LMFDB - Embedded newform 729.4.a.a.1.8

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			1.8
Root			$1.45995$ 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)

$f(q)$	$=$	$q+0.459945 q^{2} -7.78845 q^{4} +11.1167 q^{5} -7.10397 q^{7} -7.26182 q^{8} +O(q^{10})$	\(q+0.459945 q^{2} -7.78845 q^{4} +11.1167 q^{5} -7.10397 q^{7} -7.26182 q^{8} +5.11307 q^{10} +4.86293 q^{11} +7.82754 q^{13} -3.26744 q^{14} +58.9676 q^{16} -102.280 q^{17} +120.959 q^{19} -86.5818 q^{20} +2.23668 q^{22} -63.8485 q^{23} -1.41923 q^{25} +3.60024 q^{26} +55.3289 q^{28} +136.281 q^{29} -217.939 q^{31} +85.2164 q^{32} -47.0432 q^{34} -78.9727 q^{35} +121.549 q^{37} +55.6344 q^{38} -80.7274 q^{40} +87.3695 q^{41} -208.031 q^{43} -37.8747 q^{44} -29.3668 q^{46} -470.414 q^{47} -292.534 q^{49} -0.652769 q^{50} -60.9644 q^{52} +496.852 q^{53} +54.0596 q^{55} +51.5878 q^{56} +62.6820 q^{58} -832.796 q^{59} +395.570 q^{61} -100.240 q^{62} -432.546 q^{64} +87.0163 q^{65} -789.333 q^{67} +796.603 q^{68} -36.3231 q^{70} +665.848 q^{71} -687.195 q^{73} +55.9059 q^{74} -942.082 q^{76} -34.5461 q^{77} +96.0375 q^{79} +655.524 q^{80} +40.1852 q^{82} +44.0012 q^{83} -1137.02 q^{85} -95.6830 q^{86} -35.3137 q^{88} -1492.59 q^{89} -55.6066 q^{91} +497.281 q^{92} -216.365 q^{94} +1344.66 q^{95} -246.479 q^{97} -134.549 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{2} + 36 q^{4} - 12 q^{5} - 42 q^{7} - 21 q^{8}+O(q^{10}) $ 12 * q - 6 * q^2 + 36 * q^4 - 12 * q^5 - 42 * q^7 - 21 * q^8	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.459945	0.162615	0.0813076	−	0.996689i	$-0.474090\pi$
$2$	0.459945	0.162615	0.0813076	+	0.996689i	$0.474090\pi$
$3$	0	0
$3$	0	0
$4$	−7.78845	−0.973556
$5$	11.1167	0.994307	0.497153	−	0.867663i	$-0.334379\pi$
$5$	11.1167	0.994307	0.497153	+	0.867663i	$0.334379\pi$
$6$	0	0
$7$	−7.10397	−0.383578	−0.191789	−	0.981436i	$-0.561429\pi$
$7$	−7.10397	−0.383578	−0.191789	+	0.981436i	$0.561429\pi$
$8$	−7.26182	−0.320930
$9$	0	0
$10$	5.11307	0.161689
$11$	4.86293	0.133293	0.0666467	−	0.997777i	$-0.478770\pi$
$11$	4.86293	0.133293	0.0666467	+	0.997777i	$0.478770\pi$
$12$	0	0
$13$	7.82754	0.166998	0.0834988	−	0.996508i	$-0.473391\pi$
$13$	7.82754	0.166998	0.0834988	+	0.996508i	$0.473391\pi$
$14$	−3.26744	−0.0623757
$15$	0	0
$16$	58.9676	0.921368
$17$	−102.280	−1.45921	−0.729604	−	0.683870i	$-0.760296\pi$
$17$	−102.280	−1.45921	−0.729604	+	0.683870i	$0.760296\pi$
$18$	0	0
$19$	120.959	1.46052	0.730259	−	0.683170i	$-0.239399\pi$
$19$	120.959	1.46052	0.730259	+	0.683170i	$0.239399\pi$
$20$	−86.5818	−0.968014
$21$	0	0
$22$	2.23668	0.0216755
$23$	−63.8485	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$23$	−63.8485	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$24$	0	0
$25$	−1.41923	−0.0113538
$26$	3.60024	0.0271564
$27$	0	0
$28$	55.3289	0.373435
$29$	136.281	0.872649	0.436324	−	0.899789i	$-0.356280\pi$
$29$	136.281	0.872649	0.436324	+	0.899789i	$0.356280\pi$
$30$	0	0
$31$	−217.939	−1.26268	−0.631338	−	0.775508i	$-0.717494\pi$
$31$	−217.939	−1.26268	−0.631338	+	0.775508i	$0.717494\pi$
$32$	85.2164	0.470759
$33$	0	0
$34$	−47.0432	−0.237290
$35$	−78.9727	−0.381395
$36$	0	0
$37$	121.549	0.540069	0.270034	−	0.962851i	$-0.412965\pi$
$37$	121.549	0.540069	0.270034	+	0.962851i	$0.412965\pi$
$38$	55.6344	0.237503
$39$	0	0
$40$	−80.7274	−0.319103
$41$	87.3695	0.332801	0.166400	−	0.986058i	$-0.446786\pi$
$41$	87.3695	0.332801	0.166400	+	0.986058i	$0.446786\pi$
$42$	0	0
$43$	−208.031	−0.737779	−0.368889	−	0.929473i	$-0.620262\pi$
$43$	−208.031	−0.737779	−0.368889	+	0.929473i	$0.620262\pi$
$44$	−37.8747	−0.129769
$45$	0	0
$46$	−29.3668	−0.0941283
$47$	−470.414	−1.45993	−0.729967	−	0.683482i	$-0.760465\pi$
$47$	−470.414	−1.45993	−0.729967	+	0.683482i	$0.760465\pi$
$48$	0	0
$49$	−292.534	−0.852868
$50$	−0.652769	−0.00184631
$51$	0	0
$52$	−60.9644	−0.162582
$53$	496.852	1.28770	0.643848	−	0.765153i	$-0.277337\pi$
$53$	496.852	1.28770	0.643848	+	0.765153i	$0.277337\pi$
$54$	0	0
$55$	54.0596	0.132535
$56$	51.5878	0.123102
$57$	0	0
$58$	62.6820	0.141906
$59$	−832.796	−1.83764	−0.918820	−	0.394676i	$-0.870857\pi$
$59$	−832.796	−1.83764	−0.918820	+	0.394676i	$0.870857\pi$
$60$	0	0
$61$	395.570	0.830288	0.415144	−	0.909756i	$-0.363731\pi$
$61$	395.570	0.830288	0.415144	+	0.909756i	$0.363731\pi$
$62$	−100.240	−0.205330
$63$	0	0
$64$	−432.546	−0.844816
$65$	87.0163	0.166047
$66$	0	0
$67$	−789.333	−1.43929	−0.719645	−	0.694342i	$-0.755695\pi$
$67$	−789.333	−1.43929	−0.719645	+	0.694342i	$0.755695\pi$
$68$	796.603	1.42062
$69$	0	0
$70$	−36.3231	−0.0620206
$71$	665.848	1.11298	0.556491	−	0.830854i	$-0.312148\pi$
$71$	665.848	1.11298	0.556491	+	0.830854i	$0.312148\pi$
$72$	0	0
$73$	−687.195	−1.10178	−0.550891	−	0.834577i	$-0.685712\pi$
$73$	−687.195	−1.10178	−0.550891	+	0.834577i	$0.685712\pi$
$74$	55.9059	0.0878234
$75$	0	0
$76$	−942.082	−1.42190
$77$	−34.5461	−0.0511285
$78$	0	0
$79$	96.0375	0.136773	0.0683865	−	0.997659i	$-0.478215\pi$
