635$	3.92063	0.155585
$636$	0	0
$637$	−21.0071	−0.832331
$638$	−13.5658	−0.537076
$639$	0	0
$640$	−45.0973	−1.78263
$641$	5.30366	0.209482	0.104741	−	0.994500i	$-0.466599\pi$
$641$	5.30366	0.209482	0.104741	+	0.994500i	$0.466599\pi$
$642$	0	0
$643$	−0.823869	−0.0324902	−0.0162451	−	0.999868i	$-0.505171\pi$
$643$	−0.823869	−0.0324902	−0.0162451	+	0.999868i	$0.505171\pi$
$644$	12.3665	0.487309
$645$	0	0
$646$	41.7828	1.64392
$647$	−40.8373	−1.60548	−0.802740	−	0.596329i	$-0.796626\pi$
$647$	−40.8373	−1.60548	−0.802740	+	0.596329i	$0.796626\pi$
$648$	0	0
$649$	4.20232	0.164955
$650$	18.5489	0.727547
$651$	0	0
$652$	45.1838	1.76953
$653$	1.06936	0.0418472	0.0209236	−	0.999781i	$-0.493339\pi$
$653$	1.06936	0.0418472	0.0209236	+	0.999781i	$0.493339\pi$
$654$	0	0
$655$	28.6018	1.11756
$656$	157.514	6.14988
$657$	0	0
$658$	9.25175	0.360671
$659$	−28.6643	−1.11660	−0.558301	−	0.829638i	$-0.688547\pi$
$659$	−28.6643	−1.11660	−0.558301	+	0.829638i	$0.688547\pi$
$660$	0	0
$661$	−4.49572	−0.174863	−0.0874316	−	0.996171i	$-0.527866\pi$
$661$	−4.49572	−0.174863	−0.0874316	+	0.996171i	$0.527866\pi$
$662$	39.4135	1.53185
$663$	0	0
$664$	−24.4816	−0.950072
$665$	4.85283	0.188185
$666$	0	0
$667$	12.1493	0.470422
$668$	−123.054	−4.76112
$669$	0	0
$670$	−56.4747	−2.18181
$671$	13.1178	0.506409
$672$	0	0
$673$	18.0586	0.696107	0.348054	−	0.937475i	$-0.386843\pi$
$673$	18.0586	0.696107	0.348054	+	0.937475i	$0.386843\pi$
$674$	55.7643	2.14796
$675$	0	0
$676$	−17.6151	−0.677505
$677$	15.7392	0.604906	0.302453	−	0.953164i	$-0.402195\pi$
$677$	15.7392	0.604906	0.302453	+	0.953164i	$0.402195\pi$
$678$	0	0
$679$	−3.44864	−0.132347
$680$	39.9901	1.53355
$681$	0	0
$682$	23.9385	0.916652
$683$	−2.76118	−0.105654	−0.0528268	−	0.998604i	$-0.516823\pi$
$683$	−2.76118	−0.105654	−0.0528268	+	0.998604i	$0.516823\pi$
$684$	0	0
$685$	−23.2916	−0.889925
$686$	18.6185	0.710858
$687$	0	0
$688$	−122.817	−4.68234
$689$	16.9105	0.644240
$690$	0	0
$691$	−34.5255	−1.31341	−0.656707	−	0.754146i	$-0.728051\pi$
$691$	−34.5255	−1.31341	−0.656707	+	0.754146i	$0.728051\pi$
$692$	13.7107	0.521204
$693$	0	0
$694$	−57.3029	−2.17519
$695$	13.2397	0.502209
$696$	0	0
$697$	−30.7721	−1.16557
$698$	12.5136	0.473645
$699$	0	0
$700$	−5.86436	−0.221652
$701$	−20.7410	−0.783378	−0.391689	−	0.920098i	$-0.628109\pi$
$701$	−20.7410	−0.783378	−0.391689	+	0.920098i	$0.628109\pi$
$702$	0	0
$703$	−28.1640	−1.06222
$704$	45.9842	1.73310
$705$	0	0
$706$	−36.9063	−1.38899
$707$	−1.86703	−0.0702168
$708$	0	0
$709$	27.7071	1.04056	0.520281	−	0.853995i	$-0.325827\pi$
$709$	27.7071	1.04056	0.520281	+	0.853995i	$0.325827\pi$
$710$	−12.8284	−0.481441
$711$	0	0
