LMFDB - Embedded newform 729.4.a.a.1.7

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			1.7
Root			$0.713877$ 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)

$f(q)$	$=$	\(q-0.286123 q^{2} -7.91813 q^{4} -19.8713 q^{5} -14.8337 q^{7} +4.55455 q^{8} +5.68565 q^{10} +54.2813 q^{11} +79.8034 q^{13} +4.24426 q^{14} +62.0419 q^{16} -37.9070 q^{17} -124.017 q^{19} +157.344 q^{20} -15.5312 q^{22} +101.116 q^{23} +269.869 q^{25} -22.8336 q^{26} +117.455 q^{28} +58.0729 q^{29} -58.6421 q^{31} -54.1880 q^{32} +10.8461 q^{34} +294.764 q^{35} -148.063 q^{37} +35.4842 q^{38} -90.5049 q^{40} +184.306 q^{41} +14.9230 q^{43} -429.807 q^{44} -28.9316 q^{46} -205.825 q^{47} -122.963 q^{49} -77.2158 q^{50} -631.894 q^{52} +285.547 q^{53} -1078.64 q^{55} -67.5607 q^{56} -16.6160 q^{58} -355.222 q^{59} +146.381 q^{61} +16.7789 q^{62} -480.831 q^{64} -1585.80 q^{65} -455.666 q^{67} +300.152 q^{68} -84.3389 q^{70} +127.438 q^{71} +222.372 q^{73} +42.3642 q^{74} +981.983 q^{76} -805.191 q^{77} -117.399 q^{79} -1232.85 q^{80} -52.7343 q^{82} +622.293 q^{83} +753.261 q^{85} -4.26982 q^{86} +247.227 q^{88} +1044.81 q^{89} -1183.78 q^{91} -800.649 q^{92} +58.8912 q^{94} +2464.38 q^{95} -834.977 q^{97} +35.1825 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.286123	−0.101160	−0.0505800	−	0.998720i	$-0.516107\pi$
$2$	−0.286123	−0.101160	−0.0505800	+	0.998720i	$0.516107\pi$
$3$	0	0
$3$	0	0
$4$	−7.91813	−0.989767
$5$	−19.8713	−1.77734	−0.888672	−	0.458544i	$-0.848371\pi$
$5$	−19.8713	−1.77734	−0.888672	+	0.458544i	$0.848371\pi$
$6$	0	0
$7$	−14.8337	−0.800942	−0.400471	−	0.916309i	$-0.631154\pi$
$7$	−14.8337	−0.800942	−0.400471	+	0.916309i	$0.631154\pi$
$8$	4.55455	0.201285
$9$	0	0
$10$	5.68565	0.179796
$11$	54.2813	1.48786	0.743929	−	0.668259i	$-0.232960\pi$
$11$	54.2813	1.48786	0.743929	+	0.668259i	$0.232960\pi$
$12$	0	0
$13$	79.8034	1.70258	0.851288	−	0.524699i	$-0.175822\pi$
$13$	79.8034	1.70258	0.851288	+	0.524699i	$0.175822\pi$
$14$	4.24426	0.0810232
$15$	0	0
$16$	62.0419	0.969405
$17$	−37.9070	−0.540811	−0.270406	−	0.962746i	$-0.587158\pi$
$17$	−37.9070	−0.540811	−0.270406	+	0.962746i	$0.587158\pi$
$18$	0	0
$19$	−124.017	−1.49745	−0.748723	−	0.662883i	$-0.769332\pi$
$19$	−124.017	−1.49745	−0.748723	+	0.662883i	$0.769332\pi$
$20$	157.344	1.75916
$21$	0	0
$22$	−15.5312	−0.150512
$23$	101.116	0.916701	0.458351	−	0.888772i	$-0.348440\pi$
$23$	101.116	0.916701	0.458351	+	0.888772i	$0.348440\pi$
$24$	0	0
$25$	269.869	2.15895
$26$	−22.8336	−0.172232
$27$	0	0
$28$	117.455	0.792746
$29$	58.0729	0.371858	0.185929	−	0.982563i	$-0.440471\pi$
$29$	58.0729	0.371858	0.185929	+	0.982563i	$0.440471\pi$
$30$	0	0
$31$	−58.6421	−0.339756	−0.169878	−	0.985465i	$-0.554337\pi$
$31$	−58.6421	−0.339756	−0.169878	+	0.985465i	$0.554337\pi$
$32$	−54.1880	−0.299349
$33$	0	0
$34$	10.8461	0.0547084
$35$	294.764	1.42355
$36$	0	0
$37$	−148.063	−0.657875	−0.328937	−	0.944352i	$-0.606691\pi$
$37$	−148.063	−0.657875	−0.328937	+	0.944352i	$0.606691\pi$
$38$	35.4842	0.151481
$39$	0	0
$40$	−90.5049	−0.357752
$41$	184.306	0.702044	0.351022	−	0.936367i	$-0.385834\pi$
$41$	184.306	0.702044	0.351022	+	0.936367i	$0.385834\pi$
$42$	0	0
$43$	14.9230	0.0529241	0.0264620	−	0.999650i	$-0.491576\pi$
$43$	14.9230	0.0529241	0.0264620	+	0.999650i	$0.491576\pi$
$44$	−429.807	−1.47263
$45$	0	0
$46$	−28.9316	−0.0927334
$47$	−205.825	−0.638779	−0.319389	−	0.947624i	$-0.603478\pi$
$47$	−205.825	−0.638779	−0.319389	+	0.947624i	$0.603478\pi$
$48$	0	0
$49$	−122.963	−0.358491
$50$	−77.2158	−0.218399
$51$	0	0
$52$	−631.894	−1.68515
$53$	285.547	0.740053	0.370027	−	0.929021i	$-0.379349\pi$
$53$	285.547	0.740053	0.370027	+	0.929021i	$0.379349\pi$
$54$	0	0
$55$	−1078.64	−2.64444
$56$	−67.5607	−0.161217
$57$	0	0
$58$	−16.6160	−0.0376171
$59$	−355.222	−0.783830	−0.391915	−	0.920001i	$-0.628187\pi$
$59$	−355.222	−0.783830	−0.391915	+	0.920001i	$0.628187\pi$
$60$	0	0
$61$	146.381	0.307249	0.153624	−	0.988129i	$-0.450905\pi$
$61$	146.381	0.307249	0.153624	+	0.988129i	$0.450905\pi$
$62$	16.7789	0.0343697
$63$	0	0
$64$	−480.831	−0.939123
$65$	−1585.80	−3.02606
$66$	0	0
$67$	−455.666	−0.830873	−0.415436	−	0.909622i	$-0.636371\pi$
$67$	−455.666	−0.830873	−0.415436	+	0.909622i	$0.636371\pi$
$68$	300.152	0.535277
$69$	0	0
$70$	−84.3389	−0.144006
$71$	127.438	0.213016	0.106508	−	0.994312i	$-0.466033\pi$
$71$	127.438	0.213016	0.106508	+	0.994312i	$0.466033\pi$
$72$	0	0
$73$	222.372	0.356530	0.178265	−	0.983983i	$-0.442952\pi$
$73$	222.372	0.356530	0.178265	+	0.983983i	$0.442952\pi$
$74$	42.3642	0.0665505
$75$	0	0
$76$	981.983	1.48212
$77$	−805.191	−1.19169
$78$	0	0
$79$	−117.399	−0.167196	−0.0835978	−	0.996500i	$-0.526641\pi$
$79$	−117.399	−0.167196	−0.0835978	+	0.996500i	$0.526641\pi$
$80$	−1232.85	−1.72297
$81$	0	0
$82$	−52.7343	−0.0710187
