LMFDB - Embedded newform 81.5.b.a.80.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,5,Mod(80,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.80"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	81.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$8.37296700979$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.39400128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 $ x^6 - x^5 + 11x^4 + 14x^3 + 98x^2 + 20x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			80.1
Root			$1.89154 + 3.27625i$ of defining polynomial
Character	$\chi$	$=$	81.80
Dual form			81.5.b.a.80.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-6.55250i q^{2} -26.9353 q^{4} -11.7830i q^{5} -53.2728 q^{7} +71.6534i q^{8} -77.2081 q^{10} +124.971i q^{11} -74.6104 q^{13} +349.070i q^{14} +38.5445 q^{16} -7.70989i q^{17} -54.1307 q^{19} +317.378i q^{20} +818.874 q^{22} -399.954i q^{23} +486.161 q^{25} +488.885i q^{26} +1434.92 q^{28} -540.653i q^{29} -1532.17 q^{31} +893.891i q^{32} -50.5190 q^{34} +627.714i q^{35} -1719.10 q^{37} +354.691i q^{38} +844.292 q^{40} -1259.63i q^{41} -2607.79 q^{43} -3366.13i q^{44} -2620.70 q^{46} -800.034i q^{47} +436.996 q^{49} -3185.57i q^{50} +2009.65 q^{52} -4229.81i q^{53} +1472.54 q^{55} -3817.18i q^{56} -3542.63 q^{58} +3333.19i q^{59} -15.0169 q^{61} +10039.5i q^{62} +6473.94 q^{64} +879.134i q^{65} +5182.39 q^{67} +207.668i q^{68} +4113.10 q^{70} -1924.16i q^{71} +949.554 q^{73} +11264.4i q^{74} +1458.02 q^{76} -6657.57i q^{77} -237.980 q^{79} -454.169i q^{80} -8253.75 q^{82} -13165.9i q^{83} -90.8456 q^{85} +17087.5i q^{86} -8954.61 q^{88} -575.925i q^{89} +3974.71 q^{91} +10772.9i q^{92} -5242.22 q^{94} +637.822i q^{95} +15123.4 q^{97} -2863.41i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 30 q^{4} - 24 q^{7} - 36 q^{10} + 12 q^{13} - 30 q^{16} - 258 q^{19} + 738 q^{22} + 546 q^{25} + 1308 q^{28} - 2580 q^{31} - 1026 q^{34} + 12 q^{37} + 2628 q^{40} + 570 q^{43} - 5760 q^{46} + 3726 q^{49}+ \cdots + 57918 q^{97}+O(q^{100}) $ 6 * q - 30 * q^4 - 24 * q^7 - 36 * q^10 + 12 * q^13 - 30 * q^16 - 258 * q^19 + 738 * q^22 + 546 * q^25 + 1308 * q^28 - 2580 * q^31 - 1026 * q^34 + 12 * q^37 + 2628 * q^40 + 570 * q^43 - 5760 * q^46 + 3726 * q^49 + 480 * q^52 + 2016 * q^55 - 12924 * q^58 - 7260 * q^61 + 15450 * q^64 + 10110 * q^67 + 19368 * q^70 - 14622 * q^73 + 8094 * q^76 - 9528 * q^79 - 9702 * q^82 - 24732 * q^85 - 29574 * q^88 + 34836 * q^91 + 25416 * q^94 + 57918 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 6.55250i	− 1.63813i	−0.573704	−	0.819063i	$-0.694494\pi$
$2$	− 6.55250i	− 1.63813i	0.573704	−	0.819063i	$-0.305506\pi$
$3$	0	0
$3$	0	0
$4$	−26.9353	−1.68345
$5$	− 11.7830i	− 0.471320i	−0.971836	−	0.235660i	$-0.924275\pi$
$5$	− 11.7830i	− 0.471320i	0.971836	−	0.235660i	$-0.0757252\pi$
