LMFDB - Embedded newform 729.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,4,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	729.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$43.0123923942$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 48 x^{10} + 269 x^{9} + 900 x^{8} - 4059 x^{7} - 8325 x^{6} + 23940 x^{5} + \cdots - 3392 $ x^12 - 6x^11 - 48x^10 + 269x^9 + 900x^8 - 4059x^7 - 8325x^6 + 23940x^5 + 32976x^4 - 55288x^3 - 40224x^2 + 45408*x - 3392
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.67624$ 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)

$f(q)$	$=$	\(q-4.67624 q^{2} +13.8672 q^{4} +4.95489 q^{5} +6.52395 q^{7} -27.4363 q^{8} -23.1702 q^{10} +57.3020 q^{11} -78.5988 q^{13} -30.5075 q^{14} +17.3611 q^{16} +78.3576 q^{17} +8.61844 q^{19} +68.7103 q^{20} -267.958 q^{22} -93.6380 q^{23} -100.449 q^{25} +367.547 q^{26} +90.4687 q^{28} -241.900 q^{29} -47.6877 q^{31} +138.306 q^{32} -366.418 q^{34} +32.3254 q^{35} -325.243 q^{37} -40.3018 q^{38} -135.944 q^{40} -208.403 q^{41} +445.599 q^{43} +794.617 q^{44} +437.873 q^{46} +24.6137 q^{47} -300.438 q^{49} +469.724 q^{50} -1089.94 q^{52} +208.946 q^{53} +283.925 q^{55} -178.993 q^{56} +1131.18 q^{58} -628.898 q^{59} +116.895 q^{61} +222.999 q^{62} -785.638 q^{64} -389.448 q^{65} +466.665 q^{67} +1086.60 q^{68} -151.161 q^{70} +372.746 q^{71} +90.8124 q^{73} +1520.91 q^{74} +119.513 q^{76} +373.835 q^{77} -301.801 q^{79} +86.0225 q^{80} +974.540 q^{82} +1073.01 q^{83} +388.253 q^{85} -2083.73 q^{86} -1572.15 q^{88} +1075.23 q^{89} -512.775 q^{91} -1298.49 q^{92} -115.099 q^{94} +42.7034 q^{95} -1589.67 q^{97} +1404.92 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} + 36 q^{4} + 12 q^{5} - 42 q^{7} + 21 q^{8} - 60 q^{10} + 42 q^{11} - 78 q^{13} - 312 q^{14} + 48 q^{16} - 18 q^{17} - 228 q^{19} - 69 q^{20} - 309 q^{22} - 114 q^{23} - 18 q^{25} + 30 q^{26}+ \cdots + 9567 q^{98}+O(q^{100}) $ 12 * q + 6 * q^2 + 36 * q^4 + 12 * q^5 - 42 * q^7 + 21 * q^8 - 60 * q^10 + 42 * q^11 - 78 * q^13 - 312 * q^14 + 48 * q^16 - 18 * q^17 - 228 * q^19 - 69 * q^20 - 309 * q^22 - 114 * q^23 - 18 * q^25 + 30 * q^26 - 813 * q^28 - 660 * q^29 - 708 * q^31 + 729 * q^32 - 972 * q^34 - 624 * q^35 - 354 * q^37 - 213 * q^38 - 1335 * q^40 - 1032 * q^41 - 744 * q^43 + 1653 * q^44 - 1077 * q^46 + 942 * q^47 - 192 * q^49 + 1905 * q^50 - 1371 * q^52 - 828 * q^53 - 1554 * q^55 - 3324 * q^56 - 831 * q^58 - 24 * q^59 - 1698 * q^61 - 2184 * q^62 - 933 * q^64 + 294 * q^65 - 1266 * q^67 - 4734 * q^68 - 1137 * q^70 + 3888 * q^71 - 1164 * q^73 + 5448 * q^74 - 2385 * q^76 + 3018 * q^77 - 2382 * q^79 - 4677 * q^80 - 276 * q^82 - 4008 * q^83 - 1116 * q^85 - 1176 * q^86 - 2769 * q^88 - 3582 * q^89 - 3222 * q^91 - 2958 * q^92 - 3324 * q^94 - 2784 * q^95 - 2958 * q^97 + 9567 * 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 $

$2$	−4.67624	−1.65330	−0.826649	−	0.562717i	$-0.809756\pi$
$2$	−4.67624	−1.65330	−0.826649	+	0.562717i	$0.809756\pi$
$3$	0	0
$3$	0	0
$4$	13.8672	1.73340
$5$	4.95489	0.443179	0.221589	−	0.975140i	$-0.428876\pi$
$5$	4.95489	0.443179	0.221589	+	0.975140i	$0.428876\pi$
$6$	0	0
$7$	6.52395	0.352260	0.176130	−	0.984367i	$-0.443642\pi$
$7$	6.52395	0.352260	0.176130	+	0.984367i	$0.443642\pi$
$8$	−27.4363	−1.21252
$9$	0	0
$10$	−23.1702	−0.732707
$11$	57.3020	1.57065	0.785327	−	0.619081i	$-0.212495\pi$
$11$	57.3020	1.57065	0.785327	+	0.619081i	$0.212495\pi$
$12$	0	0
$13$	−78.5988	−1.67688	−0.838438	−	0.544997i	$-0.816531\pi$
$13$	−78.5988	−1.67688	−0.838438	+	0.544997i	$0.816531\pi$
$14$	−30.5075	−0.582391
$15$	0	0
$16$	17.3611	0.271268
$17$	78.3576	1.11791	0.558956	−	0.829197i	$-0.311202\pi$
$17$	78.3576	1.11791	0.558956	+	0.829197i	$0.311202\pi$
$18$	0	0
$19$	8.61844	0.104063	0.0520317	−	0.998645i	$-0.483430\pi$
$19$	8.61844	0.104063	0.0520317	+	0.998645i	$0.483430\pi$
$20$	68.7103	0.768205
$21$	0	0
$22$	−267.958	−2.59676
$23$	−93.6380	−0.848907	−0.424454	−	0.905450i	$-0.639534\pi$
$23$	−93.6380	−0.848907	−0.424454	+	0.905450i	$0.639534\pi$
$24$	0	0
$25$	−100.449	−0.803593
$26$	367.547	2.77238
$27$	0	0
$28$	90.4687	0.610606
$29$	−241.900	−1.54895	−0.774477	−	0.632602i	$-0.781987\pi$
$29$	−241.900	−1.54895	−0.774477	+	0.632602i	$0.781987\pi$
$30$	0	0
$31$	−47.6877	−0.276289	−0.138145	−	0.990412i	$-0.544114\pi$
$31$	−47.6877	−0.276289	−0.138145	+	0.990412i	$0.544114\pi$
$32$	138.306	0.764037
$33$	0	0
$34$	−366.418	−1.84824
$35$	32.3254	0.156114
$36$	0	0
$37$	−325.243	−1.44512	−0.722562	−	0.691306i	$-0.757036\pi$
$37$	−325.243	−1.44512	−0.722562	+	0.691306i	$0.757036\pi$
$38$	−40.3018	−0.172048
$39$	0	0
$40$	−135.944	−0.537365
$41$	−208.403	−0.793830	−0.396915	−	0.917855i	$-0.629919\pi$
$41$	−208.403	−0.793830	−0.396915	+	0.917855i	$0.629919\pi$
$42$	0	0
$43$	445.599	1.58031	0.790153	−	0.612909i	$-0.210001\pi$
$43$	445.599	1.58031	0.790153	+	0.612909i	$0.210001\pi$
$44$	794.617	2.72257
$45$	0	0
$46$	437.873	1.40350
$47$	24.6137	0.0763889	0.0381944	−	0.999270i	$-0.487839\pi$
$47$	24.6137	0.0763889	0.0381944	+	0.999270i	$0.487839\pi$
$48$	0	0
$49$	−300.438	−0.875913
$50$	469.724	1.32858