$83$	622.293	0.822958	0.411479	−	0.911419i	$-0.365012\pi$
$83$	622.293	0.822958	0.411479	+	0.911419i	$0.365012\pi$
$84$	0	0
$85$	753.261	0.961208
$86$	−4.26982	−0.00535379
$87$	0	0
$88$	247.227	0.299483
$89$	1044.81	1.24437	0.622187	−	0.782868i	$-0.286244\pi$
$89$	1044.81	1.24437	0.622187	+	0.782868i	$0.286244\pi$
$90$	0	0
$91$	−1183.78	−1.36367
$92$	−800.649	−0.907320
$93$	0	0
$94$	58.8912	0.0646188
$95$	2464.38	2.66147
$96$	0	0
$97$	−834.977	−0.874011	−0.437005	−	0.899459i	$-0.643961\pi$
$97$	−834.977	−0.874011	−0.437005	+	0.899459i	$0.643961\pi$
$98$	35.1825	0.0362650
$99$	0	0
$100$	−2136.86	−2.13686
$101$	1552.74	1.52973	0.764867	−	0.644188i	$-0.222804\pi$
$101$	1552.74	1.52973	0.764867	+	0.644188i	$0.222804\pi$
$102$	0	0
$103$	−1279.19	−1.22371	−0.611856	−	0.790969i	$-0.709577\pi$
$103$	−1279.19	−1.22371	−0.611856	+	0.790969i	$0.709577\pi$
$104$	363.469	0.342702
$105$	0	0
$106$	−81.7016	−0.0748637
$107$	1090.37	0.985140	0.492570	−	0.870273i	$-0.336058\pi$
$107$	1090.37	0.985140	0.492570	+	0.870273i	$0.336058\pi$
$108$	0	0
$109$	−822.350	−0.722632	−0.361316	−	0.932444i	$-0.617672\pi$
$109$	−822.350	−0.722632	−0.361316	+	0.932444i	$0.617672\pi$
$110$	308.625	0.267511
$111$	0	0
$112$	−920.308	−0.776437
$113$	−1192.71	−0.992930	−0.496465	−	0.868057i	$-0.665369\pi$
$113$	−1192.71	−0.992930	−0.496465	+	0.868057i	$0.665369\pi$
$114$	0	0
$115$	−2009.31	−1.62929
$116$	−459.829	−0.368052
$117$	0	0
$118$	101.637	0.0792922
$119$	562.299	0.433159
$120$	0	0
$121$	1615.46	1.21372
$122$	−41.8830	−0.0310813
$123$	0	0
$124$	464.336	0.336279
$125$	−2878.73	−2.05985
$126$	0	0
$127$	−1701.92	−1.18914	−0.594571	−	0.804043i	$-0.702678\pi$
$127$	−1701.92	−1.18914	−0.594571	+	0.804043i	$0.702678\pi$
$128$	571.081	0.394351
$129$	0	0
$130$	453.734	0.306116
$131$	−609.473	−0.406488	−0.203244	−	0.979128i	$-0.565148\pi$
$131$	−609.473	−0.406488	−0.203244	+	0.979128i	$0.565148\pi$
$132$	0	0
$133$	1839.63	1.19937
$134$	130.377	0.0840510
$135$	0	0
$136$	−172.649	−0.108857
$137$	2947.05	1.83784	0.918918	−	0.394449i	$-0.129065\pi$
$137$	2947.05	1.83784	0.918918	+	0.394449i	$0.129065\pi$
$138$	0	0
$139$	−840.966	−0.513164	−0.256582	−	0.966522i	$-0.582596\pi$
$139$	−840.966	−0.513164	−0.256582	+	0.966522i	$0.582596\pi$
$140$	−2333.98	−1.40898
$141$	0	0
$142$	−36.4630	−0.0215486
$143$	4331.84	2.53319
$144$	0	0
$145$	−1153.98	−0.660919
$146$	−63.6259	−0.0360665
$147$	0	0
$148$	1172.38	0.651142
$149$	−2130.41	−1.17134	−0.585670	−	0.810549i	$-0.699169\pi$
$149$	−2130.41	−1.17134	−0.585670	+	0.810549i	$0.699169\pi$
$150$	0	0
$151$	2378.26	1.28172	0.640861	−	0.767657i	$-0.278577\pi$
$151$	2378.26	1.28172	0.640861	+	0.767657i	$0.278577\pi$
$152$	−564.842	−0.301413
$153$	0	0
$154$	230.384	0.120551
$155$	1165.29	0.603863
$156$	0	0
$157$	−1850.68	−0.940766	−0.470383	−	0.882462i	$-0.655884\pi$
$157$	−1850.68	−0.940766	−0.470383	+	0.882462i	$0.655884\pi$
$158$	33.5907	0.0169135
$159$	0	0
$160$	1076.79	0.532047
$161$	−1499.92	−0.734225
$162$	0	0
$163$	−968.412	−0.465349	−0.232675	−	0.972555i	$-0.574748\pi$
$163$	−968.412	−0.465349	−0.232675	+	0.972555i	$0.574748\pi$
$164$	−1459.36	−0.694860
$165$	0	0
$166$	−178.053	−0.0832504
$167$	1750.09	0.810937	0.405468	−	0.914109i	$-0.367108\pi$
$167$	1750.09	0.810937	0.405468	+	0.914109i	$0.367108\pi$
$168$	0	0
$169$	4171.59	1.89876
$170$	−215.526	−0.0972357
$171$	0	0
$172$	−118.162	−0.0523825
$173$	−2051.85	−0.901732	−0.450866	−	0.892592i	$-0.648885\pi$
$173$	−2051.85	−0.901732	−0.450866	+	0.892592i	$0.648885\pi$
$174$	0	0
$175$	−4003.14	−1.72920
$176$	3367.72	1.44234
$177$	0	0
$178$	−298.944	−0.125881
$179$	−710.848	−0.296823	−0.148411	−	0.988926i	$-0.547416\pi$
$179$	−710.848	−0.296823	−0.148411	+	0.988926i	$0.547416\pi$
$180$	0	0
$181$	−2195.53	−0.901615	−0.450807	−	0.892621i	$-0.648864\pi$
$181$	−2195.53	−0.901615	−0.450807	+	0.892621i	$0.648864\pi$
$182$	338.706	0.137948
$183$	0	0
$184$	460.538	0.184518
$185$	2942.20	1.16927
$186$	0	0
$187$	−2057.64	−0.804650
$188$	1629.75	0.632242
$189$	0	0
$190$	−705.117	−0.269235
$191$	−4883.06	−1.84987	−0.924936	−	0.380123i	$-0.875882\pi$
$191$	−4883.06	−1.84987	−0.924936	+	0.380123i	$0.875882\pi$
$192$	0	0
$193$	1101.21	0.410710	0.205355	−	0.978688i	$-0.434165\pi$
$193$	1101.21	0.410710	0.205355	+	0.978688i	$0.434165\pi$
$194$	238.906	0.0884149
$195$	0	0
$196$	973.634	0.354823
$197$	−2146.89	−0.776445	−0.388222	−	0.921566i	$-0.626911\pi$
$197$	−2146.89	−0.776445	−0.388222	+	0.921566i	$0.626911\pi$
$198$	0	0
$199$	−1058.00	−0.376882	−0.188441	−	0.982085i	$-0.560343\pi$
$199$	−1058.00	−0.376882	−0.188441	+	0.982085i	$0.560343\pi$
$200$	1229.13	0.434564
$201$	0	0
$202$	−444.275	−0.154748
$203$	−861.434	−0.297836
$204$	0	0
$205$	−3662.41	−1.24777
$206$	366.006	0.123791
$207$	0	0
$208$	4951.16	1.65049
$209$	−6731.81	−2.22799
$210$	0	0
$211$	5103.67	1.66517	0.832586	−	0.553896i	$-0.186859\pi$