$79$	96.0375	0.136773	0.0683865	+	0.997659i	$0.478215\pi$
$80$	655.524	0.916123
$81$	0	0
$82$	40.1852	0.0541184
$83$	44.0012	0.0581899	0.0290950	−	0.999577i	$-0.490737\pi$
$83$	44.0012	0.0581899	0.0290950	+	0.999577i	$0.490737\pi$
$84$	0	0
$85$	−1137.02	−1.45090
$86$	−95.6830	−0.119974
$87$	0	0
$88$	−35.3137	−0.0427779
$89$	−1492.59	−1.77769	−0.888843	−	0.458212i	$-0.848490\pi$
$89$	−1492.59	−1.77769	−0.888843	+	0.458212i	$0.848490\pi$
$90$	0	0
$91$	−55.6066	−0.0640567
$92$	497.281	0.563534
$93$	0	0
$94$	−216.365	−0.237408
$95$	1344.66	1.45220
$96$	0	0
$97$	−246.479	−0.258002	−0.129001	−	0.991644i	$-0.541177\pi$
$97$	−246.479	−0.258002	−0.129001	+	0.991644i	$0.541177\pi$
$98$	−134.549	−0.138689
$99$	0	0
$100$	11.0536	0.0110536
$101$	−1248.57	−1.23007	−0.615034	−	0.788500i	$-0.710858\pi$
$101$	−1248.57	−1.23007	−0.615034	+	0.788500i	$0.710858\pi$
$102$	0	0
$103$	−549.924	−0.526074	−0.263037	−	0.964786i	$-0.584724\pi$
$103$	−549.924	−0.526074	−0.263037	+	0.964786i	$0.584724\pi$
$104$	−56.8422	−0.0535946
$105$	0	0
$106$	228.525	0.209399
$107$	1115.38	1.00773	0.503866	−	0.863782i	$-0.331910\pi$
$107$	1115.38	1.00773	0.503866	+	0.863782i	$0.331910\pi$
$108$	0	0
$109$	−1138.53	−1.00047	−0.500235	−	0.865890i	$-0.666753\pi$
$109$	−1138.53	−1.00047	−0.500235	+	0.865890i	$0.666753\pi$
$110$	24.8645	0.0215521
$111$	0	0
$112$	−418.904	−0.353417
$113$	−2010.83	−1.67401	−0.837005	−	0.547195i	$-0.815696\pi$
$113$	−2010.83	−1.67401	−0.837005	+	0.547195i	$0.815696\pi$
$114$	0	0
$115$	−709.784	−0.575545
$116$	−1061.42	−0.849573
$117$	0	0
$118$	−383.041	−0.298828
$119$	726.594	0.559721
$120$	0	0
$121$	−1307.35	−0.982233
$122$	181.941	0.135018
$123$	0	0
$124$	1697.40	1.22929
$125$	−1405.36	−1.00560
$126$	0	0
$127$	1649.61	1.15259	0.576296	−	0.817241i	$-0.304498\pi$
$127$	1649.61	1.15259	0.576296	+	0.817241i	$0.304498\pi$
$128$	−880.679	−0.608139
$129$	0	0
$130$	40.0227	0.0270017
$131$	−1078.01	−0.718979	−0.359489	−	0.933149i	$-0.617049\pi$
$131$	−1078.01	−0.718979	−0.359489	+	0.933149i	$0.617049\pi$
$132$	0	0
$133$	−859.288	−0.560223
$134$	−363.050	−0.234050
$135$	0	0
$136$	742.739	0.468304
$137$	3176.28	1.98079	0.990393	−	0.138283i	$-0.0441585\pi$
$137$	3176.28	1.98079	0.990393	+	0.138283i	$0.0441585\pi$
$138$	0	0
$139$	−2808.11	−1.71353	−0.856764	−	0.515708i	$-0.827529\pi$
$139$	−2808.11	−1.71353	−0.856764	+	0.515708i	$0.827529\pi$
$140$	615.075	0.371309
$141$	0	0
$142$	306.254	0.180988
$143$	38.0647	0.0222597
$144$	0	0
$145$	1515.00	0.867681
$146$	−316.072	−0.179167
$147$	0	0
$148$	−946.679	−0.525787
$149$	738.254	0.405907	0.202953	−	0.979188i	$-0.434946\pi$
$149$	738.254	0.405907	0.202953	+	0.979188i	$0.434946\pi$
$150$	0	0
$151$	−1027.19	−0.553588	−0.276794	−	0.960929i	$-0.589272\pi$
$151$	−1027.19	−0.553588	−0.276794	+	0.960929i	$0.589272\pi$
$152$	−878.382	−0.468725
$153$	0	0
$154$	−15.8893	−0.00831427
$155$	−2422.76	−1.25549
$156$	0	0
$157$	−1840.76	−0.935722	−0.467861	−	0.883802i	$-0.654975\pi$
$157$	−1840.76	−0.935722	−0.467861	+	0.883802i	$0.654975\pi$
$158$	44.1720	0.0222414
$159$	0	0
$160$	947.325	0.468079
$161$	453.578	0.222031
$162$	0	0
$163$	739.726	0.355459	0.177730	−	0.984079i	$-0.443125\pi$
$163$	739.726	0.355459	0.177730	+	0.984079i	$0.443125\pi$
$164$	−680.473	−0.324000
$165$	0	0
$166$	20.2382	0.00946257
$167$	1012.24	0.469039	0.234520	−	0.972111i	$-0.424648\pi$
$167$	1012.24	0.469039	0.234520	+	0.972111i	$0.424648\pi$
$168$	0	0
$169$	−2135.73	−0.972112
$170$	−522.965	−0.235939
$171$	0	0
$172$	1620.24	0.718269
$173$	2048.23	0.900140	0.450070	−	0.892993i	$-0.351399\pi$
$173$	2048.23	0.900140	0.450070	+	0.892993i	$0.351399\pi$
$174$	0	0
$175$	10.0822	0.00435509
$176$	286.755	0.122812
$177$	0	0
$178$	−686.509	−0.289079
$179$	499.102	0.208406	0.104203	−	0.994556i	$-0.466771\pi$
$179$	499.102	0.208406	0.104203	+	0.994556i	$0.466771\pi$
$180$	0	0
$181$	−411.248	−0.168883	−0.0844415	−	0.996428i	$-0.526911\pi$
$181$	−411.248	−0.168883	−0.0844415	+	0.996428i	$0.526911\pi$
$182$	−25.5760	−0.0104166
$183$	0	0
$184$	463.657	0.185767
$185$	1351.22	0.536994
$186$	0	0
$187$	−497.380	−0.194503
$188$	3663.80	1.42133
$189$	0	0
$190$	618.471	0.236150
$191$	919.739	0.348429	0.174215	−	0.984708i	$-0.444261\pi$
$191$	919.739	0.348429	0.174215	+	0.984708i	$0.444261\pi$
$192$	0	0
$193$	4411.86	1.64545	0.822727	−	0.568437i	$-0.192452\pi$
$193$	4411.86	1.64545	0.822727	+	0.568437i	$0.192452\pi$
$194$	−113.367	−0.0419550
$195$	0	0
$196$	2278.38	0.830315
$197$	2903.11	1.04994	0.524969	−	0.851121i	$-0.324077\pi$
$197$	2903.11	1.04994	0.524969	+	0.851121i	$0.324077\pi$
$198$	0	0
$199$	−1511.10	−0.538286	−0.269143	−	0.963100i	$-0.586740\pi$
$199$	−1511.10	−0.538286	−0.269143	+	0.963100i	$0.586740\pi$
$200$	10.3062	0.00364379
$201$	0	0
$202$	−574.272	−0.200028
$203$	−968.139	−0.334729
$204$	0	0
$205$	971.260	0.330906
$206$	−252.935	−0.0855476
$207$	0	0
$208$	461.571	0.153866
$209$	588.214	0.194677
$210$	0	0
$211$	−720.035	−0.234926	−0.117463	−	0.993077i	$-0.537476\pi$