$51$	0	0
$52$	−1089.94	−2.90669
$53$	208.946	0.541527	0.270763	−	0.962646i	$-0.412724\pi$
$53$	208.946	0.541527	0.270763	+	0.962646i	$0.412724\pi$
$54$	0	0
$55$	283.925	0.696081
$56$	−178.993	−0.427124
$57$	0	0
$58$	1131.18	2.56088
$59$	−628.898	−1.38772	−0.693861	−	0.720109i	$-0.744092\pi$
$59$	−628.898	−1.38772	−0.693861	+	0.720109i	$0.744092\pi$
$60$	0	0
$61$	116.895	0.245358	0.122679	−	0.992446i	$-0.460851\pi$
$61$	116.895	0.245358	0.122679	+	0.992446i	$0.460851\pi$
$62$	222.999	0.456789
$63$	0	0
$64$	−785.638	−1.53445
$65$	−389.448	−0.743156
$66$	0	0
$67$	466.665	0.850928	0.425464	−	0.904975i	$-0.360111\pi$
$67$	466.665	0.850928	0.425464	+	0.904975i	$0.360111\pi$
$68$	1086.60	1.93778
$69$	0	0
$70$	−151.161	−0.258103
$71$	372.746	0.623054	0.311527	−	0.950237i	$-0.399160\pi$
$71$	372.746	0.623054	0.311527	+	0.950237i	$0.399160\pi$
$72$	0	0
$73$	90.8124	0.145600	0.0727999	−	0.997347i	$-0.476807\pi$
$73$	90.8124	0.145600	0.0727999	+	0.997347i	$0.476807\pi$
$74$	1520.91	2.38922
$75$	0	0
$76$	119.513	0.180383
$77$	373.835	0.553279
$78$	0	0
$79$	−301.801	−0.429813	−0.214907	−	0.976635i	$-0.568945\pi$
$79$	−301.801	−0.429813	−0.214907	+	0.976635i	$0.568945\pi$
$80$	86.0225	0.120220
$81$	0	0
$82$	974.540	1.31244
$83$	1073.01	1.41901	0.709505	−	0.704700i	$-0.248918\pi$
$83$	1073.01	1.41901	0.709505	+	0.704700i	$0.248918\pi$
$84$	0	0
$85$	388.253	0.495435
$86$	−2083.73	−2.61272
$87$	0	0
$88$	−1572.15	−1.90446
$89$	1075.23	1.28061	0.640304	−	0.768122i	$-0.278808\pi$
$89$	1075.23	1.28061	0.640304	+	0.768122i	$0.278808\pi$
$90$	0	0
$91$	−512.775	−0.590697
$92$	−1298.49	−1.47149
$93$	0	0
$94$	−115.099	−0.126294
$95$	42.7034	0.0461187
$96$	0	0
$97$	−1589.67	−1.66399	−0.831993	−	0.554786i	$-0.812800\pi$
$97$	−1589.67	−1.66399	−0.831993	+	0.554786i	$0.812800\pi$
$98$	1404.92	1.44815
$99$	0	0
$100$	−1392.94	−1.39294
$101$	−709.897	−0.699380	−0.349690	−	0.936865i	$-0.613713\pi$
$101$	−709.897	−0.699380	−0.349690	+	0.936865i	$0.613713\pi$
$102$	0	0
$103$	348.002	0.332909	0.166454	−	0.986049i	$-0.446768\pi$
$103$	348.002	0.332909	0.166454	+	0.986049i	$0.446768\pi$
$104$	2156.46	2.03325
$105$	0	0
$106$	−977.079	−0.895305
$107$	352.787	0.318741	0.159370	−	0.987219i	$-0.449054\pi$
$107$	352.787	0.318741	0.159370	+	0.987219i	$0.449054\pi$
$108$	0	0
$109$	−1340.07	−1.17757	−0.588786	−	0.808289i	$-0.700394\pi$
$109$	−1340.07	−1.17757	−0.588786	+	0.808289i	$0.700394\pi$
$110$	−1327.70	−1.15083
$111$	0	0
$112$	113.263	0.0955568
$113$	−187.760	−0.156310	−0.0781549	−	0.996941i	$-0.524903\pi$
$113$	−187.760	−0.156310	−0.0781549	+	0.996941i	$0.524903\pi$
$114$	0	0
$115$	−463.966	−0.376218
$116$	−3354.47	−2.68495
$117$	0	0
$118$	2940.88	2.29432
$119$	511.201	0.393796
$120$	0	0
$121$	1952.52	1.46695
$122$	−546.628	−0.405651
$123$	0	0
$124$	−661.294	−0.478919
$125$	−1117.08	−0.799314
$126$	0	0
$127$	2397.26	1.67498	0.837488	−	0.546455i	$-0.184023\pi$
$127$	2397.26	1.67498	0.837488	+	0.546455i	$0.184023\pi$
$128$	2567.39	1.77287
$129$	0	0
$130$	1821.15	1.22866
$131$	−2006.77	−1.33842	−0.669208	−	0.743075i	$-0.733366\pi$
$131$	−2006.77	−1.33842	−0.669208	+	0.743075i	$0.733366\pi$
$132$	0	0
$133$	56.2262	0.0366574
$134$	−2182.23	−1.40684
$135$	0	0
$136$	−2149.84	−1.35550
$137$	−1930.91	−1.20415	−0.602075	−	0.798440i	$-0.705659\pi$
$137$	−1930.91	−1.20415	−0.602075	+	0.798440i	$0.705659\pi$
$138$	0	0
$139$	−3111.97	−1.89895	−0.949473	−	0.313848i	$-0.898382\pi$
$139$	−3111.97	−1.89895	−0.949473	+	0.313848i	$0.898382\pi$
$140$	448.262	0.270608
$141$	0	0
$142$	−1743.05	−1.03009
$143$	−4503.87	−2.63379
$144$	0	0
$145$	−1198.59	−0.686463
$146$	−424.660	−0.240720
$147$	0	0
$148$	−4510.20	−2.50497
$149$	−712.220	−0.391593	−0.195797	−	0.980645i	$-0.562729\pi$
$149$	−712.220	−0.391593	−0.195797	+	0.980645i	$0.562729\pi$
$150$	0	0
$151$	−2325.67	−1.25338	−0.626690	−	0.779269i	$-0.715591\pi$
$151$	−2325.67	−1.25338	−0.626690	+	0.779269i	$0.715591\pi$
$152$	−236.458	−0.126179
$153$	0	0
$154$	−1748.14	−0.914735
$155$	−236.287	−0.122446
$156$	0	0
$157$	−3007.53	−1.52884	−0.764418	−	0.644720i	$-0.776974\pi$
$157$	−3007.53	−1.52884	−0.764418	+	0.644720i	$0.776974\pi$
$158$	1411.29	0.710610
$159$	0	0
$160$	685.289	0.338605
$161$	−610.889	−0.299036
$162$	0	0
$163$	757.816	0.364152	0.182076	−	0.983284i	$-0.441718\pi$
$163$	757.816	0.364152	0.182076	+	0.983284i	$0.441718\pi$
$164$	−2889.96	−1.37602
$165$	0	0
$166$	−5017.64	−2.34605
$167$	−2046.23	−0.948157	−0.474079	−	0.880483i	$-0.657219\pi$
$167$	−2046.23	−0.948157	−0.474079	+	0.880483i	$0.657219\pi$
$168$	0	0
$169$	3980.78	1.81191
$170$	−1815.56	−0.819102
$171$	0	0
$172$	6179.20	2.73930
$173$	857.921	0.377032	0.188516	−	0.982070i	$-0.439632\pi$
$173$	857.921	0.377032	0.188516	+	0.982070i	$0.439632\pi$
$174$	0	0
$175$	−655.324	−0.283074
$176$	994.827	0.426068
$177$	0	0
$178$	−5028.03	−2.11723
$179$	−1188.07	−0.496092	−0.248046	−	0.968748i	$-0.579788\pi$
$179$	−1188.07	−0.496092	−0.248046	+	0.968748i	$0.579788\pi$
$180$	0	0
$181$	−3062.18	−1.25751	−0.628757	−	0.777602i	$-0.716436\pi$
$181$	−3062.18	−1.25751	−0.628757	+	0.777602i	$0.716436\pi$