$211$	5103.67	1.66517	0.832586	+	0.553896i	$0.186859\pi$
$212$	−2261.00	−0.732480
$213$	0	0
$214$	−311.980	−0.0996566
$215$	−296.539	−0.0940643
$216$	0	0
$217$	869.877	0.272125
$218$	235.294	0.0731013
$219$	0	0
$220$	8540.83	2.61737
$221$	−3025.11	−0.920772
$222$	0	0
$223$	−2636.41	−0.791691	−0.395846	−	0.918317i	$-0.629548\pi$
$223$	−2636.41	−0.791691	−0.395846	+	0.918317i	$0.629548\pi$
$224$	803.807	0.239762
$225$	0	0
$226$	341.264	0.100445
$227$	−2580.54	−0.754523	−0.377261	−	0.926107i	$-0.623134\pi$
$227$	−2580.54	−0.754523	−0.377261	+	0.926107i	$0.623134\pi$
$228$	0	0
$229$	−1364.19	−0.393659	−0.196830	−	0.980438i	$-0.563065\pi$
$229$	−1364.19	−0.393659	−0.196830	+	0.980438i	$0.563065\pi$
$230$	574.909	0.164819
$231$	0	0
$232$	264.496	0.0748492
$233$	599.263	0.168494	0.0842468	−	0.996445i	$-0.473152\pi$
$233$	599.263	0.168494	0.0842468	+	0.996445i	$0.473152\pi$
$234$	0	0
$235$	4090.00	1.13533
$236$	2812.69	0.775809
$237$	0	0
$238$	−160.887	−0.0438183
$239$	−847.081	−0.229260	−0.114630	−	0.993408i	$-0.536568\pi$
$239$	−847.081	−0.229260	−0.114630	+	0.993408i	$0.536568\pi$
$240$	0	0
$241$	−7191.07	−1.92206	−0.961032	−	0.276437i	$-0.910846\pi$
$241$	−7191.07	−1.92206	−0.961032	+	0.276437i	$0.910846\pi$
$242$	−462.222	−0.122780
$243$	0	0
$244$	−1159.06	−0.304105
$245$	2443.43	0.637163
$246$	0	0
$247$	−9896.98	−2.54951
$248$	−267.088	−0.0683876
$249$	0	0
$250$	823.673	0.208375
$251$	−7344.97	−1.84705	−0.923527	−	0.383533i	$-0.874707\pi$
$251$	−7344.97	−1.84705	−0.923527	+	0.383533i	$0.874707\pi$
$252$	0	0
$253$	5488.71	1.36392
$254$	486.960	0.120294
$255$	0	0
$256$	3683.25	0.899230
$257$	−5427.99	−1.31747	−0.658733	−	0.752376i	$-0.728907\pi$
$257$	−5427.99	−1.31747	−0.658733	+	0.752376i	$0.728907\pi$
$258$	0	0
$259$	2196.31	0.526920
$260$	12556.6	2.99510
$261$	0	0
$262$	174.384	0.0411203
$263$	−589.088	−0.138117	−0.0690584	−	0.997613i	$-0.521999\pi$
$263$	−589.088	−0.138117	−0.0690584	+	0.997613i	$0.521999\pi$
$264$	0	0
$265$	−5674.18	−1.31533
$266$	−526.360	−0.121328
$267$	0	0
$268$	3608.03	0.822370
$269$	−936.001	−0.212152	−0.106076	−	0.994358i	$-0.533829\pi$
$269$	−936.001	−0.212152	−0.106076	+	0.994358i	$0.533829\pi$
$270$	0	0
$271$	4218.71	0.945639	0.472820	−	0.881159i	$-0.343236\pi$
$271$	4218.71	0.945639	0.472820	+	0.881159i	$0.343236\pi$
$272$	−2351.82	−0.524265
$273$	0	0
$274$	−843.220	−0.185915
$275$	14648.8	3.21221
$276$	0	0
$277$	−1968.60	−0.427009	−0.213504	−	0.976942i	$-0.568488\pi$
$277$	−1968.60	−0.427009	−0.213504	+	0.976942i	$0.568488\pi$
$278$	240.620	0.0519116
$279$	0	0
$280$	1342.52	0.286539
$281$	6529.79	1.38625	0.693123	−	0.720820i	$-0.256234\pi$
$281$	6529.79	1.38625	0.693123	+	0.720820i	$0.256234\pi$
$282$	0	0
$283$	3078.28	0.646589	0.323295	−	0.946298i	$-0.395210\pi$
$283$	3078.28	0.646589	0.323295	+	0.946298i	$0.395210\pi$
$284$	−1009.07	−0.210836
$285$	0	0
$286$	−1239.44	−0.256257
$287$	−2733.94	−0.562297
$288$	0	0
$289$	−3476.06	−0.707523
$290$	330.182	0.0668585
$291$	0	0
$292$	−1760.77	−0.352881
$293$	−6491.20	−1.29427	−0.647133	−	0.762377i	$-0.724032\pi$
$293$	−6491.20	−1.29427	−0.647133	+	0.762377i	$0.724032\pi$
$294$	0	0
$295$	7058.73	1.39314
$296$	−674.359	−0.132420
$297$	0	0
$298$	609.559	0.118493
$299$	8069.40	1.56075
$300$	0	0
$301$	−221.363	−0.0423891
$302$	−680.476	−0.129659
$303$	0	0
$304$	−7694.25	−1.45163
$305$	−2908.78	−0.546087
$306$	0	0
$307$	3557.55	0.661369	0.330685	−	0.943741i	$-0.392720\pi$
$307$	3557.55	0.661369	0.330685	+	0.943741i	$0.392720\pi$
$308$	6375.61	1.17949
$309$	0	0
$310$	−333.418	−0.0610867
$311$	−5026.06	−0.916404	−0.458202	−	0.888848i	$-0.651506\pi$
$311$	−5026.06	−0.916404	−0.458202	+	0.888848i	$0.651506\pi$
$312$	0	0
$313$	−478.201	−0.0863564	−0.0431782	−	0.999067i	$-0.513748\pi$
$313$	−478.201	−0.0863564	−0.0431782	+	0.999067i	$0.513748\pi$
$314$	529.522	0.0951678
$315$	0	0
$316$	929.584	0.165485
$317$	−2179.60	−0.386178	−0.193089	−	0.981181i	$-0.561851\pi$
$317$	−2179.60	−0.386178	−0.193089	+	0.981181i	$0.561851\pi$
$318$	0	0
$319$	3152.28	0.553271
$320$	9554.74	1.66914
$321$	0	0
$322$	429.162	0.0742741
$323$	4701.11	0.809835
$324$	0	0
$325$	21536.5	3.67578
$326$	277.085	0.0470747
$327$	0	0
$328$	839.432	0.141311
$329$	3053.13	0.511625
$330$	0	0
$331$	−1471.99	−0.244435	−0.122217	−	0.992503i	$-0.539000\pi$
$331$	−1471.99	−0.244435	−0.122217	+	0.992503i	$0.539000\pi$
$332$	−4927.40	−0.814537
$333$	0	0
$334$	−500.743	−0.0820343
$335$	9054.68	1.47675
$336$	0	0
$337$	−5385.97	−0.870601	−0.435301	−	0.900285i	$-0.643358\pi$
$337$	−5385.97	−0.870601	−0.435301	+	0.900285i	$0.643358\pi$
$338$	−1193.59	−0.192079
$339$	0	0
$340$	−5964.42	−0.951371
$341$	−3183.17	−0.505508
$342$	0	0
$343$	6911.93	1.08807
$344$	67.9675	0.0106528
$345$	0	0
$346$	587.084	0.0912191
$347$	10340.9	1.59979	0.799896	−	0.600139i	$-0.204888\pi$