$211$	−720.035	−0.234926	−0.117463	+	0.993077i	$0.537476\pi$
$212$	−3869.71	−1.25365
$213$	0	0
$214$	513.012	0.163873
$215$	−2312.62	−0.733578
$216$	0	0
$217$	1548.23	0.484335
$218$	−523.660	−0.162692
$219$	0	0
$220$	−421.041	−0.129030
$221$	−800.601	−0.243684
$222$	0	0
$223$	3062.84	0.919744	0.459872	−	0.887985i	$-0.347895\pi$
$223$	3062.84	0.919744	0.459872	+	0.887985i	$0.347895\pi$
$224$	−605.375	−0.180573
$225$	0	0
$226$	−924.873	−0.272220
$227$	−3616.10	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$227$	−3616.10	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$228$	0	0
$229$	−126.572	−0.0365246	−0.0182623	−	0.999833i	$-0.505813\pi$
$229$	−126.572	−0.0365246	−0.0182623	+	0.999833i	$0.505813\pi$
$230$	−326.462	−0.0935924
$231$	0	0
$232$	−989.651	−0.280059
$233$	5065.08	1.42414	0.712070	−	0.702108i	$-0.247758\pi$
$233$	5065.08	1.42414	0.712070	+	0.702108i	$0.247758\pi$
$234$	0	0
$235$	−5229.44	−1.45162
$236$	6486.19	1.78905
$237$	0	0
$238$	334.194	0.0910192
$239$	4549.84	1.23140	0.615700	−	0.787980i	$-0.288873\pi$
$239$	4549.84	1.23140	0.615700	+	0.787980i	$0.288873\pi$
$240$	0	0
$241$	−1235.80	−0.330311	−0.165155	−	0.986268i	$-0.552813\pi$
$241$	−1235.80	−0.330311	−0.165155	+	0.986268i	$0.552813\pi$
$242$	−601.310	−0.159726
$243$	0	0
$244$	−3080.88	−0.808332
$245$	−3252.00	−0.848012
$246$	0	0
$247$	946.810	0.243903
$248$	1582.63	0.405231
$249$	0	0
$250$	−646.390	−0.163525
$251$	857.660	0.215677	0.107839	−	0.994168i	$-0.465607\pi$
$251$	857.660	0.215677	0.107839	+	0.994168i	$0.465607\pi$
$252$	0	0
$253$	−310.491	−0.0771556
$254$	758.730	0.187429
$255$	0	0
$256$	3055.30	0.745923
$257$	−6.79502	−0.00164927	−0.000824634	−	1.00000i	$-0.500262\pi$
$257$	−6.79502	−0.00164927	−0.000824634		1.00000i	$0.500262\pi$
$258$	0	0
$259$	−863.481	−0.207159
$260$	−677.722	−0.161656
$261$	0	0
$262$	−495.826	−0.116917
$263$	1282.90	0.300787	0.150394	−	0.988626i	$-0.451946\pi$
$263$	1282.90	0.300787	0.150394	+	0.988626i	$0.451946\pi$
$264$	0	0
$265$	5523.35	1.28037
$266$	−395.226	−0.0911009
$267$	0	0
$268$	6147.68	1.40123
$269$	−6329.27	−1.43458	−0.717290	−	0.696774i	$-0.754618\pi$
$269$	−6329.27	−1.43458	−0.717290	+	0.696774i	$0.754618\pi$
$270$	0	0
$271$	−3310.31	−0.742018	−0.371009	−	0.928629i	$-0.620988\pi$
$271$	−3310.31	−0.742018	−0.371009	+	0.928629i	$0.620988\pi$
$272$	−6031.20	−1.34447
$273$	0	0
$274$	1460.91	0.322106
$275$	−6.90162	−0.00151339
$276$	0	0
$277$	−7760.10	−1.68325	−0.841623	−	0.540066i	$-0.818399\pi$
$277$	−7760.10	−1.68325	−0.841623	+	0.540066i	$0.818399\pi$
$278$	−1291.58	−0.278646
$279$	0	0
$280$	573.486	0.122401
$281$	−1714.24	−0.363925	−0.181963	−	0.983305i	$-0.558245\pi$
$281$	−1714.24	−0.363925	−0.181963	+	0.983305i	$0.558245\pi$
$282$	0	0
$283$	7547.55	1.58535	0.792677	−	0.609642i	$-0.208687\pi$
$283$	7547.55	1.58535	0.792677	+	0.609642i	$0.208687\pi$
$284$	−5185.93	−1.08355
$285$	0	0
$286$	17.5077	0.00361976
$287$	−620.671	−0.127655
$288$	0	0
$289$	5548.20	1.12929
$290$	696.816	0.141098
$291$	0	0
$292$	5352.19	1.07265
$293$	5236.74	1.04414	0.522071	−	0.852902i	$-0.325160\pi$
$293$	5236.74	1.04414	0.522071	+	0.852902i	$0.325160\pi$
$294$	0	0
$295$	−9257.93	−1.82718
$296$	−882.668	−0.173324
$297$	0	0
$298$	339.556	0.0660066
$299$	−499.777	−0.0966650
$300$	0	0
$301$	1477.85	0.282996
$302$	−472.453	−0.0900218
$303$	0	0
$304$	7132.65	1.34568
$305$	4397.43	0.825561
$306$	0	0
$307$	−6667.93	−1.23961	−0.619803	−	0.784758i	$-0.712787\pi$
$307$	−6667.93	−1.23961	−0.619803	+	0.784758i	$0.712787\pi$
$308$	269.061	0.0497764
$309$	0	0
$310$	−1114.34	−0.204161
$311$	−4667.46	−0.851021	−0.425511	−	0.904953i	$-0.639906\pi$
$311$	−4667.46	−0.851021	−0.425511	+	0.904953i	$0.639906\pi$
$312$	0	0
$313$	1259.12	0.227379	0.113690	−	0.993516i	$-0.463733\pi$
$313$	1259.12	0.227379	0.113690	+	0.993516i	$0.463733\pi$
$314$	−846.648	−0.152163
$315$	0	0
$316$	−747.983	−0.133156
$317$	7867.19	1.39390	0.696949	−	0.717121i	$-0.254541\pi$
$317$	7867.19	1.39390	0.696949	+	0.717121i	$0.254541\pi$
$318$	0	0
$319$	662.726	0.116318
$320$	−4808.47	−0.840006
$321$	0	0
$322$	208.621	0.0361056
$323$	−12371.7	−2.13120
$324$	0	0
$325$	−11.1091	−0.00189607
$326$	340.234	0.0578030
$327$	0	0
$328$	−634.462	−0.106806
$329$	3341.81	0.559999
$330$	0	0
$331$	−3495.07	−0.580381	−0.290191	−	0.956969i	$-0.593719\pi$
$331$	−3495.07	−0.580381	−0.290191	+	0.956969i	$0.593719\pi$
$332$	−342.701	−0.0566512
$333$	0	0
$334$	465.576	0.0762729
$335$	−8774.77	−1.43110
$336$	0	0
$337$	7630.86	1.23347	0.616735	−	0.787171i	$-0.288455\pi$
$337$	7630.86	1.23347	0.616735	+	0.787171i	$0.288455\pi$
$338$	−982.319	−0.158080
$339$	0	0
$340$	8855.59	1.41253
$341$	−1059.82	−0.168306
$342$	0	0
$343$	4514.81	0.710720
$344$	1510.69	0.236776
$345$	0	0
$346$	942.074	0.146376
$347$	2270.78	0.351302	0.175651	−	0.984453i	$-0.443797\pi$
$347$	2270.78	0.351302	0.175651	+	0.984453i	$0.443797\pi$
$348$	0	0
$349$	3421.14	0.524726	0.262363	−	0.964969i	$-0.415498\pi$