$182$	2397.85	0.976598
$183$	0	0
$184$	2569.08	1.02932
$185$	−1611.54	−0.640449
$186$	0	0
$187$	4490.04	1.75585
$188$	341.322	0.132412
$189$	0	0
$190$	−199.691	−0.0762480
$191$	1143.09	0.433041	0.216520	−	0.976278i	$-0.430529\pi$
$191$	1143.09	0.433041	0.216520	+	0.976278i	$0.430529\pi$
$192$	0	0
$193$	−402.366	−0.150067	−0.0750336	−	0.997181i	$-0.523906\pi$
$193$	−402.366	−0.150067	−0.0750336	+	0.997181i	$0.523906\pi$
$194$	7433.68	2.75107
$195$	0	0
$196$	−4166.23	−1.51830
$197$	−1142.27	−0.413115	−0.206558	−	0.978434i	$-0.566226\pi$
$197$	−1142.27	−0.413115	−0.206558	+	0.978434i	$0.566226\pi$
$198$	0	0
$199$	−1985.63	−0.707323	−0.353662	−	0.935373i	$-0.615064\pi$
$199$	−1985.63	−0.707323	−0.353662	+	0.935373i	$0.615064\pi$
$200$	2755.95	0.974375
$201$	0	0
$202$	3319.65	1.15628
$203$	−1578.14	−0.545635
$204$	0	0
$205$	−1032.61	−0.351809
$206$	−1627.34	−0.550398
$207$	0	0
$208$	−1364.56	−0.454882
$209$	493.853	0.163448
$210$	0	0
$211$	4396.49	1.43444	0.717220	−	0.696847i	$-0.245414\pi$
$211$	4396.49	1.43444	0.717220	+	0.696847i	$0.245414\pi$
$212$	2897.49	0.938680
$213$	0	0
$214$	−1649.72	−0.526973
$215$	2207.89	0.700358
$216$	0	0
$217$	−311.112	−0.0973257
$218$	6266.48	1.94688
$219$	0	0
$220$	3937.24	1.20658
$221$	−6158.81	−1.87460
$222$	0	0
$223$	3098.83	0.930551	0.465276	−	0.885166i	$-0.345955\pi$
$223$	3098.83	0.930551	0.465276	+	0.885166i	$0.345955\pi$
$224$	902.298	0.269140
$225$	0	0
$226$	878.012	0.258427
$227$	1069.74	0.312782	0.156391	−	0.987695i	$-0.450014\pi$
$227$	1069.74	0.312782	0.156391	+	0.987695i	$0.450014\pi$
$228$	0	0
$229$	3833.71	1.10628	0.553141	−	0.833088i	$-0.313429\pi$
$229$	3833.71	1.10628	0.553141	+	0.833088i	$0.313429\pi$
$230$	2169.61	0.622000
$231$	0	0
$232$	6636.83	1.87814
$233$	1039.53	0.292284	0.146142	−	0.989264i	$-0.453314\pi$
$233$	1039.53	0.292284	0.146142	+	0.989264i	$0.453314\pi$
$234$	0	0
$235$	121.958	0.0338539
$236$	−8721.04	−2.40547
$237$	0	0
$238$	−2390.49	−0.651062
$239$	−946.437	−0.256150	−0.128075	−	0.991764i	$-0.540880\pi$
$239$	−946.437	−0.256150	−0.128075	+	0.991764i	$0.540880\pi$
$240$	0	0
$241$	−4085.70	−1.09205	−0.546023	−	0.837770i	$-0.683859\pi$
$241$	−4085.70	−1.09205	−0.546023	+	0.837770i	$0.683859\pi$
$242$	−9130.43	−2.42531
$243$	0	0
$244$	1621.00	0.425304
$245$	−1488.64	−0.388186
$246$	0	0
$247$	−677.399	−0.174501
$248$	1308.37	0.335007
$249$	0	0
$250$	5223.71	1.32150
$251$	7193.10	1.80886	0.904431	−	0.426620i	$-0.140296\pi$
$251$	7193.10	1.80886	0.904431	+	0.426620i	$0.140296\pi$
$252$	0	0
$253$	−5365.64	−1.33334
$254$	−11210.1	−2.76924
$255$	0	0
$256$	−5720.59	−1.39663
$257$	−518.281	−0.125796	−0.0628978	−	0.998020i	$-0.520034\pi$
$257$	−518.281	−0.125796	−0.0628978	+	0.998020i	$0.520034\pi$
$258$	0	0
$259$	−2121.87	−0.509060
$260$	−5400.55	−1.28818
$261$	0	0
$262$	9384.13	2.21280
$263$	−2727.63	−0.639516	−0.319758	−	0.947499i	$-0.603602\pi$
$263$	−2727.63	−0.639516	−0.319758	+	0.947499i	$0.603602\pi$
$264$	0	0
$265$	1035.30	0.239993
$266$	−262.927	−0.0606056
$267$	0	0
$268$	6471.32	1.47500
$269$	2110.73	0.478415	0.239208	−	0.970968i	$-0.423112\pi$
$269$	2110.73	0.478415	0.239208	+	0.970968i	$0.423112\pi$
$270$	0	0
$271$	−3783.81	−0.848156	−0.424078	−	0.905626i	$-0.639402\pi$
$271$	−3783.81	−0.848156	−0.424078	+	0.905626i	$0.639402\pi$
$272$	1360.38	0.303253
$273$	0	0
$274$	9029.38	1.99082
$275$	−5755.93	−1.26217
$276$	0	0
$277$	−2422.91	−0.525555	−0.262777	−	0.964856i	$-0.584638\pi$
$277$	−2422.91	−0.525555	−0.262777	+	0.964856i	$0.584638\pi$
$278$	14552.3	3.13953
$279$	0	0
$280$	−886.890	−0.189292
$281$	2644.49	0.561413	0.280707	−	0.959794i	$-0.409431\pi$
$281$	2644.49	0.561413	0.280707	+	0.959794i	$0.409431\pi$
$282$	0	0
$283$	7274.58	1.52802	0.764008	−	0.645207i	$-0.223229\pi$
$283$	7274.58	1.52802	0.764008	+	0.645207i	$0.223229\pi$
$284$	5168.94	1.08000
$285$	0	0
$286$	21061.1	4.35445
$287$	−1359.61	−0.279635
$288$	0	0
$289$	1226.91	0.249727
$290$	5604.87	1.13493
$291$	0	0
$292$	1259.31	0.252382
$293$	−7627.23	−1.52078	−0.760389	−	0.649468i	$-0.774992\pi$
$293$	−7627.23	−1.52078	−0.760389	+	0.649468i	$0.774992\pi$
$294$	0	0
$295$	−3116.12	−0.615009
$296$	8923.46	1.75225
$297$	0	0
$298$	3330.51	0.647420
$299$	7359.84	1.42351
$300$	0	0
$301$	2907.06	0.556679
$302$	10875.4	2.07221
$303$	0	0
$304$	149.626	0.0282290
$305$	579.202	0.108738
$306$	0	0
$307$	−9095.70	−1.69094	−0.845471	−	0.534022i	$-0.820680\pi$
$307$	−9095.70	−1.69094	−0.845471	+	0.534022i	$0.820680\pi$
$308$	5184.04	0.959052
$309$	0	0
$310$	1104.94	0.202439
$311$	−101.672	−0.0185380	−0.00926899	−	0.999957i	$-0.502950\pi$
$311$	−101.672	−0.0185380	−0.00926899	+	0.999957i	$0.502950\pi$
$312$	0	0
$313$	−178.202	−0.0321807	−0.0160904	−	0.999871i	$-0.505122\pi$
$313$	−178.202	−0.0321807	−0.0160904	+	0.999871i	$0.505122\pi$
$314$	14063.9	2.52762
$315$	0	0
$316$	−4185.13	−0.745037
$317$	−6284.14	−1.11341	−0.556707	−	0.830709i	$-0.687936\pi$
$317$	−6284.14	−1.11341	−0.556707	+	0.830709i	$0.687936\pi$
$318$	0	0
$319$	−13861.3	−2.43287
$320$	−3892.75	−0.680036
$321$	0	0
$322$	2856.66	0.494396
$323$	675.320	0.116334
$324$	0	0
$325$	7895.18	1.34753