$347$	10340.9	1.59979	0.799896	+	0.600139i	$0.204888\pi$
$348$	0	0
$349$	542.711	0.0832397	0.0416198	−	0.999134i	$-0.486748\pi$
$349$	542.711	0.0832397	0.0416198	+	0.999134i	$0.486748\pi$
$350$	1145.39	0.174925
$351$	0	0
$352$	−2941.40	−0.445390
$353$	−2542.45	−0.383346	−0.191673	−	0.981459i	$-0.561391\pi$
$353$	−2542.45	−0.383346	−0.191673	+	0.981459i	$0.561391\pi$
$354$	0	0
$355$	−2532.36	−0.378602
$356$	−8272.92	−1.23164
$357$	0	0
$358$	203.390	0.0300266
$359$	−2307.58	−0.339246	−0.169623	−	0.985509i	$-0.554255\pi$
$359$	−2307.58	−0.339246	−0.169623	+	0.985509i	$0.554255\pi$
$360$	0	0
$361$	8521.22	1.24234
$362$	628.192	0.0912072
$363$	0	0
$364$	9373.30	1.34971
$365$	−4418.82	−0.633676
$366$	0	0
$367$	11695.0	1.66342	0.831709	−	0.555212i	$-0.187363\pi$
$367$	11695.0	1.66342	0.831709	+	0.555212i	$0.187363\pi$
$368$	6273.42	0.888654
$369$	0	0
$370$	−841.832	−0.118283
$371$	−4235.70	−0.592740
$372$	0	0
$373$	−7607.91	−1.05609	−0.528047	−	0.849215i	$-0.677075\pi$
$373$	−7607.91	−1.05609	−0.528047	+	0.849215i	$0.677075\pi$
$374$	588.739	0.0813984
$375$	0	0
$376$	−937.439	−0.128576
$377$	4634.42	0.633116
$378$	0	0
$379$	−8796.91	−1.19226	−0.596131	−	0.802887i	$-0.703296\pi$
$379$	−8796.91	−1.19226	−0.596131	+	0.802887i	$0.703296\pi$
$380$	−19513.3	−2.63424
$381$	0	0
$382$	1397.16	0.187133
$383$	9599.82	1.28075	0.640376	−	0.768062i	$-0.278779\pi$
$383$	9599.82	1.28075	0.640376	+	0.768062i	$0.278779\pi$
$384$	0	0
$385$	16000.2	2.11804
$386$	−315.083	−0.0415474
$387$	0	0
$388$	6611.46	0.865067
$389$	6559.78	0.854997	0.427499	−	0.904016i	$-0.359395\pi$
$389$	6559.78	0.854997	0.427499	+	0.904016i	$0.359395\pi$
$390$	0	0
$391$	−3833.00	−0.495762
$392$	−560.039	−0.0721588
$393$	0	0
$394$	614.276	0.0785451
$395$	2332.88	0.297164
$396$	0	0
$397$	13432.5	1.69813	0.849066	−	0.528287i	$-0.177166\pi$
$397$	13432.5	1.69813	0.849066	+	0.528287i	$0.177166\pi$
$398$	302.718	0.0381253
$399$	0	0
$400$	16743.2	2.09290
$401$	−9473.43	−1.17975	−0.589876	−	0.807494i	$-0.700823\pi$
$401$	−9473.43	−1.17975	−0.589876	+	0.807494i	$0.700823\pi$
$402$	0	0
$403$	−4679.84	−0.578460
$404$	−12294.8	−1.51408
$405$	0	0
$406$	246.476	0.0301291
$407$	−8037.04	−0.978824
$408$	0	0
$409$	2910.41	0.351859	0.175930	−	0.984403i	$-0.443707\pi$
$409$	2910.41	0.351859	0.175930	+	0.984403i	$0.443707\pi$
$410$	1047.90	0.126225
$411$	0	0
$412$	10128.8	1.21119
$413$	5269.24	0.627802
$414$	0	0
$415$	−12365.8	−1.46268
$416$	−4324.39	−0.509665
$417$	0	0
$418$	1926.13	0.225383
$419$	−9867.43	−1.15049	−0.575245	−	0.817981i	$-0.695093\pi$
$419$	−9867.43	−1.15049	−0.575245	+	0.817981i	$0.695093\pi$
$420$	0	0
$421$	6365.62	0.736915	0.368457	−	0.929645i	$-0.379886\pi$
$421$	6365.62	0.736915	0.368457	+	0.929645i	$0.379886\pi$
$422$	−1460.28	−0.168449
$423$	0	0
$424$	1300.54	0.148961
$425$	−10229.9	−1.16759
$426$	0	0
$427$	−2171.37	−0.246088
$428$	−8633.69	−0.975058
$429$	0	0
$430$	84.8469	0.00951553
$431$	359.313	0.0401566	0.0200783	−	0.999798i	$-0.493608\pi$
$431$	359.313	0.0401566	0.0200783	+	0.999798i	$0.493608\pi$
$432$	0	0
$433$	−12657.1	−1.40476	−0.702380	−	0.711803i	$-0.747879\pi$
$433$	−12657.1	−1.40476	−0.702380	+	0.711803i	$0.747879\pi$
$434$	−248.892	−0.0275281
$435$	0	0
$436$	6511.48	0.715237
$437$	−12540.1	−1.37271
$438$	0	0
$439$	−7447.02	−0.809629	−0.404814	−	0.914399i	$-0.632664\pi$
$439$	−7447.02	−0.809629	−0.404814	+	0.914399i	$0.632664\pi$
$440$	−4912.73	−0.532284
$441$	0	0
$442$	865.554	0.0931452
$443$	−11678.0	−1.25246	−0.626231	−	0.779638i	$-0.715403\pi$
$443$	−11678.0	−1.25246	−0.626231	+	0.779638i	$0.715403\pi$
$444$	0	0
$445$	−20761.7	−2.21168
$446$	754.339	0.0800874
$447$	0	0
$448$	7132.48	0.752183
$449$	8059.04	0.847059	0.423530	−	0.905882i	$-0.360791\pi$
$449$	8059.04	0.847059	0.423530	+	0.905882i	$0.360791\pi$
$450$	0	0
$451$	10004.4	1.04454
$452$	9444.07	0.982769
$453$	0	0
$454$	738.354	0.0763274
$455$	23523.2	2.42370
$456$	0	0
$457$	5873.13	0.601167	0.300583	−	0.953756i	$-0.402819\pi$
$457$	5873.13	0.601167	0.300583	+	0.953756i	$0.402819\pi$
$458$	390.326	0.0398226
$459$	0	0
$460$	15909.9	1.61262
$461$	7559.60	0.763743	0.381872	−	0.924215i	$-0.375280\pi$
$461$	7559.60	0.763743	0.381872	+	0.924215i	$0.375280\pi$
$462$	0	0
$463$	3621.75	0.363536	0.181768	−	0.983341i	$-0.441818\pi$
$463$	3621.75	0.363536	0.181768	+	0.983341i	$0.441818\pi$
$464$	3602.95	0.360480
$465$	0	0
$466$	−171.463	−0.0170448
$467$	−3757.19	−0.372296	−0.186148	−	0.982522i	$-0.559600\pi$
$467$	−3757.19	−0.372296	−0.186148	+	0.982522i	$0.559600\pi$
$468$	0	0
$469$	6759.20	0.665481
$470$	−1170.25	−0.114850
$471$	0	0
$472$	−1617.88	−0.157773
$473$	810.040	0.0787435
$474$	0	0
$475$	−33468.3	−3.23291
$476$	−4452.36	−0.428726
$477$	0	0
$478$	242.370	0.0231919
$479$	−13501.9	−1.28793	−0.643963	−	0.765056i	$-0.722711\pi$