$6$	0	0
$7$	−53.2728	−1.08720	−0.543600	−	0.839344i	$-0.682939\pi$
$7$	−53.2728	−1.08720	−0.543600	+	0.839344i	$0.682939\pi$
$8$	71.6534i	1.11958i
$9$	0	0
$10$	−77.2081	−0.772081
$11$	124.971i	1.03282i	0.856341	+	0.516410i	$0.172732\pi$
$11$	124.971i	1.03282i	−0.856341	+	0.516410i	$0.827268\pi$
$12$	0	0
$13$	−74.6104	−0.441482	−0.220741	−	0.975332i	$-0.570848\pi$
$13$	−74.6104	−0.441482	−0.220741	+	0.975332i	$0.570848\pi$
$14$	349.070i	1.78097i
$15$	0	0
$16$	38.5445	0.150564
$17$	− 7.70989i	− 0.0266778i	−0.999911	−	0.0133389i	$-0.995754\pi$
$17$	− 7.70989i	− 0.0266778i	0.999911	−	0.0133389i	$-0.00424603\pi$
$18$	0	0
$19$	−54.1307	−0.149946	−0.0749732	−	0.997186i	$-0.523887\pi$
$19$	−54.1307	−0.149946	−0.0749732	+	0.997186i	$0.523887\pi$
$20$	317.378i	0.793446i
$21$	0	0
$22$	818.874	1.69189
$23$	− 399.954i	− 0.756057i	−0.925794	−	0.378029i	$-0.876602\pi$
$23$	− 399.954i	− 0.756057i	0.925794	−	0.378029i	$-0.123398\pi$
$24$	0	0
$25$	486.161	0.777858
$26$	488.885i	0.723202i
$27$	0	0
$28$	1434.92	1.83025
$29$	− 540.653i	− 0.642869i	−0.946932	−	0.321435i	$-0.895835\pi$
$29$	− 540.653i	− 0.642869i	0.946932	−	0.321435i	$-0.104165\pi$
$30$	0	0
$31$	−1532.17	−1.59435	−0.797173	−	0.603751i	$-0.793672\pi$
$31$	−1532.17	−1.59435	−0.797173	+	0.603751i	$0.793672\pi$
$32$	893.891i	0.872941i
$33$	0	0
$34$	−50.5190	−0.0437016
$35$	627.714i	0.512419i
$36$	0	0
$37$	−1719.10	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$37$	−1719.10	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$38$	354.691i	0.245631i
$39$	0	0
$40$	844.292	0.527682
$41$	− 1259.63i	− 0.749335i	−0.927159	−	0.374668i	$-0.877757\pi$
$41$	− 1259.63i	− 0.749335i	0.927159	−	0.374668i	$-0.122243\pi$
$42$	0	0
$43$	−2607.79	−1.41038	−0.705189	−	0.709020i	$-0.749138\pi$
$43$	−2607.79	−1.41038	−0.705189	+	0.709020i	$0.749138\pi$
$44$	− 3366.13i	− 1.73871i
$45$	0	0
$46$	−2620.70	−1.23852
$47$	− 800.034i	− 0.362170i	−0.983467	−	0.181085i	$-0.942039\pi$
$47$	− 800.034i	− 0.362170i	0.983467	−	0.181085i	$-0.0579609\pi$
$48$	0	0
$49$	436.996	0.182006
$50$	− 3185.57i	− 1.27423i
$51$	0	0
$52$	2009.65	0.743214
$53$	− 4229.81i	− 1.50581i	−0.658131	−	0.752904i	$-0.728653\pi$
$53$	− 4229.81i	− 1.50581i	0.658131	−	0.752904i	$-0.271347\pi$
$54$	0	0
$55$	1472.54	0.486789
$56$	− 3817.18i	− 1.21721i
$57$	0	0
$58$	−3542.63	−1.05310
$59$	3333.19i	0.957538i	0.877941	+	0.478769i	$0.158917\pi$
$59$	3333.19i	0.957538i	−0.877941	+	0.478769i	$0.841083\pi$
$60$	0	0
$61$	−15.0169	−0.00403570	−0.00201785	−	0.999998i	$-0.500642\pi$
$61$	−15.0169	−0.00403570	−0.00201785	+	0.999998i	$0.500642\pi$
$62$	10039.5i	2.61174i
$63$	0	0
$64$	6473.94	1.58055
$65$	879.134i	0.208079i
$66$	0	0
$67$	5182.39	1.15447	0.577233	−	0.816580i	$-0.304133\pi$
$67$	5182.39	1.15447	0.577233	+	0.816580i	$0.304133\pi$
$68$	207.668i	0.0449109i
$69$	0	0