$712$	100.675	3.77296
$713$	−21.4388	−0.802891
$714$	0	0
$715$	9.98209	0.373309
$716$	47.3041	1.76784
$717$	0	0
$718$	−76.4037	−2.85136
$719$	33.1314	1.23559	0.617797	−	0.786337i	$-0.288025\pi$
$719$	33.1314	1.23559	0.617797	+	0.786337i	$0.288025\pi$
$720$	0	0
$721$	3.84494	0.143193
$722$	−39.4966	−1.46991
$723$	0	0
$724$	42.0702	1.56353
$725$	−5.76134	−0.213971
$726$	0	0
$727$	−0.0826055	−0.00306367	−0.00153183	−	0.999999i	$-0.500488\pi$
$727$	−0.0826055	−0.00306367	−0.00153183	+	0.999999i	$0.500488\pi$
$728$	−13.9816	−0.518191
$729$	0	0
$730$	−44.9449	−1.66349
$731$	23.9936	0.887435
$732$	0	0
$733$	−31.9175	−1.17890	−0.589450	−	0.807805i	$-0.700655\pi$
$733$	−31.9175	−1.17890	−0.589450	+	0.807805i	$0.700655\pi$
$734$	−94.3532	−3.48264
$735$	0	0
$736$	−88.0342	−3.24498
$737$	23.9400	0.881842
$738$	0	0
$739$	−35.7919	−1.31663	−0.658314	−	0.752744i	$-0.728730\pi$
$739$	−35.7919	−1.31663	−0.658314	+	0.752744i	$0.728730\pi$
$740$	−43.2067	−1.58831
$741$	0	0
$742$	−7.35729	−0.270095
$743$	19.7789	0.725616	0.362808	−	0.931864i	$-0.381818\pi$
$743$	19.7789	0.725616	0.362808	+	0.931864i	$0.381818\pi$
$744$	0	0
$745$	−1.21772	−0.0446138
$746$	−8.27598	−0.303005
$747$	0	0
$748$	−27.1723	−0.993517
$749$	5.38924	0.196919
$750$	0	0
$751$	30.5806	1.11590	0.557950	−	0.829874i	$-0.311588\pi$
$751$	30.5806	1.11590	0.557950	+	0.829874i	$0.311588\pi$
$752$	−93.1899	−3.39829
$753$	0	0
$754$	−22.0172	−0.801818
$755$	7.26165	0.264278
$756$	0	0
$757$	25.4129	0.923647	0.461824	−	0.886972i	$-0.347195\pi$
$757$	25.4129	0.923647	0.461824	+	0.886972i	$0.347195\pi$
$758$	20.7669	0.754289
$759$	0	0
$760$	−86.9931	−3.15557
$761$	46.8180	1.69715	0.848575	−	0.529075i	$-0.177461\pi$
$761$	46.8180	1.69715	0.848575	+	0.529075i	$0.177461\pi$
$762$	0	0
$763$	6.12215	0.221637
$764$	84.5322	3.05827
$765$	0	0
$766$	28.0297	1.01275
$767$	6.82032	0.246267
$768$	0	0
$769$	14.5654	0.525243	0.262621	−	0.964899i	$-0.415413\pi$
$769$	14.5654	0.525243	0.262621	+	0.964899i	$0.415413\pi$
$770$	−4.34292	−0.156508
$771$	0	0
$772$	−24.3790	−0.877419
$773$	−24.3533	−0.875929	−0.437964	−	0.898992i	$-0.644300\pi$
$773$	−24.3533	−0.875929	−0.437964	+	0.898992i	$0.644300\pi$
$774$	0	0
$775$	10.1666	0.365194
$776$	61.8212	2.21925
$777$	0	0
$778$	−1.15493	−0.0414063
$779$	66.9405	2.39839
$780$	0	0
$781$	5.43804	0.194589
$782$	33.4879	1.19753
$783$	0	0
$784$	−92.0601	−3.28786
$785$	−25.9609	−0.926583
$786$	0	0
$787$	−27.5672	−0.982664	−0.491332	−	0.870972i	$-0.663490\pi$
$787$	−27.5672	−0.982664	−0.491332	+	0.870972i	$0.663490\pi$
$788$	−15.9211	−0.567166
$789$	0	0
$790$	−24.0274	−0.854858
$791$	0.973124	0.0346003
$792$	0	0
$793$	21.2901	0.756034
$794$	73.2777	2.60053