$326$	−3543.73	−0.602052
$327$	0	0
$328$	5717.80	0.962538
$329$	160.578	0.0269087
$330$	0	0
$331$	1412.84	0.234613	0.117306	−	0.993096i	$-0.462574\pi$
$331$	1412.84	0.234613	0.117306	+	0.993096i	$0.462574\pi$
$332$	14879.6	2.45971
$333$	0	0
$334$	9568.66	1.56759
$335$	2312.27	0.377113
$336$	0	0
$337$	−5832.29	−0.942745	−0.471372	−	0.881934i	$-0.656241\pi$
$337$	−5832.29	−0.942745	−0.471372	+	0.881934i	$0.656241\pi$
$338$	−18615.0	−2.99564
$339$	0	0
$340$	5383.97	0.858785
$341$	−2732.60	−0.433955
$342$	0	0
$343$	−4197.76	−0.660809
$344$	−12225.6	−1.91616
$345$	0	0
$346$	−4011.84	−0.623346
$347$	10548.2	1.63187	0.815934	−	0.578146i	$-0.196223\pi$
$347$	10548.2	1.63187	0.815934	+	0.578146i	$0.196223\pi$
$348$	0	0
$349$	1719.30	0.263702	0.131851	−	0.991270i	$-0.457908\pi$
$349$	1719.30	0.263702	0.131851	+	0.991270i	$0.457908\pi$
$350$	3064.45	0.468005
$351$	0	0
$352$	7925.18	1.20004
$353$	5719.41	0.862361	0.431180	−	0.902266i	$-0.358097\pi$
$353$	5719.41	0.862361	0.431180	+	0.902266i	$0.358097\pi$
$354$	0	0
$355$	1846.92	0.276124
$356$	14910.4	2.21980
$357$	0	0
$358$	5555.70	0.820189
$359$	−9801.99	−1.44103	−0.720514	−	0.693440i	$-0.756094\pi$
$359$	−9801.99	−1.44103	−0.720514	+	0.693440i	$0.756094\pi$
$360$	0	0
$361$	−6784.72	−0.989171
$362$	14319.5	2.07905
$363$	0	0
$364$	−7110.73	−1.02391
$365$	449.965	0.0645268
$366$	0	0
$367$	−1439.25	−0.204710	−0.102355	−	0.994748i	$-0.532638\pi$
$367$	−1439.25	−0.204710	−0.102355	+	0.994748i	$0.532638\pi$
$368$	−1625.66	−0.230281
$369$	0	0
$370$	7535.95	1.05885
$371$	1363.15	0.190758
$372$	0	0
$373$	−3048.32	−0.423153	−0.211576	−	0.977361i	$-0.567860\pi$
$373$	−3048.32	−0.423153	−0.211576	+	0.977361i	$0.567860\pi$
$374$	−20996.5	−2.90295
$375$	0	0
$376$	−675.309	−0.0926233
$377$	19013.0	2.59740
$378$	0	0
$379$	−11419.1	−1.54765	−0.773826	−	0.633399i	$-0.781659\pi$
$379$	−11419.1	−1.54765	−0.773826	+	0.633399i	$0.781659\pi$
$380$	592.175	0.0799420
$381$	0	0
$382$	−5345.34	−0.715946
$383$	4921.04	0.656537	0.328268	−	0.944585i	$-0.393535\pi$
$383$	4921.04	0.656537	0.328268	+	0.944585i	$0.393535\pi$
$384$	0	0
$385$	1852.31	0.245201
$386$	1881.56	0.248106
$387$	0	0
$388$	−22044.3	−2.88435
$389$	3228.49	0.420800	0.210400	−	0.977615i	$-0.432523\pi$
$389$	3228.49	0.420800	0.210400	+	0.977615i	$0.432523\pi$
$390$	0	0
$391$	−7337.24	−0.949004
$392$	8242.91	1.06207
$393$	0	0
$394$	5341.54	0.683003
$395$	−1495.39	−0.190484
$396$	0	0
$397$	7990.82	1.01020	0.505098	−	0.863062i	$-0.331456\pi$
$397$	7990.82	1.01020	0.505098	+	0.863062i	$0.331456\pi$
$398$	9285.25	1.16942
$399$	0	0
$400$	−1743.91	−0.217989
$401$	−1632.52	−0.203302	−0.101651	−	0.994820i	$-0.532412\pi$
$401$	−1632.52	−0.203302	−0.101651	+	0.994820i	$0.532412\pi$
$402$	0	0
$403$	3748.20	0.463303
$404$	−9844.27	−1.21230
$405$	0	0
$406$	7379.76	0.902097
$407$	−18637.1	−2.26979
$408$	0	0
$409$	−2416.60	−0.292159	−0.146080	−	0.989273i	$-0.546666\pi$
$409$	−2416.60	−0.292159	−0.146080	+	0.989273i	$0.546666\pi$
$410$	4828.74	0.581645
$411$	0	0
$412$	4825.80	0.577063
$413$	−4102.90	−0.488839
$414$	0	0
$415$	5316.63	0.628875
$416$	−10870.7	−1.28120
$417$	0	0
$418$	−2309.37	−0.270228
$419$	3847.50	0.448598	0.224299	−	0.974520i	$-0.427991\pi$
$419$	3847.50	0.448598	0.224299	+	0.974520i	$0.427991\pi$
$420$	0	0
$421$	6588.13	0.762674	0.381337	−	0.924436i	$-0.375464\pi$
$421$	6588.13	0.762674	0.381337	+	0.924436i	$0.375464\pi$
$422$	−20559.0	−2.37156
$423$	0	0
$424$	−5732.70	−0.656614
$425$	−7870.95	−0.898346
$426$	0	0
$427$	762.617	0.0864300
$428$	4892.16	0.552504
$429$	0	0
$430$	−10324.6	−1.15790
$431$	−1462.53	−0.163452	−0.0817258	−	0.996655i	$-0.526043\pi$
$431$	−1462.53	−0.163452	−0.0817258	+	0.996655i	$0.526043\pi$
$432$	0	0
$433$	−9532.02	−1.05792	−0.528961	−	0.848646i	$-0.677418\pi$
$433$	−9532.02	−1.05792	−0.528961	+	0.848646i	$0.677418\pi$
$434$	1454.83	0.160908
$435$	0	0
$436$	−18583.0	−2.04120
$437$	−807.013	−0.0883402
$438$	0	0
$439$	8021.09	0.872040	0.436020	−	0.899937i	$-0.356388\pi$
$439$	8021.09	0.872040	0.436020	+	0.899937i	$0.356388\pi$
$440$	−7789.85	−0.844014
$441$	0	0
$442$	28800.1	3.09927
$443$	3257.44	0.349358	0.174679	−	0.984625i	$-0.444111\pi$
$443$	3257.44	0.349358	0.174679	+	0.984625i	$0.444111\pi$
$444$	0	0
$445$	5327.64	0.567538
$446$	−14490.9	−1.53848
$447$	0	0
$448$	−5125.46	−0.540525
$449$	9378.18	0.985710	0.492855	−	0.870112i	$-0.335953\pi$
$449$	9378.18	0.985710	0.492855	+	0.870112i	$0.335953\pi$
$450$	0	0
$451$	−11941.9	−1.24683
$452$	−2603.71	−0.270947
$453$	0	0
$454$	−5002.38	−0.517121
$455$	−2540.74	−0.261784
$456$	0	0
$457$	4170.58	0.426896	0.213448	−	0.976954i	$-0.431531\pi$
$457$	4170.58	0.426896	0.213448	+	0.976954i	$0.431531\pi$
$458$	−17927.3	−1.82901
$459$	0	0
$460$	−6433.89	−0.652135
$461$	11233.2	1.13488	0.567441	−	0.823414i	$-0.307933\pi$
$461$	11233.2	1.13488	0.567441	+	0.823414i	$0.307933\pi$
$462$	0	0
$463$	−8526.59	−0.855862	−0.427931	−	0.903811i	$-0.640757\pi$
$463$	−8526.59	−0.855862	−0.427931	+	0.903811i	$0.640757\pi$
$464$	−4199.65	−0.420181
$465$	0	0
$466$	−4861.10	−0.483232
$467$	−1275.34	−0.126372	−0.0631861	−	0.998002i	$-0.520126\pi$