$70$	4113.10	0.839407
$71$	− 1924.16i	− 0.381701i	−0.981619	−	0.190851i	$-0.938875\pi$
$71$	− 1924.16i	− 0.381701i	0.981619	−	0.190851i	$-0.0611246\pi$
$72$	0	0
$73$	949.554	0.178186	0.0890931	−	0.996023i	$-0.471603\pi$
$73$	949.554	0.178186	0.0890931	+	0.996023i	$0.471603\pi$
$74$	11264.4i	2.05705i
$75$	0	0
$76$	1458.02	0.252428
$77$	− 6657.57i	− 1.12288i
$78$	0	0
$79$	−237.980	−0.0381318	−0.0190659	−	0.999818i	$-0.506069\pi$
$79$	−237.980	−0.0381318	−0.0190659	+	0.999818i	$0.506069\pi$
$80$	− 454.169i	− 0.0709640i
$81$	0	0
$82$	−8253.75	−1.22751
$83$	− 13165.9i	− 1.91115i	−0.294750	−	0.955574i	$-0.595236\pi$
$83$	− 13165.9i	− 1.91115i	0.294750	−	0.955574i	$-0.404764\pi$
$84$	0	0
$85$	−90.8456	−0.0125738
$86$	17087.5i	2.31038i
$87$	0	0
$88$	−8954.61	−1.15633
$89$	− 575.925i	− 0.0727086i	−0.999339	−	0.0363543i	$-0.988426\pi$
$89$	− 575.925i	− 0.0727086i	0.999339	−	0.0363543i	$-0.0115745\pi$
$90$	0	0
$91$	3974.71	0.479979
$92$	10772.9i	1.27279i
$93$	0	0
$94$	−5242.22	−0.593280
$95$	637.822i	0.0706728i
$96$	0	0
$97$	15123.4	1.60733	0.803667	−	0.595079i	$-0.202879\pi$
$97$	15123.4	1.60733	0.803667	+	0.595079i	$0.202879\pi$
$98$	− 2863.41i	− 0.298148i
$99$	0	0
$100$	−13094.9	−1.30949
$101$	11060.8i	1.08428i	0.840287	+	0.542141i	$0.182386\pi$
$101$	11060.8i	1.08428i	−0.840287	+	0.542141i	$0.817614\pi$
$102$	0	0
$103$	−3562.92	−0.335839	−0.167920	−	0.985801i	$-0.553705\pi$
$103$	−3562.92	−0.335839	−0.167920	+	0.985801i	$0.553705\pi$
$104$	− 5346.09i	− 0.494276i
$105$	0	0
$106$	−27715.9	−2.46670
$107$	11432.7i	0.998576i	0.866436	+	0.499288i	$0.166405\pi$
$107$	11432.7i	0.998576i	−0.866436	+	0.499288i	$0.833595\pi$
$108$	0	0
$109$	−4780.43	−0.402359	−0.201180	−	0.979554i	$-0.564478\pi$
$109$	−4780.43	−0.402359	−0.201180	+	0.979554i	$0.564478\pi$
$110$	− 9648.79i	− 0.797421i
$111$	0	0
$112$	−2053.37	−0.163694
$113$	− 4491.39i	− 0.351742i	−0.984413	−	0.175871i	$-0.943726\pi$
$113$	− 4491.39i	− 0.351742i	0.984413	−	0.175871i	$-0.0562741\pi$
$114$	0	0
$115$	−4712.66	−0.356345
$116$	14562.6i	1.08224i
$117$	0	0
$118$	21840.7	1.56857
$119$	410.728i	0.0290041i
$120$	0	0
$121$	−976.812	−0.0667176
$122$	98.3979i	0.00661099i
$123$	0	0
$124$	41269.3	2.68401
$125$	− 13092.8i	− 0.837940i
$126$	0	0
$127$	13521.3	0.838322	0.419161	−	0.907912i	$-0.362324\pi$
$127$	13521.3	0.838322	0.419161	+	0.907912i	$0.362324\pi$
$128$	− 28118.2i	− 1.71620i
$129$	0	0
$130$	5760.53	0.340860
$131$	− 3396.97i	− 0.197947i	−0.995090	−	0.0989735i	$-0.968444\pi$
$131$	− 3396.97i	− 0.197947i	0.995090	−	0.0989735i	$-0.0315559\pi$
$132$	0	0
$133$	2883.69	0.163022
$134$	− 33957.6i	− 1.89116i
$135$	0	0
$136$	552.439	0.0298680
$137$	3823.19i	0.203697i	0.994800	+	0.101849i	$0.0324757\pi$
$137$	3823.19i	0.203697i	−0.994800	+	0.101849i	$0.967524\pi$
$138$	0	0