$349$	3421.14	0.524726	0.262363	+	0.964969i	$0.415498\pi$
$350$	4.63725	0.000708204 0
$351$	0	0
$352$	414.401	0.0627490
$353$	5311.03	0.800787	0.400394	−	0.916343i	$-0.368873\pi$
$353$	5311.03	0.800787	0.400394	+	0.916343i	$0.368873\pi$
$354$	0	0
$355$	7402.03	1.10664
$356$	11624.9	1.73068
$357$	0	0
$358$	229.560	0.0338900
$359$	10257.5	1.50799	0.753996	−	0.656879i	$-0.228124\pi$
$359$	10257.5	1.50799	0.753996	+	0.656879i	$0.228124\pi$
$360$	0	0
$361$	7772.03	1.13311
$362$	−189.152	−0.0274629
$363$	0	0
$364$	433.089	0.0623628
$365$	−7639.33	−1.09551
$366$	0	0
$367$	−11286.9	−1.60537	−0.802684	−	0.596404i	$-0.796596\pi$
$367$	−11286.9	−1.60537	−0.802684	+	0.596404i	$0.796596\pi$
$368$	−3764.99	−0.533325
$369$	0	0
$370$	621.489	0.0873234
$371$	−3529.63	−0.493933
$372$	0	0
$373$	7156.42	0.993419	0.496709	−	0.867917i	$-0.334542\pi$
$373$	7156.42	0.993419	0.496709	+	0.867917i	$0.334542\pi$
$374$	−228.768	−0.0316291
$375$	0	0
$376$	3416.06	0.468537
$377$	1066.75	0.145730
$378$	0	0
$379$	10053.0	1.36251	0.681253	−	0.732048i	$-0.261435\pi$
$379$	10053.0	1.36251	0.681253	+	0.732048i	$0.261435\pi$
$380$	−10472.8	−1.41380
$381$	0	0
$382$	423.029	0.0566599
$383$	6679.30	0.891113	0.445557	−	0.895254i	$-0.353006\pi$
$383$	6679.30	0.891113	0.445557	+	0.895254i	$0.353006\pi$
$384$	0	0
$385$	−384.038	−0.0508374
$386$	2029.21	0.267576
$387$	0	0
$388$	1919.69	0.251179
$389$	4112.68	0.536044	0.268022	−	0.963413i	$-0.413630\pi$
$389$	4112.68	0.536044	0.268022	+	0.963413i	$0.413630\pi$
$390$	0	0
$391$	6530.43	0.844649
$392$	2124.33	0.273711
$393$	0	0
$394$	1335.27	0.170736
$395$	1067.62	0.135994
$396$	0	0
$397$	−2142.41	−0.270842	−0.135421	−	0.990788i	$-0.543239\pi$
$397$	−2142.41	−0.270842	−0.135421	+	0.990788i	$0.543239\pi$
$398$	−695.022	−0.0875335
$399$	0	0
$400$	−83.6886	−0.0104611
$401$	3200.71	0.398593	0.199296	−	0.979939i	$-0.436134\pi$
$401$	3200.71	0.398593	0.199296	+	0.979939i	$0.436134\pi$
$402$	0	0
$403$	−1705.92	−0.210864
$404$	9724.39	1.19754
$405$	0	0
$406$	−445.291	−0.0544321
$407$	591.084	0.0719876
$408$	0	0
$409$	7010.83	0.847587	0.423794	−	0.905759i	$-0.360698\pi$
$409$	7010.83	0.847587	0.423794	+	0.905759i	$0.360698\pi$
$410$	446.726	0.0538103
$411$	0	0
$412$	4283.05	0.512162
$413$	5916.16	0.704879
$414$	0	0
$415$	489.148	0.0578586
$416$	667.035	0.0786156
$417$	0	0
$418$	270.546	0.0316575
$419$	−9072.29	−1.05778	−0.528890	−	0.848690i	$-0.677392\pi$
$419$	−9072.29	−1.05778	−0.528890	+	0.848690i	$0.677392\pi$
$420$	0	0
$421$	−4555.40	−0.527356	−0.263678	−	0.964611i	$-0.584936\pi$
$421$	−4555.40	−0.527356	−0.263678	+	0.964611i	$0.584936\pi$
$422$	−331.177	−0.0382025
$423$	0	0
$424$	−3608.06	−0.413261
$425$	145.159	0.0165676
$426$	0	0
$427$	−2810.12	−0.318481
$428$	−8687.05	−0.981085
$429$	0	0
$430$	−1063.68	−0.119291
$431$	−5707.29	−0.637843	−0.318922	−	0.947781i	$-0.603321\pi$
$431$	−5707.29	−0.637843	−0.318922	+	0.947781i	$0.603321\pi$
$432$	0	0
$433$	1350.67	0.149905	0.0749526	−	0.997187i	$-0.476119\pi$
$433$	1350.67	0.149905	0.0749526	+	0.997187i	$0.476119\pi$
$434$	712.101	0.0787603
$435$	0	0
$436$	8867.36	0.974013
$437$	−7723.04	−0.845407
$438$	0	0
$439$	7301.19	0.793774	0.396887	−	0.917868i	$-0.370091\pi$
$439$	7301.19	0.793774	0.396887	+	0.917868i	$0.370091\pi$
$440$	−392.572	−0.0425344
$441$	0	0
$442$	−368.233	−0.0396268
$443$	−570.523	−0.0611881	−0.0305941	−	0.999532i	$-0.509740\pi$
$443$	−570.523	−0.0611881	−0.0305941	+	0.999532i	$0.509740\pi$
$444$	0	0
$445$	−16592.6	−1.76756
$446$	1408.74	0.149564
$447$	0	0
$448$	3072.79	0.324053
$449$	9143.49	0.961043	0.480521	−	0.876983i	$-0.340447\pi$
$449$	9143.49	0.961043	0.480521	+	0.876983i	$0.340447\pi$
$450$	0	0
$451$	424.871	0.0443601
$452$	15661.3	1.62974
$453$	0	0
$454$	−1663.21	−0.171935
$455$	−618.162	−0.0636920
$456$	0	0
$457$	−6793.72	−0.695398	−0.347699	−	0.937606i	$-0.613037\pi$
$457$	−6793.72	−0.695398	−0.347699	+	0.937606i	$0.613037\pi$
$458$	−58.2163	−0.00593945
$459$	0	0
$460$	5528.12	0.560326
$461$	9737.96	0.983822	0.491911	−	0.870645i	$-0.336299\pi$
$461$	9737.96	0.983822	0.491911	+	0.870645i	$0.336299\pi$
$462$	0	0
$463$	−11251.3	−1.12936	−0.564679	−	0.825310i	$-0.691000\pi$
$463$	−11251.3	−1.12936	−0.564679	+	0.825310i	$0.691000\pi$
$464$	8036.18	0.804031
$465$	0	0
$466$	2329.66	0.231587
$467$	1340.50	0.132829	0.0664144	−	0.997792i	$-0.478844\pi$
$467$	1340.50	0.132829	0.0664144	+	0.997792i	$0.478844\pi$
$468$	0	0
$469$	5607.40	0.552081
$470$	−2405.26	−0.236056
$471$	0	0
$472$	6047.62	0.589755
$473$	−1011.64	−0.0983410
$474$	0	0
$475$	−171.668	−0.0165825
$476$	−5659.04	−0.544920
$477$	0	0
$478$	2092.68	0.200245
$479$	−13640.3	−1.30113	−0.650564	−	0.759452i	$-0.725467\pi$
$479$	−13640.3	−1.30113	−0.650564	+	0.759452i	$0.725467\pi$
$480$	0	0
$481$	951.430	0.0901902
$482$	−568.400	−0.0537135
$483$	0	0
$484$	10182.2	0.956259
$485$	−2740.03	−0.256533
$486$	0	0
$487$	−17589.3	−1.63665	−0.818325	−	0.574756i	$-0.805097\pi$
$487$	−17589.3	−1.63665	−0.818325	+	0.574756i	$0.805097\pi$