$479$	−13501.9	−1.28793	−0.643963	+	0.765056i	$0.722711\pi$
$480$	0	0
$481$	−11815.9	−1.12008
$482$	2057.53	0.194436
$483$	0	0
$484$	−12791.5	−1.20130
$485$	16592.1	1.55342
$486$	0	0
$487$	−4509.25	−0.419577	−0.209788	−	0.977747i	$-0.567277\pi$
$487$	−4509.25	−0.419577	−0.209788	+	0.977747i	$0.567277\pi$
$488$	666.700	0.0618444
$489$	0	0
$490$	−699.122	−0.0644553
$491$	7420.57	0.682048	0.341024	−	0.940055i	$-0.389226\pi$
$491$	7420.57	0.682048	0.341024	+	0.940055i	$0.389226\pi$
$492$	0	0
$493$	−2201.37	−0.201105
$494$	2831.76	0.257909
$495$	0	0
$496$	−3638.27	−0.329361
$497$	−1890.37	−0.170613
$498$	0	0
$499$	15944.4	1.43040	0.715199	−	0.698921i	$-0.246336\pi$
$499$	15944.4	1.43040	0.715199	+	0.698921i	$0.246336\pi$
$500$	22794.2	2.03878
$501$	0	0
$502$	2101.57	0.186848
$503$	−16600.5	−1.47153	−0.735763	−	0.677239i	$-0.763176\pi$
$503$	−16600.5	−1.47153	−0.735763	+	0.677239i	$0.763176\pi$
$504$	0	0
$505$	−30854.9	−2.71886
$506$	−1570.45	−0.137974
$507$	0	0
$508$	13476.0	1.17697
$509$	−22099.1	−1.92441	−0.962205	−	0.272325i	$-0.912207\pi$
$509$	−22099.1	−1.92441	−0.962205	+	0.272325i	$0.912207\pi$
$510$	0	0
$511$	−3298.59	−0.285560
$512$	−5622.51	−0.485317
$513$	0	0
$514$	1553.08	0.133275
$515$	25419.2	2.17496
$516$	0	0
$517$	−11172.4	−0.950412
$518$	−628.416	−0.0533031
$519$	0	0
$520$	−7222.60	−0.609100
$521$	17315.3	1.45604	0.728021	−	0.685554i	$-0.240440\pi$
$521$	17315.3	1.45604	0.728021	+	0.685554i	$0.240440\pi$
$522$	0	0
$523$	−11120.5	−0.929764	−0.464882	−	0.885373i	$-0.653903\pi$
$523$	−11120.5	−0.929764	−0.464882	+	0.885373i	$0.653903\pi$
$524$	4825.89	0.402328
$525$	0	0
$526$	168.552	0.0139719
$527$	2222.94	0.183744
$528$	0	0
$529$	−1942.57	−0.159659
$530$	1623.52	0.133059
$531$	0	0
$532$	−14566.4	−1.18709
$533$	14708.3	1.19528
$534$	0	0
$535$	−21667.1	−1.75093
$536$	−2075.35	−0.167242
$537$	0	0
$538$	267.812	0.0214613
$539$	−6674.57	−0.533384
$540$	0	0
$541$	−4291.69	−0.341061	−0.170531	−	0.985352i	$-0.554548\pi$
$541$	−4291.69	−0.341061	−0.170531	+	0.985352i	$0.554548\pi$
$542$	−1207.07	−0.0956608
$543$	0	0
$544$	2054.10	0.161892
$545$	16341.2	1.28436
$546$	0	0
$547$	−18965.2	−1.48244	−0.741219	−	0.671263i	$-0.765752\pi$
$547$	−18965.2	−1.48244	−0.741219	+	0.671263i	$0.765752\pi$
$548$	−23335.1	−1.81903
$549$	0	0
$550$	−4191.38	−0.324947
$551$	−7202.03	−0.556836
$552$	0	0
$553$	1741.46	0.133914
$554$	563.261	0.0431962
$555$	0	0
$556$	6658.88	0.507913
$557$	10083.3	0.767044	0.383522	−	0.923532i	$-0.374711\pi$
$557$	10083.3	0.767044	0.383522	+	0.923532i	$0.374711\pi$
$558$	0	0
$559$	1190.91	0.0901073
$560$	18287.7	1.38000
$561$	0	0
$562$	−1868.33	−0.140232
$563$	−21595.9	−1.61662	−0.808311	−	0.588756i	$-0.799618\pi$
$563$	−21595.9	−1.61662	−0.808311	+	0.588756i	$0.799618\pi$
$564$	0	0
$565$	23700.8	1.76478
$566$	−880.768	−0.0654089
$567$	0	0
$568$	580.423	0.0428768
$569$	15711.4	1.15757	0.578783	−	0.815481i	$-0.303528\pi$
$569$	15711.4	1.15757	0.578783	+	0.815481i	$0.303528\pi$
$570$	0	0
$571$	−21001.2	−1.53918	−0.769591	−	0.638537i	$-0.779540\pi$
$571$	−21001.2	−1.53918	−0.769591	+	0.638537i	$0.779540\pi$
$572$	−34300.1	−2.50727
$573$	0	0
$574$	782.243	0.0568819
$575$	27288.0	1.97911
$576$	0	0
$577$	6914.65	0.498892	0.249446	−	0.968389i	$-0.419751\pi$
$577$	6914.65	0.498892	0.249446	+	0.968389i	$0.419751\pi$
$578$	994.583	0.0715730
$579$	0	0
$580$	9137.40	0.654155
$581$	−9230.88	−0.659142
$582$	0	0
$583$	15499.8	1.10109
$584$	1012.80	0.0717640
$585$	0	0
$586$	1857.28	0.130928
$587$	24755.6	1.74067	0.870335	−	0.492461i	$-0.163903\pi$
$587$	24755.6	1.74067	0.870335	+	0.492461i	$0.163903\pi$
$588$	0	0
$589$	7272.62	0.508766
$590$	−2019.67	−0.140929
$591$	0	0
$592$	−9186.09	−0.637747
$593$	−14262.7	−0.987687	−0.493844	−	0.869551i	$-0.664409\pi$
$593$	−14262.7	−0.987687	−0.493844	+	0.869551i	$0.664409\pi$
$594$	0	0
$595$	−11173.6	−0.769872
$596$	16868.8	1.15935
$597$	0	0
$598$	−2308.84	−0.157886
$599$	−15252.8	−1.04042	−0.520209	−	0.854039i	$-0.674146\pi$
$599$	−15252.8	−1.04042	−0.520209	+	0.854039i	$0.674146\pi$
$600$	0	0
$601$	−7861.99	−0.533606	−0.266803	−	0.963751i	$-0.585967\pi$
$601$	−7861.99	−0.533606	−0.266803	+	0.963751i	$0.585967\pi$
$602$	63.3370	0.00428808
$603$	0	0
$604$	−18831.4	−1.26861
$605$	−32101.4	−2.15720
$606$	0	0
$607$	13227.1	0.884469	0.442234	−	0.896900i	$-0.354186\pi$
$607$	13227.1	0.884469	0.442234	+	0.896900i	$0.354186\pi$
$608$	6720.24	0.448259
$609$	0	0
$610$	832.271	0.0552421
$611$	−16425.5	−1.08757
$612$	0	0
$613$	−20704.9	−1.36422	−0.682108	−	0.731251i	$-0.738937\pi$
$613$	−20704.9	−1.36422	−0.682108	+	0.731251i	$0.738937\pi$
$614$	−1017.90	−0.0669041
$615$	0	0
$616$	−3667.28	−0.239869
$617$	13026.6	0.849971	0.424985	−	0.905200i	$-0.360279\pi$
$617$	13026.6	0.849971	0.424985	+	0.905200i	$0.360279\pi$
$618$	0	0
$619$	4744.99	0.308105	0.154053	−	0.988063i	$-0.450767\pi$