$139$	3476.63	0.179940	0.0899702	−	0.995944i	$-0.471323\pi$
$139$	3476.63	0.179940	0.0899702	+	0.995944i	$0.471323\pi$
$140$	− 16907.6i	− 0.862635i
$141$	0	0
$142$	−12608.0	−0.625275
$143$	− 9324.16i	− 0.455971i
$144$	0	0
$145$	−6370.51	−0.302997
$146$	− 6221.95i	− 0.291891i
$147$	0	0
$148$	46304.5	2.11398
$149$	− 8910.94i	− 0.401376i	−0.979655	−	0.200688i	$-0.935682\pi$
$149$	− 8910.94i	− 0.401376i	0.979655	−	0.200688i	$-0.0643177\pi$
$150$	0	0
$151$	−9915.33	−0.434864	−0.217432	−	0.976076i	$-0.569768\pi$
$151$	−9915.33	−0.434864	−0.217432	+	0.976076i	$0.569768\pi$
$152$	− 3878.65i	− 0.167878i
$153$	0	0
$154$	−43623.8	−1.83942
$155$	18053.5i	0.751447i
$156$	0	0
$157$	5646.94	0.229094	0.114547	−	0.993418i	$-0.463458\pi$
$157$	5646.94	0.229094	0.114547	+	0.993418i	$0.463458\pi$
$158$	1559.37i	0.0624646i
$159$	0	0
$160$	10532.7	0.411434
$161$	21306.7i	0.821986i
$162$	0	0
$163$	−2153.55	−0.0810552	−0.0405276	−	0.999178i	$-0.512904\pi$
$163$	−2153.55	−0.0810552	−0.0405276	+	0.999178i	$0.512904\pi$
$164$	33928.6i	1.26147i
$165$	0	0
$166$	−86269.6	−3.13070
$167$	37020.2i	1.32741i	0.747994	+	0.663706i	$0.231017\pi$
$167$	37020.2i	1.32741i	−0.747994	+	0.663706i	$0.768983\pi$
$168$	0	0
$169$	−22994.3	−0.805094
$170$	595.266i	0.0205974i
$171$	0	0
$172$	70241.5	2.37431
$173$	− 48585.4i	− 1.62336i	−0.584105	−	0.811678i	$-0.698554\pi$
$173$	− 48585.4i	− 1.62336i	0.584105	−	0.811678i	$-0.301446\pi$
$174$	0	0
$175$	−25899.2	−0.845687
$176$	4816.95i	0.155506i
$177$	0	0
$178$	−3773.75	−0.119106
$179$	47717.9i	1.48928i	0.667468	+	0.744639i	$0.267378\pi$
$179$	47717.9i	1.48928i	−0.667468	+	0.744639i	$0.732622\pi$
$180$	0	0
$181$	−45767.2	−1.39700	−0.698502	−	0.715608i	$-0.746150\pi$
$181$	−45767.2	−1.39700	−0.698502	+	0.715608i	$0.746150\pi$
$182$	− 26044.3i	− 0.786266i
$183$	0	0
$184$	28658.1	0.846469
$185$	20256.2i	0.591854i
$186$	0	0
$187$	963.514	0.0275534
$188$	21549.1i	0.609697i
$189$	0	0
$190$	4179.33	0.115771
$191$	46719.5i	1.28065i	0.768103	+	0.640326i	$0.221201\pi$
$191$	46719.5i	1.28065i	−0.768103	+	0.640326i	$0.778799\pi$
$192$	0	0
$193$	−27207.8	−0.730430	−0.365215	−	0.930923i	$-0.619005\pi$
$193$	−27207.8	−0.730430	−0.365215	+	0.930923i	$0.619005\pi$
$194$	− 99096.1i	− 2.63301i
$195$	0	0
$196$	−11770.6	−0.306398
$197$	− 64665.2i	− 1.66624i	−0.553090	−	0.833122i	$-0.686551\pi$
$197$	− 64665.2i	− 1.66624i	0.553090	−	0.833122i	$-0.313449\pi$
$198$	0	0
$199$	−25841.3	−0.652542	−0.326271	−	0.945276i	$-0.605792\pi$
$199$	−25841.3	−0.652542	−0.326271	+	0.945276i	$0.605792\pi$
$200$	34835.1i	0.870877i
$201$	0	0
$202$	72475.7	1.77619
$203$	28802.1i	0.698928i
$204$	0	0
$205$	−14842.3	−0.353177
$206$	23346.0i	0.550147i
$207$	0	0
$208$	−2875.82	−0.0664714
$209$	− 6764.78i	− 0.154868i
$210$	0	0
$211$	46058.7	1.03454	0.517269	−	0.855823i	$-0.326949\pi$
$211$	46058.7	1.03454	0.517269	+	0.855823i	$0.326949\pi$