$795$	0	0
$796$	79.1677	2.80603
$797$	6.61692	0.234383	0.117192	−	0.993109i	$-0.462611\pi$
$797$	6.61692	0.234383	0.117192	+	0.993109i	$0.462611\pi$
$798$	0	0
$799$	18.2057	0.644070
$800$	41.7469	1.47598
$801$	0	0
$802$	79.4220	2.80449
$803$	19.0525	0.672347
$804$	0	0
$805$	3.88943	0.137084
$806$	38.8519	1.36850
$807$	0	0
$808$	33.4688	1.17743
$809$	8.61362	0.302839	0.151419	−	0.988470i	$-0.451616\pi$
$809$	8.61362	0.302839	0.151419	+	0.988470i	$0.451616\pi$
$810$	0	0
$811$	−9.58716	−0.336651	−0.168325	−	0.985732i	$-0.553836\pi$
$811$	−9.58716	−0.336651	−0.168325	+	0.985732i	$0.553836\pi$
$812$	6.96088	0.244279
$813$	0	0
$814$	25.2046	0.883422
$815$	14.2109	0.497786
$816$	0	0
$817$	−52.1949	−1.82607
$818$	54.5988	1.90900
$819$	0	0
$820$	102.694	3.58624
$821$	13.8635	0.483838	0.241919	−	0.970296i	$-0.422223\pi$
$821$	13.8635	0.483838	0.241919	+	0.970296i	$0.422223\pi$
$822$	0	0
$823$	47.8257	1.66710	0.833550	−	0.552444i	$-0.186305\pi$
$823$	47.8257	1.66710	0.833550	+	0.552444i	$0.186305\pi$
$824$	−68.9253	−2.40113
$825$	0	0
$826$	−2.96733	−0.103246
$827$	42.4417	1.47584	0.737921	−	0.674887i	$-0.235808\pi$
$827$	42.4417	1.47584	0.737921	+	0.674887i	$0.235808\pi$
$828$	0	0
$829$	26.0037	0.903146	0.451573	−	0.892234i	$-0.350863\pi$
$829$	26.0037	0.903146	0.451573	+	0.892234i	$0.350863\pi$
$830$	−12.3419	−0.428394
$831$	0	0
$832$	74.6319	2.58740
$833$	17.9850	0.623142
$834$	0	0
$835$	−38.7022	−1.33935
$836$	59.1096	2.04435
$837$	0	0
$838$	65.9233	2.27728
$839$	−1.25628	−0.0433715	−0.0216857	−	0.999765i	$-0.506903\pi$
$839$	−1.25628	−0.0433715	−0.0216857	+	0.999765i	$0.506903\pi$
$840$	0	0
$841$	−22.1614	−0.764186
$842$	70.8038	2.44006
$843$	0	0
$844$	73.7996	2.54029
$845$	−5.54019	−0.190588
$846$	0	0
$847$	−3.66551	−0.125948
$848$	74.1076	2.54487
$849$	0	0
$850$	−15.8804	−0.544693
$851$	−22.5728	−0.773784
$852$	0	0
$853$	1.45435	0.0497959	0.0248979	−	0.999690i	$-0.492074\pi$
$853$	1.45435	0.0497959	0.0248979	+	0.999690i	$0.492074\pi$
$854$	−9.26272	−0.316964
$855$	0	0
$856$	−96.6089	−3.30202
$857$	52.3097	1.78686	0.893432	−	0.449198i	$-0.148290\pi$
$857$	52.3097	1.78686	0.893432	+	0.449198i	$0.148290\pi$
$858$	0	0
$859$	−21.0043	−0.716659	−0.358329	−	0.933595i	$-0.616654\pi$
$859$	−21.0043	−0.716659	−0.358329	+	0.933595i	$0.616654\pi$
$860$	−80.0728	−2.73046
$861$	0	0
$862$	86.3417	2.94081
$863$	−12.9813	−0.441890	−0.220945	−	0.975286i	$-0.570914\pi$
$863$	−12.9813	−0.441890	−0.220945	+	0.975286i	$0.570914\pi$
$864$	0	0
$865$	4.31221	0.146619
$866$	−0.0332939	−0.00113137
$867$	0	0
$868$	−12.2833	−0.416922
$869$	10.1854	0.345516
$870$	0	0
$871$	38.8544	1.31653
$872$	−109.747	−3.71650