$619$	4744.99	0.308105	0.154053	+	0.988063i	$0.450767\pi$
$620$	−9226.96	−0.597683
$621$	0	0
$622$	1438.07	0.0927034
$623$	−15498.3	−0.996672
$624$	0	0
$625$	23470.6	1.50212
$626$	136.825	0.00873580
$627$	0	0
$628$	14653.9	0.931139
$629$	5612.61	0.355786
$630$	0	0
$631$	9445.90	0.595936	0.297968	−	0.954576i	$-0.403691\pi$
$631$	9445.90	0.595936	0.297968	+	0.954576i	$0.403691\pi$
$632$	−534.701	−0.0336539
$633$	0	0
$634$	623.634	0.0390657
$635$	33819.4	2.11352
$636$	0	0
$637$	−9812.83	−0.610359
$638$	−901.940	−0.0559689
$639$	0	0
$640$	−11348.1	−0.700897
$641$	−14692.1	−0.905310	−0.452655	−	0.891686i	$-0.649523\pi$
$641$	−14692.1	−0.905310	−0.452655	+	0.891686i	$0.649523\pi$
$642$	0	0
$643$	19527.3	1.19764	0.598821	−	0.800883i	$-0.295636\pi$
$643$	19527.3	1.19764	0.598821	+	0.800883i	$0.295636\pi$
$644$	11876.6	0.726711
$645$	0	0
$646$	−1345.10	−0.0819229
$647$	19872.0	1.20750	0.603748	−	0.797176i	$-0.293673\pi$
$647$	19872.0	1.20750	0.603748	+	0.797176i	$0.293673\pi$
$648$	0	0
$649$	−19281.9	−1.16623
$650$	−6162.09	−0.371841
$651$	0	0
$652$	7668.02	0.460587
$653$	−6068.53	−0.363675	−0.181838	−	0.983329i	$-0.558205\pi$
$653$	−6068.53	−0.363675	−0.181838	+	0.983329i	$0.558205\pi$
$654$	0	0
$655$	12111.0	0.722469
$656$	11434.7	0.680565
$657$	0	0
$658$	−873.573	−0.0517559
$659$	7023.26	0.415155	0.207578	−	0.978219i	$-0.433442\pi$
$659$	7023.26	0.415155	0.207578	+	0.978219i	$0.433442\pi$
$660$	0	0
$661$	−1565.22	−0.0921028	−0.0460514	−	0.998939i	$-0.514664\pi$
$661$	−1565.22	−0.0921028	−0.0460514	+	0.998939i	$0.514664\pi$
$662$	421.171	0.0247270
$663$	0	0
$664$	2834.27	0.165649
$665$	−36555.8	−2.13169
$666$	0	0
$667$	5872.09	0.340882
$668$	−13857.5	−0.802638
$669$	0	0
$670$	−2590.76	−0.149388
$671$	7945.76	0.457143
$672$	0	0
$673$	−25203.3	−1.44356	−0.721779	−	0.692124i	$-0.756675\pi$
$673$	−25203.3	−1.44356	−0.721779	+	0.692124i	$0.756675\pi$
$674$	1541.05	0.0880700
$675$	0	0
$676$	−33031.2	−1.87933
$677$	−13481.9	−0.765363	−0.382682	−	0.923880i	$-0.624999\pi$
$677$	−13481.9	−0.765363	−0.382682	+	0.923880i	$0.624999\pi$
$678$	0	0
$679$	12385.8	0.700032
$680$	3430.77	0.193476
$681$	0	0
$682$	910.780	0.0511372
$683$	28771.7	1.61189	0.805944	−	0.591992i	$-0.201658\pi$
$683$	28771.7	1.61189	0.805944	+	0.591992i	$0.201658\pi$
$684$	0	0
$685$	−58561.7	−3.26647
$686$	−1977.66	−0.110069
$687$	0	0
$688$	925.851	0.0513049
$689$	22787.6	1.26000
$690$	0	0
$691$	−18579.9	−1.02288	−0.511442	−	0.859318i	$-0.670888\pi$
$691$	−18579.9	−1.02288	−0.511442	+	0.859318i	$0.670888\pi$
$692$	16246.9	0.892504
$693$	0	0
$694$	−2958.77	−0.161835
$695$	16711.1	0.912069
$696$	0	0
$697$	−6986.49	−0.379673
$698$	−155.282	−0.00842052
$699$	0	0
$700$	31697.4	1.71150
$701$	25470.7	1.37234	0.686172	−	0.727439i	$-0.259290\pi$
$701$	25470.7	1.37234	0.686172	+	0.727439i	$0.259290\pi$
$702$	0	0
$703$	18362.3	0.985131
$704$	−26100.1	−1.39728
$705$	0	0
$706$	727.456	0.0387793
$707$	−23032.8	−1.22523
$708$	0	0
$709$	29079.5	1.54035	0.770173	−	0.637835i	$-0.220170\pi$
$709$	29079.5	1.54035	0.770173	+	0.637835i	$0.220170\pi$
$710$	724.568	0.0382993
$711$	0	0
$712$	4758.63	0.250473
$713$	−5929.65	−0.311454
$714$	0	0
$715$	−86079.3	−4.50235
$716$	5628.59	0.293785
$717$	0	0
$718$	660.252	0.0343181
$719$	−26982.8	−1.39956	−0.699782	−	0.714356i	$-0.746720\pi$
$719$	−26982.8	−1.39956	−0.699782	+	0.714356i	$0.746720\pi$
$720$	0	0
$721$	18975.1	0.980123
$722$	−2438.12	−0.125675
$723$	0	0
$724$	17384.5	0.892388
$725$	15672.1	0.802822
$726$	0	0
$727$	−14435.2	−0.736412	−0.368206	−	0.929744i	$-0.620028\pi$
$727$	−14435.2	−0.736412	−0.368206	+	0.929744i	$0.620028\pi$
$728$	−5391.57	−0.274485
$729$	0	0
$730$	1264.33	0.0641026
$731$	−565.686	−0.0286219
$732$	0	0
$733$	−21774.7	−1.09723	−0.548614	−	0.836076i	$-0.684844\pi$
$733$	−21774.7	−1.09723	−0.548614	+	0.836076i	$0.684844\pi$
$734$	−3346.21	−0.168271
$735$	0	0
$736$	−5479.27	−0.274414
$737$	−24734.2	−1.23622
$738$	0	0
$739$	10454.1	0.520381	0.260190	−	0.965557i	$-0.416215\pi$
$739$	10454.1	0.520381	0.260190	+	0.965557i	$0.416215\pi$
$740$	−23296.7	−1.15730
$741$	0	0
$742$	1211.93	0.0599615
$743$	−34626.3	−1.70971	−0.854857	−	0.518864i	$-0.826355\pi$
$743$	−34626.3	−1.70971	−0.854857	+	0.518864i	$0.826355\pi$
$744$	0	0
$745$	42334.0	2.08187
$746$	2176.80	0.106834
$747$	0	0
$748$	16292.7	0.796416
$749$	−16174.2	−0.789040
$750$	0	0
$751$	−15080.3	−0.732741	−0.366370	−	0.930469i	$-0.619400\pi$
$751$	−15080.3	−0.732741	−0.366370	+	0.930469i	$0.619400\pi$
$752$	−12769.8	−0.619235
$753$	0	0
$754$	−1326.02	−0.0640459
$755$	−47259.1	−2.27806
$756$	0	0
$757$	−27825.5	−1.33598	−0.667988	−	0.744172i	$-0.732844\pi$
$757$	−27825.5	−1.33598	−0.667988	+	0.744172i	$0.732844\pi$
$758$	2517.00	0.120609
$759$	0	0
$760$	11224.1	0.535714
$761$	2545.70	0.121264	0.0606319	−	0.998160i	$-0.480688\pi$