$212$	113931.i	2.53496i
$213$	0	0
$214$	74912.8	1.63579
$215$	30727.6i	0.664739i
$216$	0	0
$217$	81622.8	1.73337
$218$	31323.8i	0.659115i
$219$	0	0
$220$	−39663.2	−0.819487
$221$	575.238i	0.0117778i
$222$	0	0
$223$	21911.1	0.440610	0.220305	−	0.975431i	$-0.429295\pi$
$223$	21911.1	0.440610	0.220305	+	0.975431i	$0.429295\pi$
$224$	− 47620.1i	− 0.949062i
$225$	0	0
$226$	−29429.8	−0.576197
$227$	49785.7i	0.966168i	0.875574	+	0.483084i	$0.160483\pi$
$227$	49785.7i	0.966168i	−0.875574	+	0.483084i	$0.839517\pi$
$228$	0	0
$229$	28521.0	0.543868	0.271934	−	0.962316i	$-0.412337\pi$
$229$	28521.0	0.543868	0.271934	+	0.962316i	$0.412337\pi$
$230$	30879.7i	0.583737i
$231$	0	0
$232$	38739.6	0.719746
$233$	44050.2i	0.811403i	0.914006	+	0.405701i	$0.132973\pi$
$233$	44050.2i	0.811403i	−0.914006	+	0.405701i	$0.867027\pi$
$234$	0	0
$235$	−9426.80	−0.170698
$236$	− 89780.4i	− 1.61197i
$237$	0	0
$238$	2691.29	0.0475124
$239$	− 20575.2i	− 0.360203i	−0.983648	−	0.180102i	$-0.942357\pi$
$239$	− 20575.2i	− 0.360203i	0.983648	−	0.180102i	$-0.0576427\pi$
$240$	0	0
$241$	−57917.5	−0.997185	−0.498593	−	0.866836i	$-0.666150\pi$
$241$	−57917.5	−0.997185	−0.498593	+	0.866836i	$0.666150\pi$
$242$	6400.56i	0.109292i
$243$	0	0
$244$	404.483	0.00679392
$245$	− 5149.12i	− 0.0857829i
$246$	0	0
$247$	4038.71	0.0661986
$248$	− 109785.i	− 1.78500i
$249$	0	0
$250$	−85790.6	−1.37265
$251$	− 54140.4i	− 0.859357i	−0.902982	−	0.429679i	$-0.858627\pi$
$251$	− 54140.4i	− 0.859357i	0.902982	−	0.429679i	$-0.141373\pi$
$252$	0	0
$253$	49982.8	0.780871
$254$	− 88598.3i	− 1.37328i
$255$	0	0
$256$	−80661.6	−1.23080
$257$	− 113764.i	− 1.72242i	−0.508248	−	0.861211i	$-0.669707\pi$
$257$	− 113764.i	− 1.72242i	0.508248	−	0.861211i	$-0.330293\pi$
$258$	0	0
$259$	91581.5	1.36524
$260$	− 23679.7i	− 0.350292i
$261$	0	0
$262$	−22258.6	−0.324262
$263$	44363.7i	0.641381i	0.947184	+	0.320691i	$0.103915\pi$
$263$	44363.7i	0.641381i	−0.947184	+	0.320691i	$0.896085\pi$
$264$	0	0
$265$	−49839.9	−0.709717
$266$	− 18895.4i	− 0.267050i
$267$	0	0
$268$	−139589.	−1.94349
$269$	− 4429.77i	− 0.0612176i	−0.999531	−	0.0306088i	$-0.990255\pi$
$269$	− 4429.77i	− 0.0612176i	0.999531	−	0.0306088i	$-0.00974461\pi$
$270$	0	0
$271$	−86111.9	−1.17253	−0.586266	−	0.810119i	$-0.699403\pi$
$271$	−86111.9	−1.17253	−0.586266	+	0.810119i	$0.699403\pi$
$272$	− 297.174i	− 0.00401673i
$273$	0	0
$274$	25051.5	0.333681
$275$	60756.1i	0.803387i
$276$	0	0
$277$	70613.4	0.920297	0.460148	−	0.887842i	$-0.347796\pi$
$277$	70613.4	0.920297	0.460148	+	0.887842i	$0.347796\pi$
$278$	− 22780.6i	− 0.294765i
$279$	0	0
$280$	−44977.8	−0.573697
$281$	− 149254.i	− 1.89023i	−0.326740	−	0.945114i	$-0.605950\pi$
$281$	− 149254.i	− 1.89023i	0.326740	−	0.945114i	$-0.394050\pi$
$282$	0	0
$283$	−83926.7	−1.04792	−0.523959	−	0.851743i	$-0.675546\pi$