$873$	0	0
$874$	−72.8485	−2.46414
$875$	−6.03032	−0.203862
$876$	0	0
$877$	31.0457	1.04834	0.524169	−	0.851614i	$-0.324376\pi$
$877$	31.0457	1.04834	0.524169	+	0.851614i	$0.324376\pi$
$878$	−14.3498	−0.484284
$879$	0	0
$880$	43.7448	1.47464
$881$	−19.2957	−0.650087	−0.325044	−	0.945699i	$-0.605379\pi$
$881$	−19.2957	−0.650087	−0.325044	+	0.945699i	$0.605379\pi$
$882$	0	0
$883$	−9.82388	−0.330600	−0.165300	−	0.986243i	$-0.552859\pi$
$883$	−9.82388	−0.330600	−0.165300	+	0.986243i	$0.552859\pi$
$884$	−44.1003	−1.48325
$885$	0	0
$886$	110.086	3.69842
$887$	−44.1133	−1.48118	−0.740590	−	0.671958i	$-0.765454\pi$
$887$	−44.1133	−1.48118	−0.740590	+	0.671958i	$0.765454\pi$
$888$	0	0
$889$	1.17355	0.0393597
$890$	50.7533	1.70125
$891$	0	0
$892$	−36.2219	−1.21280
$893$	−39.6040	−1.32530
$894$	0	0
$895$	14.8778	0.497308
$896$	−13.4989	−0.450965
$897$	0	0
$898$	41.7338	1.39268
$899$	−12.0675	−0.402474
$900$	0	0
$901$	−14.4777	−0.482323
$902$	−59.9067	−1.99467
$903$	0	0
$904$	−17.4444	−0.580194
$905$	13.2316	0.439834
$906$	0	0
$907$	−42.4124	−1.40828	−0.704140	−	0.710061i	$-0.748667\pi$
$907$	−42.4124	−1.40828	−0.704140	+	0.710061i	$0.748667\pi$
$908$	51.7957	1.71890
$909$	0	0
$910$	−7.04851	−0.233656
$911$	−36.7104	−1.21627	−0.608134	−	0.793834i	$-0.708082\pi$
$911$	−36.7104	−1.21627	−0.608134	+	0.793834i	$0.708082\pi$
$912$	0	0
$913$	5.23182	0.173148
$914$	−6.46171	−0.213735
$915$	0	0
$916$	−74.6817	−2.46755
$917$	8.56130	0.282719
$918$	0	0
$919$	−14.0589	−0.463761	−0.231881	−	0.972744i	$-0.574488\pi$
$919$	−14.0589	−0.463761	−0.231881	+	0.972744i	$0.574488\pi$
$920$	−69.7229	−2.29870
$921$	0	0
$922$	87.8710	2.89388
$923$	8.82588	0.290508
$924$	0	0
$925$	10.7043	0.351955
$926$	−91.6436	−3.01160
$927$	0	0
$928$	−49.5528	−1.62665
$929$	18.5175	0.607539	0.303770	−	0.952746i	$-0.401755\pi$
$929$	18.5175	0.607539	0.303770	+	0.952746i	$0.401755\pi$
$930$	0	0
$931$	−39.1239	−1.28223
$932$	−28.3062	−0.927199
$933$	0	0
$934$	−37.3371	−1.22171
$935$	−8.54604	−0.279485
$936$	0	0
$937$	4.46818	0.145969	0.0729845	−	0.997333i	$-0.476748\pi$
$937$	4.46818	0.145969	0.0729845	+	0.997333i	$0.476748\pi$
$938$	−16.9044	−0.551949
$939$	0	0
$940$	−60.7570	−1.98168
$941$	2.00386	0.0653239	0.0326620	−	0.999466i	$-0.489602\pi$
$941$	2.00386	0.0653239	0.0326620	+	0.999466i	$0.489602\pi$
$942$	0	0
$943$	53.6512	1.74712
$944$	29.8889	0.972801
$945$	0	0
$946$	46.7105	1.51869
$947$	12.5838	0.408919	0.204460	−	0.978875i	$-0.434456\pi$
$947$	12.5838	0.408919	0.204460	+	0.978875i	$0.434456\pi$
$948$	0	0
$949$	30.9220	1.00377
$950$	34.5457	1.12081
$951$	0	0
$952$	11.9701	0.387955
$953$	19.9641	0.646700	0.323350	−	0.946279i	$-0.395191\pi$