$761$	2545.70	0.121264	0.0606319	+	0.998160i	$0.480688\pi$
$762$	0	0
$763$	12198.5	0.578786
$764$	38664.7	1.83094
$765$	0	0
$766$	−2746.73	−0.129561
$767$	−28347.9	−1.33453
$768$	0	0
$769$	40890.4	1.91749	0.958743	−	0.284274i	$-0.0917526\pi$
$769$	40890.4	1.91749	0.958743	+	0.284274i	$0.0917526\pi$
$770$	−4578.03	−0.214261
$771$	0	0
$772$	−8719.55	−0.406507
$773$	−10738.2	−0.499647	−0.249824	−	0.968291i	$-0.580373\pi$
$773$	−10738.2	−0.499647	−0.249824	+	0.968291i	$0.580373\pi$
$774$	0	0
$775$	−15825.7	−0.733516
$776$	−3802.94	−0.175925
$777$	0	0
$778$	−1876.91	−0.0864914
$779$	−22857.1	−1.05127
$780$	0	0
$781$	6917.51	0.316937
$782$	1096.71	0.0501513
$783$	0	0
$784$	−7628.83	−0.347523
$785$	36775.4	1.67206
$786$	0	0
$787$	−12825.1	−0.580896	−0.290448	−	0.956891i	$-0.593804\pi$
$787$	−12825.1	−0.580896	−0.290448	+	0.956891i	$0.593804\pi$
$788$	16999.4	0.768499
$789$	0	0
$790$	−667.491	−0.0300611
$791$	17692.3	0.795280
$792$	0	0
$793$	11681.7	0.523114
$794$	−3843.35	−0.171783
$795$	0	0
$796$	8377.36	0.373025
$797$	13258.6	0.589263	0.294632	−	0.955611i	$-0.404803\pi$
$797$	13258.6	0.589263	0.294632	+	0.955611i	$0.404803\pi$
$798$	0	0
$799$	7802.19	0.345459
$800$	−14623.7	−0.646281
$801$	0	0
$802$	2710.57	0.119344
$803$	12070.7	0.530466
$804$	0	0
$805$	29805.4	1.30497
$806$	1339.01	0.0585170
$807$	0	0
$808$	7072.02	0.307912
$809$	11138.9	0.484083	0.242041	−	0.970266i	$-0.422183\pi$
$809$	11138.9	0.484083	0.242041	+	0.970266i	$0.422183\pi$
$810$	0	0
$811$	6032.83	0.261210	0.130605	−	0.991434i	$-0.458308\pi$
$811$	6032.83	0.261210	0.130605	+	0.991434i	$0.458308\pi$
$812$	6820.95	0.294789
$813$	0	0
$814$	2299.59	0.0990178
$815$	19243.6	0.827085
$816$	0	0
$817$	−1850.71	−0.0792509
$818$	−832.736	−0.0355940
$819$	0	0
$820$	28999.4	1.23500
$821$	26084.1	1.10882	0.554411	−	0.832243i	$-0.312944\pi$
$821$	26084.1	1.10882	0.554411	+	0.832243i	$0.312944\pi$
$822$	0	0
$823$	33968.7	1.43873	0.719365	−	0.694633i	$-0.244433\pi$
$823$	33968.7	1.43873	0.719365	+	0.694633i	$0.244433\pi$
$824$	−5826.14	−0.246314
$825$	0	0
$826$	−1507.65	−0.0635084
$827$	43501.0	1.82911	0.914557	−	0.404458i	$-0.132540\pi$
$827$	43501.0	1.82911	0.914557	+	0.404458i	$0.132540\pi$
$828$	0	0
$829$	−10989.9	−0.460430	−0.230215	−	0.973140i	$-0.573943\pi$
$829$	−10989.9	−0.460430	−0.230215	+	0.973140i	$0.573943\pi$
$830$	3538.14	0.147965
$831$	0	0
$832$	−38371.9	−1.59893
$833$	4661.14	0.193876
$834$	0	0
$835$	−34776.7	−1.44131
$836$	53303.4	2.20519
$837$	0	0
$838$	2823.30	0.116384
$839$	−5324.22	−0.219085	−0.109543	−	0.993982i	$-0.534939\pi$
$839$	−5324.22	−0.219085	−0.109543	+	0.993982i	$0.534939\pi$
$840$	0	0
$841$	−21016.5	−0.861722
$842$	−1821.35	−0.0745462
$843$	0	0
$844$	−40411.6	−1.64813
$845$	−82894.9	−3.37476
$846$	0	0
$847$	−23963.2	−0.972121
$848$	17715.9	0.717411
$849$	0	0
$850$	2927.02	0.118113
$851$	−14971.5	−0.603074
$852$	0	0
$853$	18452.7	0.740689	0.370345	−	0.928894i	$-0.379240\pi$
$853$	18452.7	0.740689	0.370345	+	0.928894i	$0.379240\pi$
$854$	621.279	0.0248943
$855$	0	0
$856$	4966.14	0.198293
$857$	10741.0	0.428128	0.214064	−	0.976820i	$-0.431330\pi$
$857$	10741.0	0.428128	0.214064	+	0.976820i	$0.431330\pi$
$858$	0	0
$859$	−10081.0	−0.400418	−0.200209	−	0.979753i	$-0.564162\pi$
$859$	−10081.0	−0.400418	−0.200209	+	0.979753i	$0.564162\pi$
$860$	2348.04	0.0931017
$861$	0	0
$862$	−102.808	−0.00406224
$863$	−42095.0	−1.66041	−0.830203	−	0.557461i	$-0.811776\pi$
$863$	−42095.0	−1.66041	−0.830203	+	0.557461i	$0.811776\pi$
$864$	0	0
$865$	40773.0	1.60269
$866$	3621.49	0.142105
$867$	0	0
$868$	−6887.80	−0.269340
$869$	−6372.59	−0.248763
$870$	0	0
$871$	−36363.7	−1.41462
$872$	−3745.43	−0.145455
$873$	0	0
$874$	3588.01	0.138863
$875$	42702.2	1.64982
$876$	0	0
$877$	−12114.2	−0.466439	−0.233219	−	0.972424i	$-0.574926\pi$
$877$	−12114.2	−0.466439	−0.233219	+	0.972424i	$0.574926\pi$
$878$	2130.77	0.0819020
$879$	0	0
$880$	−66921.0	−2.56353
$881$	17436.1	0.666784	0.333392	−	0.942788i	$-0.391807\pi$
$881$	17436.1	0.666784	0.333392	+	0.942788i	$0.391807\pi$
$882$	0	0
$883$	−38565.8	−1.46981	−0.734906	−	0.678169i	$-0.762774\pi$
$883$	−38565.8	−1.46981	−0.734906	+	0.678169i	$0.762774\pi$
$884$	23953.2	0.911350
$885$	0	0
$886$	3341.36	0.126699
$887$	−4279.09	−0.161982	−0.0809908	−	0.996715i	$-0.525808\pi$
$887$	−4279.09	−0.161982	−0.0809908	+	0.996715i	$0.525808\pi$
$888$	0	0
$889$	25245.7	0.952435
$890$	5940.40	0.223733
$891$	0	0
$892$	20875.5	0.783590
$893$	25525.8	0.956536
$894$	0	0
$895$	14125.5	0.527556
$896$	−8471.23	−0.315852
$897$	0	0
$898$	−2305.88	−0.0856884
$899$	−3405.52	−0.126341
$900$	0	0
$901$	−10824.2	−0.400229
$902$	−2862.49	−0.105666
$903$	0	0
$904$	−5432.28	−0.199862
$905$	43628.0	1.60248
$906$	0	0
$907$	−19205.2	−0.703084	−0.351542	−	0.936172i	$-0.614343\pi$
$907$	−19205.2	−0.703084	−0.351542	+	0.936172i	$0.614343\pi$