$283$	−83926.7	−1.04792	−0.523959	+	0.851743i	$0.675546\pi$
$284$	51827.7i	0.642577i
$285$	0	0
$286$	−61096.5	−0.746938
$287$	67104.2i	0.814678i
$288$	0	0
$289$	83461.6	0.999288
$290$	41742.8i	0.496347i
$291$	0	0
$292$	−25576.5	−0.299968
$293$	35903.0i	0.418211i	0.977893	+	0.209106i	$0.0670553\pi$
$293$	35903.0i	0.418211i	−0.977893	+	0.209106i	$0.932945\pi$
$294$	0	0
$295$	39275.0	0.451307
$296$	− 123180.i	− 1.40590i
$297$	0	0
$298$	−58389.0	−0.657504
$299$	29840.7i	0.333785i
$300$	0	0
$301$	138924.	1.53336
$302$	64970.2i	0.712361i
$303$	0	0
$304$	−2086.44	−0.0225766
$305$	176.944i	0.00190211i
$306$	0	0
$307$	7054.30	0.0748475	0.0374238	−	0.999299i	$-0.488085\pi$
$307$	7054.30	0.0748475	0.0374238	+	0.999299i	$0.488085\pi$
$308$	179324.i	1.89032i
$309$	0	0
$310$	118296.	1.23096
$311$	− 85661.4i	− 0.885655i	−0.896607	−	0.442827i	$-0.853975\pi$
$311$	− 85661.4i	− 0.885655i	0.896607	−	0.442827i	$-0.146025\pi$
$312$	0	0
$313$	73442.5	0.749650	0.374825	−	0.927096i	$-0.377703\pi$
$313$	73442.5	0.749650	0.374825	+	0.927096i	$0.377703\pi$
$314$	− 37001.5i	− 0.375284i
$315$	0	0
$316$	6410.07	0.0641931
$317$	49239.6i	0.490000i	0.969523	+	0.245000i	$0.0787880\pi$
$317$	49239.6i	0.490000i	−0.969523	+	0.245000i	$0.921212\pi$
$318$	0	0
$319$	67566.1	0.663968
$320$	− 76282.4i	− 0.744945i
$321$	0	0
$322$	139612.	1.34652
$323$	417.341i	0.00400024i
$324$	0	0
$325$	−36272.7	−0.343410
$326$	14111.2i	0.132779i
$327$	0	0
$328$	90256.9	0.838944
$329$	42620.1i	0.393752i
$330$	0	0
$331$	−82011.1	−0.748543	−0.374271	−	0.927319i	$-0.622107\pi$
$331$	−82011.1	−0.748543	−0.374271	+	0.927319i	$0.622107\pi$
$332$	354627.i	3.21733i
$333$	0	0
$334$	242575.	2.17447
$335$	− 61064.1i	− 0.544122i
$336$	0	0
$337$	−46982.9	−0.413694	−0.206847	−	0.978373i	$-0.566320\pi$
$337$	−46982.9	−0.413694	−0.206847	+	0.978373i	$0.566320\pi$
$338$	150670.i	1.31884i
$339$	0	0
$340$	2446.95	0.0211674
$341$	− 191477.i	− 1.64667i
$342$	0	0
$343$	104628.	0.889324
$344$	− 186857.i	− 1.57904i
$345$	0	0
$346$	−318356.	−2.65926
$347$	− 44319.7i	− 0.368076i	−0.982919	−	0.184038i	$-0.941083\pi$
$347$	− 44319.7i	− 0.368076i	0.982919	−	0.184038i	$-0.0589170\pi$
$348$	0	0
$349$	−186463.	−1.53088	−0.765442	−	0.643504i	$-0.777480\pi$
$349$	−186463.	−1.53088	−0.765442	+	0.643504i	$0.777480\pi$
$350$	169704.i	1.38534i
$351$	0	0
$352$	−111711.	−0.901591
$353$	− 169694.i	− 1.36181i	−0.732370	−	0.680907i	$-0.761586\pi$
$353$	− 169694.i	− 1.36181i	0.732370	−	0.680907i	$-0.238414\pi$
$354$	0	0
$355$	−22672.3	−0.179903
$356$	15512.7i	0.122402i
$357$	0	0
$358$	312672.	2.43962
$359$	− 69656.7i	− 0.540473i	−0.962794	−	0.270236i	$-0.912898\pi$
$359$	− 69656.7i	− 0.540473i	0.962794	−	0.270236i	$-0.0871019\pi$
$360$	0	0
$361$	−127391.	−0.977516
$362$	299890.i	2.28847i
$363$	0	0
$364$	−107060.	−0.808023