$953$	19.9641	0.646700	0.323350	+	0.946279i	$0.395191\pi$
$954$	0	0
$955$	26.5865	0.860318
$956$	−94.5084	−3.05662
$957$	0	0
$958$	15.9359	0.514864
$959$	−6.97181	−0.225132
$960$	0	0
$961$	−9.70544	−0.313079
$962$	40.9068	1.31889
$963$	0	0
$964$	−10.6709	−0.343686
$965$	−7.66752	−0.246826
$966$	0	0
$967$	−32.2086	−1.03576	−0.517879	−	0.855454i	$-0.673278\pi$
$967$	−32.2086	−1.03576	−0.517879	+	0.855454i	$0.673278\pi$
$968$	65.7088	2.11196
$969$	0	0
$970$	31.1659	1.00068
$971$	6.62934	0.212746	0.106373	−	0.994326i	$-0.466076\pi$
$971$	6.62934	0.212746	0.106373	+	0.994326i	$0.466076\pi$
$972$	0	0
$973$	3.96300	0.127048
$974$	−78.5278	−2.51619
$975$	0	0
$976$	93.3004	2.98647
$977$	−11.8566	−0.379328	−0.189664	−	0.981849i	$-0.560740\pi$
$977$	−11.8566	−0.379328	−0.189664	+	0.981849i	$0.560740\pi$
$978$	0	0
$979$	−21.5147	−0.687612
$980$	−60.0205	−1.91728
$981$	0	0
$982$	11.8577	0.378394
$983$	52.0650	1.66062	0.830308	−	0.557305i	$-0.188165\pi$
$983$	52.0650	1.66062	0.830308	+	0.557305i	$0.188165\pi$
$984$	0	0
$985$	−5.00740	−0.159549
$986$	18.8497	0.600297
$987$	0	0
$988$	95.9343	3.05208
$989$	−41.8330	−1.33021
$990$	0	0
$991$	1.47115	0.0467326	0.0233663	−	0.999727i	$-0.492562\pi$
$991$	1.47115	0.0467326	0.0233663	+	0.999727i	$0.492562\pi$
$992$	87.4418	2.77628
$993$	0	0
$994$	−3.83989	−0.121794
$995$	24.8993	0.789361
$996$	0	0
$997$	−26.4213	−0.836771	−0.418385	−	0.908270i	$-0.637404\pi$
$997$	−26.4213	−0.836771	−0.418385	+	0.908270i	$0.637404\pi$
$998$	−95.6065	−3.02637
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.70506 q^{2} +5.31738 q^{4} +1.67238 q^{5} +0.500591 q^{7} -8.97372 q^{8} -4.52391 q^{10} +1.91772 q^{11} +3.11244 q^{13} -1.35413 q^{14} +13.6397 q^{16} -2.66467 q^{17} +5.79664 q^{19} +8.89270 q^{20} -5.18755 q^{22} +4.64587 q^{23} -2.20313 q^{25} -8.41934 q^{26} +2.66183 q^{28} +2.61507 q^{29} -4.61460 q^{31} -18.9489 q^{32} +7.20811 q^{34} +0.837181 q^{35} -4.85867 q^{37} -15.6803 q^{38} -15.0075 q^{40} +11.5482 q^{41} -9.00434 q^{43} +10.1972 q^{44} -12.5674 q^{46} -6.83224 q^{47} -6.74941 q^{49} +5.95961 q^{50} +16.5500 q^{52} +5.43322 q^{53} +3.20716 q^{55} -4.49216 q^{56} -7.07393 q^{58} +2.19131 q^{59} +6.84034 q^{61} +12.4828 q^{62} +23.9786 q^{64} +5.20519 q^{65} +12.4836 q^{67} -14.1691 q^{68} -2.26463 q^{70} +2.83568 q^{71} +9.93497 q^{73} +13.1430 q^{74} +30.8229 q^{76} +0.959993 q^{77} +5.31121 q^{79} +22.8109 q^{80} -31.2385 q^{82} +2.72815 q^{83} -4.45636 q^{85} +24.3573 q^{86} -17.2091 q^{88} -11.2189 q^{89} +1.55806 q^{91} +24.7038 q^{92} +18.4816 q^{94} +9.69421 q^{95} -6.88914 q^{97} +18.2576 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more