$467$	−1275.34	−0.126372	−0.0631861	+	0.998002i	$0.520126\pi$
$468$	0	0
$469$	3044.50	0.299748
$470$	−570.305	−0.0559706
$471$	0	0
$472$	17254.6	1.68265
$473$	25533.7	2.48212
$474$	0	0
$475$	−865.714	−0.0836246
$476$	7088.91	0.682604
$477$	0	0
$478$	4425.76	0.423493
$479$	−16872.6	−1.60945	−0.804725	−	0.593647i	$-0.797687\pi$
$479$	−16872.6	−1.60945	−0.804725	+	0.593647i	$0.797687\pi$
$480$	0	0
$481$	25563.7	2.42330
$482$	19105.7	1.80548
$483$	0	0
$484$	27075.9	2.54281
$485$	−7876.65	−0.737444
$486$	0	0
$487$	−203.706	−0.0189544	−0.00947719	−	0.999955i	$-0.503017\pi$
$487$	−203.706	−0.0189544	−0.00947719	+	0.999955i	$0.503017\pi$
$488$	−3207.16	−0.297503
$489$	0	0
$490$	6961.22	0.641787
$491$	−19041.4	−1.75016	−0.875078	−	0.483982i	$-0.839190\pi$
$491$	−19041.4	−1.75016	−0.875078	+	0.483982i	$0.839190\pi$
$492$	0	0
$493$	−18954.7	−1.73159
$494$	3167.68	0.288503
$495$	0	0
$496$	−827.913	−0.0749484
$497$	2431.78	0.219477
$498$	0	0
$499$	3576.66	0.320868	0.160434	−	0.987047i	$-0.448711\pi$
$499$	3576.66	0.320868	0.160434	+	0.987047i	$0.448711\pi$
$500$	−15490.7	−1.38553
$501$	0	0
$502$	−33636.6	−2.99059
$503$	−8180.28	−0.725130	−0.362565	−	0.931958i	$-0.618099\pi$
$503$	−8180.28	−0.725130	−0.362565	+	0.931958i	$0.618099\pi$
$504$	0	0
$505$	−3517.46	−0.309950
$506$	25091.0	2.20441
$507$	0	0
$508$	33243.2	2.90340
$509$	2577.87	0.224484	0.112242	−	0.993681i	$-0.464197\pi$
$509$	2577.87	0.224484	0.112242	+	0.993681i	$0.464197\pi$
$510$	0	0
$511$	592.455	0.0512890
$512$	6211.74	0.536177
$513$	0	0
$514$	2423.60	0.207978
$515$	1724.31	0.147538
$516$	0	0
$517$	1410.41	0.119981
$518$	9922.35	0.841628
$519$	0	0
$520$	10685.0	0.901094
$521$	−2045.61	−0.172015	−0.0860074	−	0.996294i	$-0.527411\pi$
$521$	−2045.61	−0.172015	−0.0860074	+	0.996294i	$0.527411\pi$
$522$	0	0
$523$	4431.94	0.370545	0.185273	−	0.982687i	$-0.440683\pi$
$523$	4431.94	0.370545	0.185273	+	0.982687i	$0.440683\pi$
$524$	−27828.2	−2.32001
$525$	0	0
$526$	12755.0	1.05731
$527$	−3736.69	−0.308867
$528$	0	0
$529$	−3398.93	−0.279356
$530$	−4841.32	−0.396780
$531$	0	0
$532$	779.699	0.0635418
$533$	16380.2	1.33116
$534$	0	0
$535$	1748.02	0.141259
$536$	−12803.6	−1.03177
$537$	0	0
$538$	−9870.29	−0.790964
$539$	−17215.7	−1.37576
$540$	0	0
$541$	−4557.61	−0.362194	−0.181097	−	0.983465i	$-0.557965\pi$
$541$	−4557.61	−0.362194	−0.181097	+	0.983465i	$0.557965\pi$
$542$	17694.0	1.40226
$543$	0	0
$544$	10837.3	0.854127
$545$	−6639.90	−0.521875
$546$	0	0
$547$	−9293.70	−0.726454	−0.363227	−	0.931701i	$-0.618325\pi$
$547$	−9293.70	−0.726454	−0.363227	+	0.931701i	$0.618325\pi$
$548$	−26776.2	−2.08727
$549$	0	0
$550$	26916.1	2.08674
$551$	−2084.80	−0.161189
$552$	0	0
$553$	−1968.93	−0.151406
$554$	11330.1	0.868899
$555$	0	0
$556$	−43154.2	−3.29163
$557$	−22480.6	−1.71011	−0.855057	−	0.518534i	$-0.826478\pi$
$557$	−22480.6	−1.71011	−0.855057	+	0.518534i	$0.826478\pi$
$558$	0	0
$559$	−35023.6	−2.64998
$560$	561.206	0.0423487
$561$	0	0
$562$	−12366.3	−0.928184
$563$	13341.8	0.998739	0.499370	−	0.866389i	$-0.333565\pi$
$563$	13341.8	0.998739	0.499370	+	0.866389i	$0.333565\pi$
$564$	0	0
$565$	−930.332	−0.0692732
$566$	−34017.6	−2.52627
$567$	0	0
$568$	−10226.8	−0.755468
$569$	821.627	0.0605350	0.0302675	−	0.999542i	$-0.490364\pi$
$569$	821.627	0.0605350	0.0302675	+	0.999542i	$0.490364\pi$
$570$	0	0
$571$	10415.1	0.763325	0.381663	−	0.924302i	$-0.375352\pi$
$571$	10415.1	0.763325	0.381663	+	0.924302i	$0.375352\pi$
$572$	−62455.9	−4.56541
$573$	0	0
$574$	6357.85	0.462320
$575$	9405.85	0.682176
$576$	0	0
$577$	7859.00	0.567026	0.283513	−	0.958968i	$-0.408500\pi$
$577$	7859.00	0.567026	0.283513	+	0.958968i	$0.408500\pi$
$578$	−5737.31	−0.412873
$579$	0	0
$580$	−16621.0	−1.18991
$581$	7000.24	0.499861
$582$	0	0
$583$	11973.0	0.850551
$584$	−2491.56	−0.176543
$585$	0	0
$586$	35666.7	2.51430
$587$	−13386.0	−0.941223	−0.470612	−	0.882341i	$-0.655967\pi$
$587$	−13386.0	−0.941223	−0.470612	+	0.882341i	$0.655967\pi$
$588$	0	0
$589$	−410.994	−0.0287516
$590$	14571.7	1.01679
$591$	0	0
$592$	−5646.59	−0.392016
$593$	14856.3	1.02879	0.514397	−	0.857552i	$-0.328016\pi$
$593$	14856.3	1.02879	0.514397	+	0.857552i	$0.328016\pi$
$594$	0	0
$595$	2532.94	0.174522
$596$	−9876.49	−0.678786
$597$	0	0
$598$	−34416.3	−2.35349
$599$	14760.5	1.00684	0.503421	−	0.864041i	$-0.332074\pi$
$599$	14760.5	1.00684	0.503421	+	0.864041i	$0.332074\pi$
$600$	0	0
$601$	7361.60	0.499644	0.249822	−	0.968292i	$-0.419628\pi$
$601$	7361.60	0.499644	0.249822	+	0.968292i	$0.419628\pi$
$602$	−13594.1	−0.920357
$603$	0	0
$604$	−32250.5	−2.17260
$605$	9674.50	0.650123
$606$	0	0
$607$	−25165.1	−1.68273	−0.841367	−	0.540464i	$-0.818249\pi$
$607$	−25165.1	−1.68273	−0.841367	+	0.540464i	$0.818249\pi$
$608$	1191.98	0.0795083
$609$	0	0
$610$	−2708.48	−0.179776
$611$	−1934.61	−0.128095
$612$	0	0
$613$	−15062.0	−0.992411	−0.496206	−	0.868205i	$-0.665274\pi$
$613$	−15062.0	−0.992411	−0.496206	+	0.868205i	$0.665274\pi$
$614$	42533.6	2.79563
$615$	0	0
$616$	−10256.6	−0.670864
$617$	6334.01	0.413286	0.206643	−	0.978416i	$-0.433746\pi$