$908$	20433.1	0.746801
$909$	0	0
$910$	−6730.54	−0.245181
$911$	−21719.8	−0.789912	−0.394956	−	0.918700i	$-0.629240\pi$
$911$	−21719.8	−0.789912	−0.394956	+	0.918700i	$0.629240\pi$
$912$	0	0
$913$	33778.9	1.22445
$914$	−1680.44	−0.0608140
$915$	0	0
$916$	10801.8	0.389631
$917$	9040.71	0.325573
$918$	0	0
$919$	40905.4	1.46828	0.734138	−	0.679000i	$-0.237586\pi$
$919$	40905.4	1.46828	0.734138	+	0.679000i	$0.237586\pi$
$920$	−9151.48	−0.327952
$921$	0	0
$922$	−2162.98	−0.0772602
$923$	10170.0	0.362675
$924$	0	0
$925$	−39957.5	−1.42032
$926$	−1036.27	−0.0367752
$927$	0	0
$928$	−3146.86	−0.111315
$929$	34352.8	1.21322	0.606609	−	0.795001i	$-0.292529\pi$
$929$	34352.8	1.21322	0.606609	+	0.795001i	$0.292529\pi$
$930$	0	0
$931$	15249.5	0.536821
$932$	−4745.04	−0.166769
$933$	0	0
$934$	1075.02	0.0376614
$935$	40888.0	1.43014
$936$	0	0
$937$	33701.3	1.17500	0.587499	−	0.809225i	$-0.300113\pi$
$937$	33701.3	1.17500	0.587499	+	0.809225i	$0.300113\pi$
$938$	−1933.96	−0.0673200
$939$	0	0
$940$	−32385.2	−1.12371
$941$	−9211.42	−0.319111	−0.159556	−	0.987189i	$-0.551006\pi$
$941$	−9211.42	−0.319111	−0.159556	+	0.987189i	$0.551006\pi$
$942$	0	0
$943$	18636.3	0.643565
$944$	−22038.6	−0.759848
$945$	0	0
$946$	−231.771	−0.00796569
$947$	6883.72	0.236210	0.118105	−	0.993001i	$-0.462318\pi$
$947$	6883.72	0.236210	0.118105	+	0.993001i	$0.462318\pi$
$948$	0	0
$949$	17746.1	0.607019
$950$	9576.08	0.327041
$951$	0	0
$952$	2561.02	0.0871882
$953$	3172.75	0.107844	0.0539221	−	0.998545i	$-0.482828\pi$
$953$	3172.75	0.107844	0.0539221	+	0.998545i	$0.482828\pi$
$954$	0	0
$955$	97032.7	3.28786
$956$	6707.30	0.226914
$957$	0	0
$958$	3863.21	0.130287
$959$	−43715.5	−1.47200
$960$	0	0
$961$	−26352.1	−0.884566
$962$	3380.81	0.113307
$963$	0	0
$964$	56939.8	1.90239
$965$	−21882.5	−0.729973
$966$	0	0
$967$	15198.6	0.505434	0.252717	−	0.967540i	$-0.418676\pi$
$967$	15198.6	0.505434	0.252717	+	0.967540i	$0.418676\pi$
$968$	7357.71	0.244304
$969$	0	0
$970$	−4747.38	−0.157144
$971$	−24507.1	−0.809958	−0.404979	−	0.914326i	$-0.632721\pi$
$971$	−24507.1	−0.809958	−0.404979	+	0.914326i	$0.632721\pi$
$972$	0	0
$973$	12474.6	0.411015
$974$	1290.20	0.0424443
$975$	0	0
$976$	9081.76	0.297848
$977$	43402.1	1.42124	0.710622	−	0.703574i	$-0.248414\pi$
$977$	43402.1	1.42124	0.710622	+	0.703574i	$0.248414\pi$
$978$	0	0
$979$	56713.5	1.85145
$980$	−19347.4	−0.630642
$981$	0	0
$982$	−2123.20	−0.0689959
$983$	−28088.6	−0.911382	−0.455691	−	0.890138i	$-0.650608\pi$
$983$	−28088.6	−0.911382	−0.455691	+	0.890138i	$0.650608\pi$
$984$	0	0
$985$	42661.5	1.38001
$986$	629.863	0.0203437
$987$	0	0
$988$	78365.6	2.52342
$989$	1508.95	0.0485156
$990$	0	0
$991$	−40776.3	−1.30706	−0.653532	−	0.756899i	$-0.726714\pi$
$991$	−40776.3	−1.30706	−0.653532	+	0.756899i	$0.726714\pi$
$992$	3177.70	0.101706
$993$	0	0
$994$	540.880	0.0172592
$995$	21023.8	0.669848
$996$	0	0
$997$	−4797.45	−0.152394	−0.0761970	−	0.997093i	$-0.524278\pi$
$997$	−4797.45	−0.152394	−0.0761970	+	0.997093i	$0.524278\pi$
$998$	−4562.06	−0.144699
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.4.a.a.1.7	✓	12
3.2	odd	2		729.4.a.b.1.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.4.a.a.1.7	✓	12	1.1	even	1	trivial
729.4.a.b.1.6	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)

\(f(q)\)	\(=\)	\(q-0.286123 q^{2} -7.91813 q^{4} -19.8713 q^{5} -14.8337 q^{7} +4.55455 q^{8} +5.68565 q^{10} +54.2813 q^{11} +79.8034 q^{13} +4.24426 q^{14} +62.0419 q^{16} -37.9070 q^{17} -124.017 q^{19} +157.344 q^{20} -15.5312 q^{22} +101.116 q^{23} +269.869 q^{25} -22.8336 q^{26} +117.455 q^{28} +58.0729 q^{29} -58.6421 q^{31} -54.1880 q^{32} +10.8461 q^{34} +294.764 q^{35} -148.063 q^{37} +35.4842 q^{38} -90.5049 q^{40} +184.306 q^{41} +14.9230 q^{43} -429.807 q^{44} -28.9316 q^{46} -205.825 q^{47} -122.963 q^{49} -77.2158 q^{50} -631.894 q^{52} +285.547 q^{53} -1078.64 q^{55} -67.5607 q^{56} -16.6160 q^{58} -355.222 q^{59} +146.381 q^{61} +16.7789 q^{62} -480.831 q^{64} -1585.80 q^{65} -455.666 q^{67} +300.152 q^{68} -84.3389 q^{70} +127.438 q^{71} +222.372 q^{73} +42.3642 q^{74} +981.983 q^{76} -805.191 q^{77} -117.399 q^{79} -1232.85 q^{80} -52.7343 q^{82} +622.293 q^{83} +753.261 q^{85} -4.26982 q^{86} +247.227 q^{88} +1044.81 q^{89} -1183.78 q^{91} -800.649 q^{92} +58.8912 q^{94} +2464.38 q^{95} -834.977 q^{97} +35.1825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 36 q^{4} - 12 q^{5} - 42 q^{7} - 21 q^{8} - 60 q^{10} - 42 q^{11} - 78 q^{13} + 312 q^{14} + 48 q^{16} + 18 q^{17} - 228 q^{19} + 69 q^{20} - 309 q^{22} + 114 q^{23} - 18 q^{25} - 30 q^{26}+ \cdots - 9567 q^{98}+O(q^{100}) \) 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

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Database

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