$488$	−2872.56	−0.266465
$489$	0	0
$490$	−1495.74	−0.137900
$491$	−18671.3	−1.71614	−0.858068	−	0.513535i	$-0.828336\pi$
$491$	−18671.3	−1.71614	−0.858068	+	0.513535i	$0.828336\pi$
$492$	0	0
$493$	−13938.9	−1.27338
$494$	435.481	0.0396624
$495$	0	0
$496$	−12851.3	−1.16339
$497$	−4730.17	−0.426916
$498$	0	0
$499$	−1815.82	−0.162900	−0.0814501	−	0.996677i	$-0.525955\pi$
$499$	−1815.82	−0.162900	−0.0814501	+	0.996677i	$0.525955\pi$
$500$	10945.6	0.979004
$501$	0	0
$502$	394.477	0.0350724
$503$	7340.27	0.650669	0.325334	−	0.945599i	$-0.394523\pi$
$503$	7340.27	0.650669	0.325334	+	0.945599i	$0.394523\pi$
$504$	0	0
$505$	−13879.9	−1.22307
$506$	−142.809	−0.0125467
$507$	0	0
$508$	−12847.9	−1.12211
$509$	15594.6	1.35799	0.678997	−	0.734141i	$-0.262415\pi$
$509$	15594.6	1.35799	0.678997	+	0.734141i	$0.262415\pi$
$510$	0	0
$511$	4881.82	0.422620
$512$	8450.70	0.729437
$513$	0	0
$514$	−3.12534	−0.000268196 0
$515$	−6113.33	−0.523079
$516$	0	0
$517$	−2287.59	−0.194600
$518$	−397.154	−0.0336872
$519$	0	0
$520$	−631.897	−0.0532895
$521$	−19088.8	−1.60517	−0.802587	−	0.596535i	$-0.796544\pi$
$521$	−19088.8	−1.60517	−0.802587	+	0.596535i	$0.796544\pi$
$522$	0	0
$523$	1005.56	0.0840731	0.0420366	−	0.999116i	$-0.486615\pi$
$523$	1005.56	0.0840731	0.0420366	+	0.999116i	$0.486615\pi$
$524$	8396.03	0.699966
$525$	0	0
$526$	590.064	0.0489126
$527$	22290.8	1.84251
$528$	0	0
$529$	−8090.37	−0.664944
$530$	2540.44	0.208207
$531$	0	0
$532$	6692.52	0.545409
$533$	683.888	0.0555769
$534$	0	0
$535$	12399.3	1.00200
$536$	5732.00	0.461912
$537$	0	0
$538$	−2911.12	−0.233285
$539$	−1422.57	−0.113682
$540$	0	0
$541$	−7836.19	−0.622743	−0.311372	−	0.950288i	$-0.600788\pi$
$541$	−7836.19	−0.622743	−0.311372	+	0.950288i	$0.600788\pi$
$542$	−1522.56	−0.120663
$543$	0	0
$544$	−8715.94	−0.686935
$545$	−12656.7	−0.994773
$546$	0	0
$547$	−12504.7	−0.977445	−0.488723	−	0.872439i	$-0.662537\pi$
$547$	−12504.7	−0.977445	−0.488723	+	0.872439i	$0.662537\pi$
$548$	−24738.3	−1.92841
$549$	0	0
$550$	−3.17437	−0.000246101 0
$551$	16484.4	1.27452
$552$	0	0
$553$	−682.248	−0.0524632
$554$	−3569.22	−0.273721
$555$	0	0
$556$	21870.8	1.66822
$557$	−12312.0	−0.936582	−0.468291	−	0.883574i	$-0.655130\pi$
$557$	−12312.0	−0.936582	−0.468291	+	0.883574i	$0.655130\pi$
$558$	0	0
$559$	−1628.37	−0.123207
$560$	−4656.82	−0.351405
$561$	0	0
$562$	−788.456	−0.0591798
$563$	−14827.0	−1.10992	−0.554959	−	0.831878i	$-0.687266\pi$
$563$	−14827.0	−1.10992	−0.554959	+	0.831878i	$0.687266\pi$
$564$	0	0
$565$	−22353.8	−1.66448
$566$	3471.46	0.257803
$567$	0	0
$568$	−4835.27	−0.357189
$569$	17003.7	1.25278	0.626391	−	0.779509i	$-0.284531\pi$
$569$	17003.7	1.25278	0.626391	+	0.779509i	$0.284531\pi$
$570$	0	0
$571$	−15153.9	−1.11063	−0.555317	−	0.831639i	$-0.687403\pi$
$571$	−15153.9	−1.11063	−0.555317	+	0.831639i	$0.687403\pi$
$572$	−296.465	−0.0216710
$573$	0	0
$574$	−285.475	−0.0207587
$575$	90.6158	0.00657207
$576$	0	0
$577$	−11634.7	−0.839443	−0.419722	−	0.907653i	$-0.637872\pi$
$577$	−11634.7	−0.839443	−0.419722	+	0.907653i	$0.637872\pi$
$578$	2551.87	0.183640
$579$	0	0
$580$	−11799.5	−0.844736
$581$	−312.584	−0.0223204
$582$	0	0
$583$	2416.16	0.171641
$584$	4990.29	0.353595
$585$	0	0
$586$	2408.61	0.169793
$587$	−6383.80	−0.448872	−0.224436	−	0.974489i	$-0.572054\pi$
$587$	−6383.80	−0.448872	−0.224436	+	0.974489i	$0.572054\pi$
$588$	0	0
$589$	−26361.6	−1.84416
$590$	−4258.14	−0.297127
$591$	0	0
$592$	7167.45	0.497602
$593$	−5472.02	−0.378936	−0.189468	−	0.981887i	$-0.560676\pi$
$593$	−5472.02	−0.378936	−0.189468	+	0.981887i	$0.560676\pi$
$594$	0	0
$595$	8077.32	0.556534
$596$	−5749.85	−0.395173
$597$	0	0
$598$	−229.870	−0.0157192
$599$	−9204.38	−0.627848	−0.313924	−	0.949448i	$-0.601644\pi$
$599$	−9204.38	−0.627848	−0.313924	+	0.949448i	$0.601644\pi$
$600$	0	0
$601$	6769.03	0.459425	0.229713	−	0.973258i	$-0.426221\pi$
$601$	6769.03	0.459425	0.229713	+	0.973258i	$0.426221\pi$
$602$	679.730	0.0460195
$603$	0	0
$604$	8000.24	0.538949
$605$	−14533.4	−0.976641
$606$	0	0
$607$	−24846.8	−1.66145	−0.830725	−	0.556683i	$-0.812074\pi$
$607$	−24846.8	−1.66145	−0.830725	+	0.556683i	$0.812074\pi$
$608$	10307.7	0.687552
$609$	0	0
$610$	2022.58	0.134249
$611$	−3682.18	−0.243806
$612$	0	0
$613$	18378.0	1.21090	0.605449	−	0.795884i	$-0.292994\pi$
$613$	18378.0	1.21090	0.605449	+	0.795884i	$0.292994\pi$
$614$	−3066.88	−0.201579
$615$	0	0
$616$	250.868	0.0164087
$617$	−6262.45	−0.408617	−0.204309	−	0.978907i	$-0.565495\pi$
$617$	−6262.45	−0.408617	−0.204309	+	0.978907i	$0.565495\pi$
$618$	0	0
$619$	14118.5	0.916752	0.458376	−	0.888758i	$-0.348431\pi$
$619$	14118.5	0.916752	0.458376	+	0.888758i	$0.348431\pi$
$620$	18869.5	1.22229
$621$	0	0
$622$	−2146.78	−0.138389
$623$	10603.3	0.681882
$624$	0	0
$625$	−15445.6	−0.988517
$626$	579.126	0.0369753
$627$	0	0
$628$	14336.6	0.910978
$629$	−12432.0	−0.788073
$630$	0	0
$631$	−3557.70	−0.224453	−0.112227	−	0.993683i	$-0.535798\pi$