$617$	6334.01	0.413286	0.206643	+	0.978416i	$0.433746\pi$
$618$	0	0
$619$	−16527.6	−1.07318	−0.536590	−	0.843843i	$-0.680288\pi$
$619$	−16527.6	−1.07318	−0.536590	+	0.843843i	$0.680288\pi$
$620$	−3276.64	−0.212247
$621$	0	0
$622$	475.444	0.0306488
$623$	7014.74	0.451107
$624$	0	0
$625$	7021.15	0.449354
$626$	833.314	0.0532043
$627$	0	0
$628$	−41706.0	−2.65008
$629$	−25485.2	−1.61552
$630$	0	0
$631$	13607.2	0.858466	0.429233	−	0.903194i	$-0.358784\pi$
$631$	13607.2	0.858466	0.429233	+	0.903194i	$0.358784\pi$
$632$	8280.30	0.521159
$633$	0	0
$634$	29386.1	1.84081
$635$	11878.1	0.742314
$636$	0	0
$637$	23614.1	1.46880
$638$	64818.9	4.02226
$639$	0	0
$640$	12721.1	0.785697
$641$	13413.4	0.826518	0.413259	−	0.910613i	$-0.364390\pi$
$641$	13413.4	0.826518	0.413259	+	0.910613i	$0.364390\pi$
$642$	0	0
$643$	18961.8	1.16296	0.581479	−	0.813562i	$-0.302474\pi$
$643$	18961.8	1.16296	0.581479	+	0.813562i	$0.302474\pi$
$644$	−8471.31	−0.518348
$645$	0	0
$646$	−3157.95	−0.192334
$647$	28772.7	1.74833	0.874167	−	0.485625i	$-0.161408\pi$
$647$	28772.7	1.74833	0.874167	+	0.485625i	$0.161408\pi$
$648$	0	0
$649$	−36037.1	−2.17963
$650$	−36919.7	−2.22786
$651$	0	0
$652$	10508.8	0.631219
$653$	24805.0	1.48651	0.743257	−	0.669006i	$-0.233280\pi$
$653$	24805.0	1.48651	0.743257	+	0.669006i	$0.233280\pi$
$654$	0	0
$655$	−9943.33	−0.593157
$656$	−3618.11	−0.215341
$657$	0	0
$658$	−750.903	−0.0444882
$659$	−20918.5	−1.23652	−0.618261	−	0.785973i	$-0.712163\pi$
$659$	−20918.5	−1.23652	−0.618261	+	0.785973i	$0.712163\pi$
$660$	0	0
$661$	968.836	0.0570096	0.0285048	−	0.999594i	$-0.490925\pi$
$661$	968.836	0.0570096	0.0285048	+	0.999594i	$0.490925\pi$
$662$	−6606.79	−0.387885
$663$	0	0
$664$	−29439.3	−1.72058
$665$	278.595	0.0162458
$666$	0	0
$667$	22651.0	1.31492
$668$	−28375.5	−1.64353
$669$	0	0
$670$	−10812.7	−0.623481
$671$	6698.31	0.385373
$672$	0	0
$673$	−8429.13	−0.482792	−0.241396	−	0.970427i	$-0.577605\pi$
$673$	−8429.13	−0.482792	−0.241396	+	0.970427i	$0.577605\pi$
$674$	27273.2	1.55864
$675$	0	0
$676$	55202.1	3.14077
$677$	−34150.7	−1.93873	−0.969363	−	0.245633i	$-0.921004\pi$
$677$	−34150.7	−1.93873	−0.969363	+	0.245633i	$0.921004\pi$
$678$	0	0
$679$	−10370.9	−0.586156
$680$	−10652.2	−0.600727
$681$	0	0
$682$	12778.3	0.717457
$683$	−10033.0	−0.562084	−0.281042	−	0.959695i	$-0.590680\pi$
$683$	−10033.0	−0.562084	−0.281042	+	0.959695i	$0.590680\pi$
$684$	0	0
$685$	−9567.43	−0.533654
$686$	19629.7	1.09251
$687$	0	0
$688$	7736.10	0.428686
$689$	−16422.9	−0.908073
$690$	0	0
$691$	−2765.70	−0.152261	−0.0761305	−	0.997098i	$-0.524257\pi$
$691$	−2765.70	−0.152261	−0.0761305	+	0.997098i	$0.524257\pi$
$692$	11896.9	0.653546
$693$	0	0
$694$	−49325.9	−2.69796
$695$	−15419.5	−0.841573
$696$	0	0
$697$	−16329.9	−0.887432
$698$	−8039.86	−0.435979
$699$	0	0
$700$	−9087.50	−0.490679
$701$	19939.8	1.07434	0.537172	−	0.843473i	$-0.319493\pi$
$701$	19939.8	1.07434	0.537172	+	0.843473i	$0.319493\pi$
$702$	0	0
$703$	−2803.09	−0.150385
$704$	−45018.6	−2.41009
$705$	0	0
$706$	−26745.3	−1.42574
$707$	−4631.33	−0.246364
$708$	0	0
$709$	−15126.7	−0.801260	−0.400630	−	0.916240i	$-0.631209\pi$
$709$	−15126.7	−0.801260	−0.400630	+	0.916240i	$0.631209\pi$
$710$	−8636.62	−0.456516
$711$	0	0
$712$	−29500.3	−1.55277
$713$	4465.38	0.234544
$714$	0	0
$715$	−22316.2	−1.16724
$716$	−16475.2	−0.859925
$717$	0	0
$718$	45836.4	2.38245
$719$	33913.8	1.75907	0.879534	−	0.475836i	$-0.157855\pi$
$719$	33913.8	1.75907	0.879534	+	0.475836i	$0.157855\pi$
$720$	0	0
$721$	2270.34	0.117270
$722$	31727.0	1.63539
$723$	0	0
$724$	−42463.8	−2.17977
$725$	24298.6	1.24473
$726$	0	0
$727$	−2032.84	−0.103705	−0.0518527	−	0.998655i	$-0.516513\pi$
$727$	−2032.84	−0.103705	−0.0518527	+	0.998655i	$0.516513\pi$
$728$	14068.6	0.716234
$729$	0	0
$730$	−2104.14	−0.106682
$731$	34916.0	1.76664
$732$	0	0
$733$	21827.0	1.09986	0.549931	−	0.835210i	$-0.314654\pi$
$733$	21827.0	1.09986	0.549931	+	0.835210i	$0.314654\pi$
$734$	6730.29	0.338446
$735$	0	0
$736$	−12950.7	−0.648597
$737$	26740.8	1.33651
$738$	0	0
$739$	−7036.04	−0.350237	−0.175118	−	0.984547i	$-0.556031\pi$
$739$	−7036.04	−0.350237	−0.175118	+	0.984547i	$0.556031\pi$
$740$	−22347.5	−1.11015
$741$	0	0
$742$	−6374.41	−0.315380
$743$	10684.1	0.527541	0.263770	−	0.964585i	$-0.415034\pi$
$743$	10684.1	0.527541	0.263770	+	0.964585i	$0.415034\pi$
$744$	0	0
$745$	−3528.97	−0.173546
$746$	14254.7	0.699598
$747$	0	0
$748$	62264.2	3.04359
$749$	2301.57	0.112280
$750$	0	0
$751$	−24333.5	−1.18235	−0.591173	−	0.806544i	$-0.701335\pi$
$751$	−24333.5	−1.18235	−0.591173	+	0.806544i	$0.701335\pi$
$752$	427.322	0.0207218
$753$	0	0
$754$	−88909.4	−4.29429
$755$	−11523.4	−0.555471
$756$	0	0
$757$	38024.8	1.82567	0.912837	−	0.408325i	$-0.133887\pi$
$757$	38024.8	1.82567	0.912837	+	0.408325i	$0.133887\pi$
$758$	53398.4	2.55873
$759$	0	0
$760$	−1171.62	−0.0559200
$761$	23023.4	1.09671	0.548357	−	0.836244i	$-0.315254\pi$
$761$	23023.4	1.09671	0.548357	+	0.836244i	$0.315254\pi$
$762$	0	0
$763$	−8742.55	−0.414812
$764$	15851.4	0.750631
$765$	0	0
$766$	−23012.0	−1.08545
$767$	49430.7	2.32704
$768$	0	0