$365$	− 11188.6i	− 0.0839827i
$366$	0	0
$367$	−176168.	−1.30796	−0.653982	−	0.756510i	$-0.726903\pi$
$367$	−176168.	−1.30796	−0.653982	+	0.756510i	$0.726903\pi$
$368$	− 15416.0i	− 0.113835i
$369$	0	0
$370$	132729.	0.969531
$371$	225334.i	1.63712i
$372$	0	0
$373$	67187.1	0.482912	0.241456	−	0.970412i	$-0.422375\pi$
$373$	67187.1	0.482912	0.241456	+	0.970412i	$0.422375\pi$
$374$	− 6313.43i	− 0.0451359i
$375$	0	0
$376$	57325.1	0.405480
$377$	40338.3i	0.283815i
$378$	0	0
$379$	43976.9	0.306158	0.153079	−	0.988214i	$-0.451081\pi$
$379$	43976.9	0.306158	0.153079	+	0.988214i	$0.451081\pi$
$380$	− 17179.9i	− 0.118974i
$381$	0	0
$382$	306129.	2.09787
$383$	185140.i	1.26212i	0.775733	+	0.631062i	$0.217381\pi$
$383$	185140.i	1.26212i	−0.775733	+	0.631062i	$0.782619\pi$
$384$	0	0
$385$	−78446.2	−0.529237
$386$	178279.i	1.19654i
$387$	0	0
$388$	−407353.	−2.70587
$389$	− 30442.6i	− 0.201179i	−0.994928	−	0.100589i	$-0.967927\pi$
$389$	− 30442.6i	− 0.201179i	0.994928	−	0.100589i	$-0.0320728\pi$
$390$	0	0
$391$	−3083.60	−0.0201699
$392$	31312.2i	0.203771i
$393$	0	0
$394$	−423719.	−2.72952
$395$	2804.12i	0.0179723i
$396$	0	0
$397$	−235724.	−1.49563	−0.747813	−	0.663910i	$-0.768896\pi$
$397$	−235724.	−1.49563	−0.747813	+	0.663910i	$0.768896\pi$
$398$	169325.i	1.06895i
$399$	0	0
$400$	18738.8	0.117118
$401$	− 234208.i	− 1.45651i	−0.685308	−	0.728254i	$-0.740332\pi$
$401$	− 234208.i	− 1.45651i	0.685308	−	0.728254i	$-0.259668\pi$
$402$	0	0
$403$	114316.	0.703874
$404$	− 297925.i	− 1.82534i
$405$	0	0
$406$	188726.	1.14493
$407$	− 214839.i	− 1.29695i
$408$	0	0
$409$	−297786.	−1.78015	−0.890076	−	0.455812i	$-0.849349\pi$
$409$	−297786.	−1.78015	−0.890076	+	0.455812i	$0.849349\pi$
$410$	97253.9i	0.578548i
$411$	0	0
$412$	95968.2	0.565370
$413$	− 177569.i	− 1.04104i
$414$	0	0
$415$	−155134.	−0.900763
$416$	− 66693.6i	− 0.385387i
$417$	0	0
$418$	−44326.2	−0.253693
$419$	36874.2i	0.210036i	0.994470	+	0.105018i	$0.0334901\pi$
$419$	36874.2i	0.210036i	−0.994470	+	0.105018i	$0.966510\pi$
$420$	0	0
$421$	240610.	1.35753	0.678764	−	0.734356i	$-0.262516\pi$
$421$	240610.	1.35753	0.678764	+	0.734356i	$0.262516\pi$
$422$	− 301799.i	− 1.69470i
$423$	0	0
$424$	303080.	1.68588
$425$	− 3748.25i	− 0.0207515i
$426$	0	0
$427$	799.990	0.00438762
$428$	− 307943.i	− 1.68106i
$429$	0	0
$430$	201342.	1.08893
$431$	162758.i	0.876169i	0.898934	+	0.438084i	$0.144343\pi$
$431$	162758.i	0.876169i	−0.898934	+	0.438084i	$0.855657\pi$
$432$	0	0
$433$	266637.	1.42215	0.711073	−	0.703118i	$-0.248209\pi$
$433$	266637.	1.42215	0.711073	+	0.703118i	$0.248209\pi$
$434$	− 534834.i	− 2.83948i
$435$	0	0
$436$	128762.	0.677354
$437$	21649.8i	0.113368i
$438$	0	0
$439$	−36817.5	−0.191040	−0.0955201	−	0.995428i	$-0.530451\pi$
$439$	−36817.5	−0.191040	−0.0955201	+	0.995428i	$0.530451\pi$
$440$	105512.i	0.545001i