$631$	−3557.70	−0.224453	−0.112227	+	0.993683i	$0.535798\pi$
$632$	−697.408	−0.0438946
$633$	0	0
$634$	3618.48	0.226669
$635$	18338.2	1.14603
$636$	0	0
$637$	−2289.82	−0.142427
$638$	304.818	0.0189151
$639$	0	0
$640$	−9790.23	−0.604676
$641$	−20879.0	−1.28654	−0.643269	−	0.765640i	$-0.722422\pi$
$641$	−20879.0	−1.28654	−0.643269	+	0.765640i	$0.722422\pi$
$642$	0	0
$643$	−9481.59	−0.581520	−0.290760	−	0.956796i	$-0.593908\pi$
$643$	−9481.59	−0.581520	−0.290760	+	0.956796i	$0.593908\pi$
$644$	−3532.67	−0.216159
$645$	0	0
$646$	−5690.29	−0.346566
$647$	20313.7	1.23434	0.617168	−	0.786831i	$-0.288280\pi$
$647$	20313.7	1.23434	0.617168	+	0.786831i	$0.288280\pi$
$648$	0	0
$649$	−4049.82	−0.244945
$650$	−5.10957	−0.000308329 0
$651$	0	0
$652$	−5761.32	−0.346059
$653$	−10806.6	−0.647619	−0.323810	−	0.946122i	$-0.604964\pi$
$653$	−10806.6	−0.647619	−0.323810	+	0.946122i	$0.604964\pi$
$654$	0	0
$655$	−11983.9	−0.714886
$656$	5151.97	0.306632
$657$	0	0
$658$	1537.05	0.0910644
$659$	21257.1	1.25654	0.628269	−	0.777996i	$-0.283764\pi$
$659$	21257.1	1.25654	0.628269	+	0.777996i	$0.283764\pi$
$660$	0	0
$661$	989.827	0.0582448	0.0291224	−	0.999576i	$-0.490729\pi$
$661$	989.827	0.0582448	0.0291224	+	0.999576i	$0.490729\pi$
$662$	−1607.54	−0.0943788
$663$	0	0
$664$	−319.529	−0.0186749
$665$	−9552.44	−0.557034
$666$	0	0
$667$	−8701.36	−0.505125
$668$	−7883.79	−0.456636
$669$	0	0
$670$	−4035.92	−0.232718
$671$	1923.63	0.110672
$672$	0	0
$673$	21092.8	1.20813	0.604063	−	0.796937i	$-0.293548\pi$
$673$	21092.8	1.20813	0.604063	+	0.796937i	$0.293548\pi$
$674$	3509.78	0.200581
$675$	0	0
$676$	16634.0	0.946406
$677$	15100.5	0.857251	0.428626	−	0.903482i	$-0.358998\pi$
$677$	15100.5	0.857251	0.428626	+	0.903482i	$0.358998\pi$
$678$	0	0
$679$	1750.98	0.0989639
$680$	8256.80	0.465638
$681$	0	0
$682$	−487.459	−0.0273692
$683$	7001.86	0.392267	0.196134	−	0.980577i	$-0.437161\pi$
$683$	7001.86	0.392267	0.196134	+	0.980577i	$0.437161\pi$
$684$	0	0
$685$	35309.7	1.96951
$686$	2076.57	0.115574
$687$	0	0
$688$	−12267.1	−0.679766
$689$	3889.13	0.215042
$690$	0	0
$691$	−33946.2	−1.86885	−0.934425	−	0.356160i	$-0.884086\pi$
$691$	−33946.2	−1.86885	−0.934425	+	0.356160i	$0.884086\pi$
$692$	−15952.5	−0.876336
$693$	0	0
$694$	1044.43	0.0571270
$695$	−31216.8	−1.70377
$696$	0	0
$697$	−8936.15	−0.485625
$698$	1573.54	0.0853285
$699$	0	0
$700$	−78.5245	−0.00423993
$701$	30431.8	1.63965	0.819825	−	0.572614i	$-0.194071\pi$
$701$	30431.8	1.63965	0.819825	+	0.572614i	$0.194071\pi$
$702$	0	0
$703$	14702.4	0.788780
$704$	−2103.44	−0.112608
$705$	0	0
$706$	2442.79	0.130220
$707$	8869.78	0.471828
$708$	0	0
$709$	22190.4	1.17543	0.587713	−	0.809069i	$-0.300028\pi$
$709$	22190.4	1.17543	0.587713	+	0.809069i	$0.300028\pi$
$710$	3404.53	0.179957
$711$	0	0
$712$	10838.9	0.570513
$713$	13915.1	0.730888
$714$	0	0
$715$	423.154	0.0221329
$716$	−3887.23	−0.202895
$717$	0	0
$718$	4717.88	0.245223
$719$	10671.9	0.553538	0.276769	−	0.960937i	$-0.410736\pi$
$719$	10671.9	0.553538	0.276769	+	0.960937i	$0.410736\pi$
$720$	0	0
$721$	3906.64	0.201791
$722$	3574.71	0.184262
$723$	0	0
$724$	3202.98	0.164417
$725$	−193.415	−0.00990792
$726$	0	0
$727$	−34608.4	−1.76555	−0.882774	−	0.469799i	$-0.844327\pi$
$727$	−34608.4	−1.76555	−0.882774	+	0.469799i	$0.844327\pi$
$728$	403.805	0.0205577
$729$	0	0
$730$	−3513.68	−0.178147
$731$	21277.4	1.07657
$732$	0	0
$733$	−1097.10	−0.0552827	−0.0276414	−	0.999618i	$-0.508800\pi$
$733$	−1097.10	−0.0552827	−0.0276414	+	0.999618i	$0.508800\pi$
$734$	−5191.35	−0.261057
$735$	0	0
$736$	−5440.94	−0.272494
$737$	−3838.47	−0.191848
$738$	0	0
$739$	−39868.4	−1.98455	−0.992275	−	0.124056i	$-0.960410\pi$
$739$	−39868.4	−1.98455	−0.992275	+	0.124056i	$0.960410\pi$
$740$	−10523.9	−0.522794
$741$	0	0
$742$	−1623.44	−0.0803210
$743$	−2587.55	−0.127763	−0.0638815	−	0.997957i	$-0.520348\pi$
$743$	−2587.55	−0.127763	−0.0638815	+	0.997957i	$0.520348\pi$
$744$	0	0
$745$	8206.94	0.403596
$746$	3291.56	0.161545
$747$	0	0
$748$	3873.82	0.189360
$749$	−7923.60	−0.386545
$750$	0	0
$751$	−25125.4	−1.22082	−0.610411	−	0.792085i	$-0.708996\pi$
$751$	−25125.4	−1.22082	−0.610411	+	0.792085i	$0.708996\pi$
$752$	−27739.2	−1.34514
$753$	0	0
$754$	490.646	0.0236980
$755$	−11419.0	−0.550436
$756$	0	0
$757$	36022.2	1.72953	0.864763	−	0.502181i	$-0.167469\pi$
$757$	36022.2	1.72953	0.864763	+	0.502181i	$0.167469\pi$
$758$	4623.85	0.221564
$759$	0	0
$760$	−9764.69	−0.466056
$761$	2304.78	0.109788	0.0548938	−	0.998492i	$-0.482518\pi$
$761$	2304.78	0.109788	0.0548938	+	0.998492i	$0.482518\pi$
$762$	0	0
$763$	8088.07	0.383758
$764$	−7163.34	−0.339215
$765$	0	0
$766$	3072.11	0.144909
$767$	−6518.74	−0.306882
$768$	0	0
$769$	−30373.1	−1.42430	−0.712148	−	0.702029i	$-0.752278\pi$
$769$	−30373.1	−1.42430	−0.712148	+	0.702029i	$0.752278\pi$
$770$	−176.637	−0.00826693
$771$	0	0
$772$	−34361.5	−1.60194
$773$	27192.3	1.26525	0.632626	−	0.774457i	$-0.281977\pi$
$773$	27192.3	1.26525	0.632626	+	0.774457i	$0.281977\pi$
$774$	0	0