$769$	−15275.7	−0.716326	−0.358163	−	0.933659i	$-0.616597\pi$
$769$	−15275.7	−0.716326	−0.358163	+	0.933659i	$0.616597\pi$
$770$	−8661.84	−0.405391
$771$	0	0
$772$	−5579.68	−0.260126
$773$	−11756.3	−0.547017	−0.273509	−	0.961870i	$-0.588184\pi$
$773$	−11756.3	−0.547017	−0.273509	+	0.961870i	$0.588184\pi$
$774$	0	0
$775$	4790.19	0.222024
$776$	43614.7	2.01762
$777$	0	0
$778$	−15097.2	−0.695708
$779$	−1796.11	−0.0826087
$780$	0	0
$781$	21359.1	0.978603
$782$	34310.7	1.56899
$783$	0	0
$784$	−5215.95	−0.237607
$785$	−14902.0	−0.677548
$786$	0	0
$787$	7815.34	0.353986	0.176993	−	0.984212i	$-0.443363\pi$
$787$	7815.34	0.353986	0.176993	+	0.984212i	$0.443363\pi$
$788$	−15840.1	−0.716093
$789$	0	0
$790$	6992.79	0.314927
$791$	−1224.94	−0.0550617
$792$	0	0
$793$	−9187.81	−0.411436
$794$	−37366.9	−1.67016
$795$	0	0
$796$	−27535.0	−1.22607
$797$	35027.6	1.55677	0.778383	−	0.627789i	$-0.216040\pi$
$797$	35027.6	1.55677	0.778383	+	0.627789i	$0.216040\pi$
$798$	0	0
$799$	1928.67	0.0853960
$800$	−13892.7	−0.613975
$801$	0	0
$802$	7634.03	0.336119
$803$	5203.73	0.228687
$804$	0	0
$805$	−3026.89	−0.132526
$806$	−17527.5	−0.765978
$807$	0	0
$808$	19476.9	0.848016
$809$	−10686.8	−0.464437	−0.232219	−	0.972664i	$-0.574598\pi$
$809$	−10686.8	−0.464437	−0.232219	+	0.972664i	$0.574598\pi$
$810$	0	0
$811$	6783.48	0.293712	0.146856	−	0.989158i	$-0.453085\pi$
$811$	6783.48	0.293712	0.146856	+	0.989158i	$0.453085\pi$
$812$	−21884.4	−0.945801
$813$	0	0
$814$	87151.3	3.75264
$815$	3754.89	0.161384
$816$	0	0
$817$	3840.37	0.164452
$818$	11300.6	0.483027
$819$	0	0
$820$	−14319.4	−0.609824
$821$	−17162.6	−0.729571	−0.364785	−	0.931092i	$-0.618858\pi$
$821$	−17162.6	−0.729571	−0.364785	+	0.931092i	$0.618858\pi$
$822$	0	0
$823$	−678.971	−0.0287575	−0.0143788	−	0.999897i	$-0.504577\pi$
$823$	−678.971	−0.0287575	−0.0143788	+	0.999897i	$0.504577\pi$
$824$	−9547.87	−0.403660
$825$	0	0
$826$	19186.1	0.808197
$827$	13266.8	0.557839	0.278920	−	0.960314i	$-0.410024\pi$
$827$	13266.8	0.557839	0.278920	+	0.960314i	$0.410024\pi$
$828$	0	0
$829$	−38747.2	−1.62334	−0.811668	−	0.584119i	$-0.801440\pi$
$829$	−38747.2	−1.62334	−0.811668	+	0.584119i	$0.801440\pi$
$830$	−24861.8	−1.03972
$831$	0	0
$832$	61750.3	2.57308
$833$	−23541.6	−0.979193
$834$	0	0
$835$	−10138.9	−0.420203
$836$	6848.35	0.283320
$837$	0	0
$838$	−17991.8	−0.741667
$839$	6016.94	0.247590	0.123795	−	0.992308i	$-0.460494\pi$
$839$	6016.94	0.247590	0.123795	+	0.992308i	$0.460494\pi$
$840$	0	0
$841$	34126.5	1.39926
$842$	−30807.6	−1.26093
$843$	0	0
$844$	60966.9	2.48645
$845$	19724.3	0.803002
$846$	0	0
$847$	12738.1	0.516750
$848$	3627.54	0.146899
$849$	0	0
$850$	36806.4	1.48523
$851$	30455.1	1.22678
$852$	0	0
$853$	13360.3	0.536281	0.268141	−	0.963380i	$-0.413591\pi$
$853$	13360.3	0.536281	0.268141	+	0.963380i	$0.413591\pi$
$854$	−3566.17	−0.142895
$855$	0	0
$856$	−9679.17	−0.386481
$857$	−3628.56	−0.144631	−0.0723157	−	0.997382i	$-0.523039\pi$
$857$	−3628.56	−0.144631	−0.0723157	+	0.997382i	$0.523039\pi$
$858$	0	0
$859$	−18432.0	−0.732119	−0.366060	−	0.930591i	$-0.619293\pi$
$859$	−18432.0	−0.732119	−0.366060	+	0.930591i	$0.619293\pi$
$860$	30617.2	1.21400
$861$	0	0
$862$	6839.13	0.270234
$863$	−21402.6	−0.844209	−0.422105	−	0.906547i	$-0.638709\pi$
$863$	−21402.6	−0.844209	−0.422105	+	0.906547i	$0.638709\pi$
$864$	0	0
$865$	4250.90	0.167092
$866$	44574.0	1.74906
$867$	0	0
$868$	−4314.25	−0.168704
$869$	−17293.8	−0.675088
$870$	0	0
$871$	−36679.3	−1.42690
$872$	36766.5	1.42784
$873$	0	0
$874$	3773.78	0.146053
$875$	−7287.74	−0.281566
$876$	0	0
$877$	37687.2	1.45109	0.725546	−	0.688174i	$-0.241587\pi$
$877$	37687.2	1.45109	0.725546	+	0.688174i	$0.241587\pi$
$878$	−37508.5	−1.44174
$879$	0	0
$880$	4929.26	0.188824
$881$	4916.17	0.188002	0.0940011	−	0.995572i	$-0.470034\pi$
$881$	4916.17	0.188002	0.0940011	+	0.995572i	$0.470034\pi$
$882$	0	0
$883$	30231.7	1.15218	0.576091	−	0.817386i	$-0.304577\pi$
$883$	30231.7	1.15218	0.576091	+	0.817386i	$0.304577\pi$
$884$	−85405.3	−3.24943
$885$	0	0
$886$	−15232.5	−0.577593
$887$	28730.0	1.08755	0.543776	−	0.839230i	$-0.316994\pi$
$887$	28730.0	1.08755	0.543776	+	0.839230i	$0.316994\pi$
$888$	0	0
$889$	15639.6	0.590027
$890$	−24913.3	−0.938310
$891$	0	0
$892$	42972.0	1.61301
$893$	212.132	0.00794929
$894$	0	0
$895$	−5886.76	−0.219858
$896$	16749.5	0.624510
$897$	0	0
$898$	−43854.6	−1.62967
$899$	11535.7	0.427959
$900$	0	0
$901$	16372.5	0.605379
$902$	55843.1	2.06139
$903$	0	0
$904$	5151.45	0.189529
$905$	−15172.8	−0.557304
$906$	0	0
$907$	33898.0	1.24097	0.620487	−	0.784217i	$-0.286935\pi$
$907$	33898.0	1.24097	0.620487	+	0.784217i	$0.286935\pi$
$908$	14834.3	0.542175
$909$	0	0
$910$	11881.1	0.432807
$911$	22863.4	0.831501	0.415751	−	0.909479i	$-0.363519\pi$
$911$	22863.4	0.831501	0.415751	+	0.909479i	$0.363519\pi$
$912$	0	0
$913$	61485.5	2.22878
$914$	−19502.6	−0.705786
$915$	0	0
$916$	53162.7	1.91762
$917$	−13092.1	−0.471470
$918$	0	0
$919$	2266.44	0.0813524	0.0406762	−	0.999172i	$-0.487049\pi$
$919$	2266.44	0.0813524	0.0406762	+	0.999172i	$0.487049\pi$
$920$	12729.5	0.456173
$921$	0	0
$922$	−52528.9	−1.87630