$775$	309.305	0.0143362
$776$	1789.89	0.0828005
$777$	0	0
$778$	1891.61	0.0871689
$779$	10568.1	0.486061
$780$	0	0
$781$	3237.97	0.148353
$782$	3003.64	0.137353
$783$	0	0
$784$	−17250.0	−0.785805
$785$	−20463.1	−0.930395
$786$	0	0
$787$	11585.2	0.524737	0.262368	−	0.964968i	$-0.415496\pi$
$787$	11585.2	0.524737	0.262368	+	0.964968i	$0.415496\pi$
$788$	−22610.7	−1.02217
$789$	0	0
$790$	491.046	0.0221147
$791$	14284.9	0.642114
$792$	0	0
$793$	3096.34	0.138656
$794$	−985.390	−0.0440430
$795$	0	0
$796$	11769.1	0.524052
$797$	15752.2	0.700089	0.350044	−	0.936733i	$-0.386166\pi$
$797$	15752.2	0.700089	0.350044	+	0.936733i	$0.386166\pi$
$798$	0	0
$799$	48113.9	2.13035
$800$	−120.942	−0.00534492
$801$	0	0
$802$	1472.15	0.0648173
$803$	−3341.78	−0.146860
$804$	0	0
$805$	5042.29	0.220767
$806$	−784.632	−0.0342897
$807$	0	0
$808$	9066.87	0.394766
$809$	−11596.3	−0.503961	−0.251981	−	0.967732i	$-0.581082\pi$
$809$	−11596.3	−0.503961	−0.251981	+	0.967732i	$0.581082\pi$
$810$	0	0
$811$	−34806.7	−1.50706	−0.753531	−	0.657412i	$-0.771651\pi$
$811$	−34806.7	−1.50706	−0.753531	+	0.657412i	$0.771651\pi$
$812$	7540.30	0.325878
$813$	0	0
$814$	271.866	0.0117063
$815$	8223.30	0.353435
$816$	0	0
$817$	−25163.2	−1.07754
$818$	3224.60	0.137831
$819$	0	0
$820$	−7564.61	−0.322156
$821$	−33943.9	−1.44294	−0.721468	−	0.692448i	$-0.756532\pi$
$821$	−33943.9	−1.44294	−0.721468	+	0.692448i	$0.756532\pi$
$822$	0	0
$823$	23212.7	0.983162	0.491581	−	0.870832i	$-0.336419\pi$
$823$	23212.7	0.983162	0.491581	+	0.870832i	$0.336419\pi$
$824$	3993.45	0.168833
$825$	0	0
$826$	2721.11	0.114624
$827$	−37918.9	−1.59440	−0.797201	−	0.603714i	$-0.793687\pi$
$827$	−37918.9	−1.59440	−0.797201	+	0.603714i	$0.793687\pi$
$828$	0	0
$829$	9417.51	0.394552	0.197276	−	0.980348i	$-0.436790\pi$
$829$	9417.51	0.394552	0.197276	+	0.980348i	$0.436790\pi$
$830$	224.981	0.00940870
$831$	0	0
$832$	−3385.77	−0.141082
$833$	29920.3	1.24451
$834$	0	0
$835$	11252.8	0.466369
$836$	−4581.27	−0.189529
$837$	0	0
$838$	−4172.76	−0.172011
$839$	−13603.7	−0.559776	−0.279888	−	0.960033i	$-0.590297\pi$
$839$	−13603.7	−0.559776	−0.279888	+	0.960033i	$0.590297\pi$
$840$	0	0
$841$	−5816.39	−0.238484
$842$	−2095.24	−0.0857561
$843$	0	0
$844$	5607.96	0.228713
$845$	−23742.2	−0.966577
$846$	0	0
$847$	9287.39	0.376763
$848$	29298.2	1.18644
$849$	0	0
$850$	66.7652	0.00269415
$851$	−7760.72	−0.312614
$852$	0	0
$853$	−11862.8	−0.476170	−0.238085	−	0.971244i	$-0.576520\pi$
$853$	−11862.8	−0.476170	−0.238085	+	0.971244i	$0.576520\pi$
$854$	−1292.50	−0.0517898
$855$	0	0
$856$	−8099.66	−0.323412
$857$	33355.2	1.32951	0.664757	−	0.747060i	$-0.268535\pi$
$857$	33355.2	1.32951	0.664757	+	0.747060i	$0.268535\pi$
$858$	0	0
$859$	44045.0	1.74947	0.874735	−	0.484602i	$-0.161035\pi$
$859$	44045.0	1.74947	0.874735	+	0.484602i	$0.161035\pi$
$860$	18011.7	0.714180
$861$	0	0
$862$	−2625.04	−0.103723
$863$	−22274.1	−0.878586	−0.439293	−	0.898344i	$-0.644771\pi$
$863$	−22274.1	−0.878586	−0.439293	+	0.898344i	$0.644771\pi$
$864$	0	0
$865$	22769.5	0.895015
$866$	621.233	0.0243769
$867$	0	0
$868$	−12058.3	−0.471528
$869$	467.023	0.0182309
$870$	0	0
$871$	−6178.54	−0.240358
$872$	8267.79	0.321081
$873$	0	0
$874$	−3552.18	−0.137476
$875$	9983.66	0.385725
$876$	0	0
$877$	44962.4	1.73121	0.865606	−	0.500725i	$-0.166933\pi$
$877$	44962.4	1.73121	0.865606	+	0.500725i	$0.166933\pi$
$878$	3358.15	0.129080
$879$	0	0
$880$	3187.76	0.122113
$881$	9974.44	0.381439	0.190719	−	0.981645i	$-0.438918\pi$
$881$	9974.44	0.381439	0.190719	+	0.981645i	$0.438918\pi$
$882$	0	0
$883$	−30814.0	−1.17438	−0.587188	−	0.809451i	$-0.699765\pi$
$883$	−30814.0	−1.17438	−0.587188	+	0.809451i	$0.699765\pi$
$884$	6235.44	0.237240
$885$	0	0
$886$	−262.409	−0.00995012
$887$	−11030.9	−0.417567	−0.208784	−	0.977962i	$-0.566950\pi$
$887$	−11030.9	−0.417567	−0.208784	+	0.977962i	$0.566950\pi$
$888$	0	0
$889$	−11718.8	−0.442109
$890$	−7631.70	−0.287433
$891$	0	0
$892$	−23854.8	−0.895423
$893$	−56900.7	−2.13226
$894$	0	0
$895$	5548.37	0.207219
$896$	6256.32	0.233269
$897$	0	0
$898$	4205.51	0.156280
$899$	−29701.0	−1.10187
$900$	0	0
$901$	−50818.1	−1.87902
$902$	195.418	0.00721363
$903$	0	0
$904$	14602.3	0.537241
$905$	−4571.72	−0.167922
$906$	0	0
$907$	8747.08	0.320223	0.160111	−	0.987099i	$-0.448815\pi$
$907$	8747.08	0.320223	0.160111	+	0.987099i	$0.448815\pi$
$908$	28163.8	1.02935
$909$	0	0
$910$	−284.320	−0.0103573
$911$	43088.9	1.56707	0.783534	−	0.621348i	$-0.213415\pi$
$911$	43088.9	1.56707	0.783534	+	0.621348i	$0.213415\pi$
$912$	0	0
$913$	213.975	0.00775633
$914$	−3124.74	−0.113082
$915$	0	0
$916$	985.802	0.0355587
$917$	7658.16	0.275785
$918$	0	0
$919$	−40894.9	−1.46790	−0.733948	−	0.679205i	$-0.762325\pi$
$919$	−40894.9	−1.46790	−0.733948	+	0.679205i	$0.762325\pi$
$920$	5154.33	0.184710
$921$	0	0
$922$	4478.93	0.159984
$923$	5211.95	0.185865
$924$	0	0
$925$	−172.506	−0.00613186
$926$	−5174.99	−0.183651
$927$	0	0
$928$	11613.4	0.410807
$929$	−22735.7	−0.802943	−0.401471	−	0.915872i	$-0.631501\pi$
$929$	−22735.7	−0.802943	−0.401471	+	0.915872i	$0.631501\pi$
$930$	0	0
$931$	−35384.5	−1.24563
$932$	−39449.2	−1.38648
$933$	0	0
$934$	616.558	0.0216000
$935$	−5529.22	−0.193396
$936$	0	0
$937$	42183.1	1.47072	0.735359	−	0.677678i	$-0.237014\pi$
$937$	42183.1	1.47072	0.735359	+	0.677678i	$0.237014\pi$
$938$	2579.10	0.0897767
$939$	0	0
$940$	40729.3	1.41324
$941$	−16983.1	−0.588344	−0.294172	−	0.955752i	$-0.595044\pi$
$941$	−16983.1	−0.588344	−0.294172	+	0.955752i	$0.595044\pi$
$942$	0	0
$943$	−5578.41	−0.192638
$944$	−49107.9	−1.69314
$945$	0	0
$946$	−465.299	−0.0159917
$947$	5643.02	0.193636	0.0968181	−	0.995302i	$-0.469133\pi$
$947$	5643.02	0.193636	0.0968181	+	0.995302i	$0.469133\pi$
$948$	0	0
$949$	−5379.05	−0.183995
$950$	−78.9581	−0.00269657
$951$	0	0
$952$	−5276.40	−0.179631
$953$	18215.5	0.619157	0.309579	−	0.950874i	$-0.399812\pi$
$953$	18215.5	0.619157	0.309579	+	0.950874i	$0.399812\pi$
$954$	0	0
$955$	10224.4	0.346445
$956$	−35436.2	−1.19884
$957$	0	0
$958$	−6273.78	−0.211583
$959$	−22564.2	−0.759787
$960$	0	0
$961$	17706.3	0.594349
$962$	437.606	0.0146663
$963$	0	0
$964$	9624.96	0.321576
$965$	49045.3	1.63609
$966$	0	0
$967$	−14864.3	−0.494317	−0.247159	−	0.968975i	$-0.579497\pi$
$967$	−14864.3	−0.494317	−0.247159	+	0.968975i	$0.579497\pi$
$968$	9493.76	0.315228
$969$	0	0
$970$	−1260.26	−0.0417161
$971$	−11703.2	−0.386790	−0.193395	−	0.981121i	$-0.561950\pi$
$971$	−11703.2	−0.386790	−0.193395	+	0.981121i	$0.561950\pi$
$972$	0	0
$973$	19948.7	0.657273
$974$	−8090.12	−0.266144
$975$	0	0
$976$	23325.8	0.765001
$977$	18306.8	0.599473	0.299737	−	0.954022i	$-0.403101\pi$
$977$	18306.8	0.599473	0.299737	+	0.954022i	$0.403101\pi$
$978$	0	0
$979$	−7258.34	−0.236954
$980$	25328.1	0.825587
$981$	0	0
$982$	−8587.77	−0.279070
$983$	31834.8	1.03293	0.516466	−	0.856308i	$-0.327247\pi$
$983$	31834.8	1.03293	0.516466	+	0.856308i	$0.327247\pi$
$984$	0	0
$985$	32273.0	1.04396
$986$	−6411.11	−0.207070
$987$	0	0
$988$	−7374.18	−0.237453
$989$	13282.5	0.427056
$990$	0	0
$991$	−39271.3	−1.25882	−0.629412	−	0.777072i	$-0.716704\pi$
$991$	−39271.3	−1.25882	−0.629412	+	0.777072i	$0.716704\pi$
$992$	−18572.0	−0.594416
$993$	0	0
$994$	−2175.62	−0.0694230
$995$	−16798.4	−0.535221
$996$	0	0
$997$	30945.2	0.982994	0.491497	−	0.870879i	$-0.336450\pi$
$997$	30945.2	0.982994	0.491497	+	0.870879i	$0.336450\pi$
$998$	−835.177	−0.0264901
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.4.a.a.1.8	✓	12
3.2	odd	2		729.4.a.b.1.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.4.a.a.1.8	✓	12	1.1	even	1	trivial
729.4.a.b.1.5	yes	12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.459945 q^{2} -7.78845 q^{4} +11.1167 q^{5} -7.10397 q^{7} -7.26182 q^{8} +O(q^{10})\)	\(q+0.459945 q^{2} -7.78845 q^{4} +11.1167 q^{5} -7.10397 q^{7} -7.26182 q^{8} +5.11307 q^{10} +4.86293 q^{11} +7.82754 q^{13} -3.26744 q^{14} +58.9676 q^{16} -102.280 q^{17} +120.959 q^{19} -86.5818 q^{20} +2.23668 q^{22} -63.8485 q^{23} -1.41923 q^{25} +3.60024 q^{26} +55.3289 q^{28} +136.281 q^{29} -217.939 q^{31} +85.2164 q^{32} -47.0432 q^{34} -78.9727 q^{35} +121.549 q^{37} +55.6344 q^{38} -80.7274 q^{40} +87.3695 q^{41} -208.031 q^{43} -37.8747 q^{44} -29.3668 q^{46} -470.414 q^{47} -292.534 q^{49} -0.652769 q^{50} -60.9644 q^{52} +496.852 q^{53} +54.0596 q^{55} +51.5878 q^{56} +62.6820 q^{58} -832.796 q^{59} +395.570 q^{61} -100.240 q^{62} -432.546 q^{64} +87.0163 q^{65} -789.333 q^{67} +796.603 q^{68} -36.3231 q^{70} +665.848 q^{71} -687.195 q^{73} +55.9059 q^{74} -942.082 q^{76} -34.5461 q^{77} +96.0375 q^{79} +655.524 q^{80} +40.1852 q^{82} +44.0012 q^{83} -1137.02 q^{85} -95.6830 q^{86} -35.3137 q^{88} -1492.59 q^{89} -55.6066 q^{91} +497.281 q^{92} -216.365 q^{94} +1344.66 q^{95} -246.479 q^{97} -134.549 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 36 q^{4} - 12 q^{5} - 42 q^{7} - 21 q^{8}+O(q^{10}) \) 12 * q - 6 * q^2 + 36 * q^4 - 12 * q^5 - 42 * q^7 - 21 * q^8	\( 12 q - 6 q^{2} + 36 q^{4} - 12 q^{5} - 42 q^{7} - 21 q^{8} - 60 q^{10} - 42 q^{11} - 78 q^{13} + 312 q^{14} + 48 q^{16} + 18 q^{17} - 228 q^{19} + 69 q^{20} - 309 q^{22} + 114 q^{23} - 18 q^{25} - 30 q^{26} - 813 q^{28} + 660 q^{29} - 708 q^{31} - 729 q^{32} - 972 q^{34} + 624 q^{35} - 354 q^{37} + 213 q^{38} - 1335 q^{40} + 1032 q^{41} - 744 q^{43} - 1653 q^{44} - 1077 q^{46} - 942 q^{47} - 192 q^{49} - 1905 q^{50} - 1371 q^{52} + 828 q^{53} - 1554 q^{55} + 3324 q^{56} - 831 q^{58} + 24 q^{59} - 1698 q^{61} + 2184 q^{62} - 933 q^{64} - 294 q^{65} - 1266 q^{67} + 4734 q^{68} - 1137 q^{70} - 3888 q^{71} - 1164 q^{73} - 5448 q^{74} - 2385 q^{76} - 3018 q^{77} - 2382 q^{79} + 4677 q^{80} - 276 q^{82} + 4008 q^{83} - 1116 q^{85} + 1176 q^{86} - 2769 q^{88} + 3582 q^{89} - 3222 q^{91} + 2958 q^{92} - 3324 q^{94} + 2784 q^{95} - 2958 q^{97} - 9567 q^{98}+O(q^{100}) \) 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

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