$923$	−29297.4	−1.04478
$924$	0	0
$925$	32670.4	1.16129
$926$	39872.3	1.41500
$927$	0	0
$928$	−33456.1	−1.18346
$929$	−53382.9	−1.88529	−0.942646	−	0.333795i	$-0.891671\pi$
$929$	−53382.9	−1.88529	−0.942646	+	0.333795i	$0.891671\pi$
$930$	0	0
$931$	−2589.31	−0.0911505
$932$	14415.4	0.506644
$933$	0	0
$934$	5963.80	0.208931
$935$	22247.7	0.778157
$936$	0	0
$937$	−12337.6	−0.430152	−0.215076	−	0.976597i	$-0.569000\pi$
$937$	−12337.6	−0.430152	−0.215076	+	0.976597i	$0.569000\pi$
$938$	−14236.8	−0.495573
$939$	0	0
$940$	1691.21	0.0586823
$941$	−3928.15	−0.136083	−0.0680414	−	0.997682i	$-0.521675\pi$
$941$	−3928.15	−0.136083	−0.0680414	+	0.997682i	$0.521675\pi$
$942$	0	0
$943$	19514.4	0.673888
$944$	−10918.4	−0.376444
$945$	0	0
$946$	−119402.	−4.10368
$947$	32673.6	1.12117	0.560585	−	0.828097i	$-0.310576\pi$
$947$	32673.6	1.12117	0.560585	+	0.828097i	$0.310576\pi$
$948$	0	0
$949$	−7137.75	−0.244153
$950$	4048.28	0.138256
$951$	0	0
$952$	−14025.4	−0.477487
$953$	−12329.1	−0.419074	−0.209537	−	0.977801i	$-0.567196\pi$
$953$	−12329.1	−0.419074	−0.209537	+	0.977801i	$0.567196\pi$
$954$	0	0
$955$	5663.86	0.191914
$956$	−13124.4	−0.444010
$957$	0	0
$958$	78900.1	2.66090
$959$	−12597.1	−0.424174
$960$	0	0
$961$	−27516.9	−0.923664
$962$	−119542.	−4.00643
$963$	0	0
$964$	−56657.1	−1.89295
$965$	−1993.68	−0.0665066
$966$	0	0
$967$	3119.92	0.103754	0.0518769	−	0.998653i	$-0.483480\pi$
$967$	3119.92	0.103754	0.0518769	+	0.998653i	$0.483480\pi$
$968$	−53569.8	−1.77872
$969$	0	0
$970$	36833.1	1.21921
$971$	5428.62	0.179416	0.0897079	−	0.995968i	$-0.471407\pi$
$971$	5428.62	0.179416	0.0897079	+	0.995968i	$0.471407\pi$
$972$	0	0
$973$	−20302.3	−0.668923
$974$	952.575	0.0313373
$975$	0	0
$976$	2029.43	0.0665578
$977$	2624.01	0.0859259	0.0429629	−	0.999077i	$-0.486320\pi$
$977$	2624.01	0.0859259	0.0429629	+	0.999077i	$0.486320\pi$
$978$	0	0
$979$	61612.8	2.01139
$980$	−20643.2	−0.672880
$981$	0	0
$982$	89042.1	2.89353
$983$	2133.68	0.0692307	0.0346154	−	0.999401i	$-0.488979\pi$
$983$	2133.68	0.0692307	0.0346154	+	0.999401i	$0.488979\pi$
$984$	0	0
$985$	−5659.84	−0.183084
$986$	88636.5	2.86284
$987$	0	0
$988$	−9393.61	−0.302480
$989$	−41725.0	−1.34153
$990$	0	0
$991$	36080.1	1.15653	0.578266	−	0.815848i	$-0.303730\pi$
$991$	36080.1	1.15653	0.578266	+	0.815848i	$0.303730\pi$
$992$	−6595.48	−0.211095
$993$	0	0
$994$	−11371.6	−0.362861
$995$	−9838.56	−0.313471
$996$	0	0
$997$	−52575.8	−1.67010	−0.835051	−	0.550172i	$-0.814562\pi$
$997$	−52575.8	−1.67010	−0.835051	+	0.550172i	$0.814562\pi$
$998$	−16725.3	−0.530491
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.4.a.b.1.1	yes	12
3.2	odd	2		729.4.a.a.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.4.a.a.1.12	✓	12	3.2	odd	2
729.4.a.b.1.1	yes	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q-4.67624 q^{2} +13.8672 q^{4} +4.95489 q^{5} +6.52395 q^{7} -27.4363 q^{8} -23.1702 q^{10} +57.3020 q^{11} -78.5988 q^{13} -30.5075 q^{14} +17.3611 q^{16} +78.3576 q^{17} +8.61844 q^{19} +68.7103 q^{20} -267.958 q^{22} -93.6380 q^{23} -100.449 q^{25} +367.547 q^{26} +90.4687 q^{28} -241.900 q^{29} -47.6877 q^{31} +138.306 q^{32} -366.418 q^{34} +32.3254 q^{35} -325.243 q^{37} -40.3018 q^{38} -135.944 q^{40} -208.403 q^{41} +445.599 q^{43} +794.617 q^{44} +437.873 q^{46} +24.6137 q^{47} -300.438 q^{49} +469.724 q^{50} -1089.94 q^{52} +208.946 q^{53} +283.925 q^{55} -178.993 q^{56} +1131.18 q^{58} -628.898 q^{59} +116.895 q^{61} +222.999 q^{62} -785.638 q^{64} -389.448 q^{65} +466.665 q^{67} +1086.60 q^{68} -151.161 q^{70} +372.746 q^{71} +90.8124 q^{73} +1520.91 q^{74} +119.513 q^{76} +373.835 q^{77} -301.801 q^{79} +86.0225 q^{80} +974.540 q^{82} +1073.01 q^{83} +388.253 q^{85} -2083.73 q^{86} -1572.15 q^{88} +1075.23 q^{89} -512.775 q^{91} -1298.49 q^{92} -115.099 q^{94} +42.7034 q^{95} -1589.67 q^{97} +1404.92 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 36 q^{4} + 12 q^{5} - 42 q^{7} + 21 q^{8} - 60 q^{10} + 42 q^{11} - 78 q^{13} - 312 q^{14} + 48 q^{16} - 18 q^{17} - 228 q^{19} - 69 q^{20} - 309 q^{22} - 114 q^{23} - 18 q^{25} + 30 q^{26}+ \cdots + 9567 q^{98}+O(q^{100}) \) 12 * q + 6 * q^2 + 36 * q^4 + 12 * q^5 - 42 * q^7 + 21 * q^8 - 60 * q^10 + 42 * q^11 - 78 * q^13 - 312 * q^14 + 48 * q^16 - 18 * q^17 - 228 * q^19 - 69 * q^20 - 309 * q^22 - 114 * q^23 - 18 * q^25 + 30 * q^26 - 813 * q^28 - 660 * q^29 - 708 * q^31 + 729 * q^32 - 972 * q^34 - 624 * q^35 - 354 * q^37 - 213 * q^38 - 1335 * q^40 - 1032 * q^41 - 744 * q^43 + 1653 * q^44 - 1077 * q^46 + 942 * q^47 - 192 * q^49 + 1905 * q^50 - 1371 * q^52 - 828 * q^53 - 1554 * q^55 - 3324 * q^56 - 831 * q^58 - 24 * q^59 - 1698 * q^61 - 2184 * q^62 - 933 * q^64 + 294 * q^65 - 1266 * q^67 - 4734 * q^68 - 1137 * q^70 + 3888 * q^71 - 1164 * q^73 + 5448 * q^74 - 2385 * q^76 + 3018 * q^77 - 2382 * q^79 - 4677 * q^80 - 276 * q^82 - 4008 * q^83 - 1116 * q^85 - 1176 * q^86 - 2769 * q^88 - 3582 * q^89 - 3222 * q^91 - 2958 * q^92 - 3324 * q^94 - 2784 * q^95 - 2958 * q^97 + 9567 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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