$441$	0	0
$442$	3769.25	0.0192935
$443$	− 137544.i	− 0.700865i	−0.936588	−	0.350432i	$-0.886035\pi$
$443$	− 137544.i	− 0.700865i	0.936588	−	0.350432i	$-0.113965\pi$
$444$	0	0
$445$	−6786.12	−0.0342690
$446$	− 143572.i	− 0.721774i
$447$	0	0
$448$	−344885.	−1.71838
$449$	174405.i	0.865101i	0.901610	+	0.432550i	$0.142386\pi$
$449$	174405.i	0.865101i	−0.901610	+	0.432550i	$0.857614\pi$
$450$	0	0
$451$	157418.	0.773929
$452$	120977.i	0.592141i
$453$	0	0
$454$	326221.	1.58270
$455$	− 46834.0i	− 0.226224i
$456$	0	0
$457$	115309.	0.552119	0.276059	−	0.961141i	$-0.410971\pi$
$457$	115309.	0.552119	0.276059	+	0.961141i	$0.410971\pi$
$458$	− 186884.i	− 0.890923i
$459$	0	0
$460$	126937.	0.599890
$461$	405463.i	1.90787i	0.300011	+	0.953936i	$0.403010\pi$
$461$	405463.i	1.90787i	−0.300011	+	0.953936i	$0.596990\pi$
$462$	0	0
$463$	72586.7	0.338606	0.169303	−	0.985564i	$-0.445848\pi$
$463$	72586.7	0.338606	0.169303	+	0.985564i	$0.445848\pi$
$464$	− 20839.2i	− 0.0967932i
$465$	0	0
$466$	288639.	1.32918
$467$	358806.i	1.64523i	0.568602	+	0.822613i	$0.307485\pi$
$467$	358806.i	1.64523i	−0.568602	+	0.822613i	$0.692515\pi$
$468$	0	0
$469$	−276081.	−1.25514
$470$	61769.1i	0.279625i
$471$	0	0
$472$	−238834.	−1.07204
$473$	− 325899.i	− 1.45667i
$474$	0	0
$475$	−26316.2	−0.116637
$476$	− 11063.1i	− 0.0488271i
$477$	0	0
$478$	−134819.	−0.590058
$479$	− 25599.5i	− 0.111573i	−0.998443	−	0.0557867i	$-0.982233\pi$
$479$	− 25599.5i	− 0.111573i	0.998443	−	0.0557867i	$-0.0177667\pi$
$480$	0	0
$481$	128263.	0.554385
$482$	379505.i	1.63351i
$483$	0	0
$484$	26310.7	0.112316
$485$	− 178199.i	− 0.757569i
$486$	0	0
$487$	247261.	1.04255	0.521277	−	0.853388i	$-0.325456\pi$
$487$	247261.	1.04255	0.521277	+	0.853388i	$0.325456\pi$
$488$	− 1076.01i	− 0.00451831i
$489$	0	0
$490$	−33739.6	−0.140523
$491$	342256.i	1.41967i	0.704366	+	0.709837i	$0.251231\pi$
$491$	342256.i	1.41967i	−0.704366	+	0.709837i	$0.748769\pi$
$492$	0	0
$493$	−4168.37	−0.0171503
$494$	− 26463.7i	− 0.108442i
$495$	0	0
$496$	−59056.5	−0.240052
$497$	102505.i	0.414986i
$498$	0	0
$499$	−197838.	−0.794527	−0.397264	−	0.917704i	$-0.630040\pi$
$499$	−197838.	−0.794527	−0.397264	+	0.917704i	$0.630040\pi$
$500$	352658.i	1.41063i
$501$	0	0
$502$	−354755.	−1.40773
$503$	− 187051.i	− 0.739305i	−0.929170	−	0.369652i	$-0.879477\pi$
$503$	− 187051.i	− 0.739305i	0.929170	−	0.369652i	$-0.120523\pi$
$504$	0	0
$505$	130329.	0.511044
$506$	− 327512.i	− 1.27916i
$507$	0	0
$508$	−364200.	−1.41128
$509$	28880.8i	0.111474i	0.998445	+	0.0557370i	$0.0177508\pi$
$509$	28880.8i	0.111474i	−0.998445	+	0.0557370i	$0.982249\pi$
$510$	0	0
$511$	−50585.4	−0.193724
$512$	78644.0i	0.300003i
$513$	0	0
$514$	−745440.	−2.82154
$515$	41981.9i	0.158288i
$516$	0	0
$517$	99981.2	0.374057
$518$	− 600088.i	− 2.23643i
$519$	0	0
$520$	−62992.9	−0.232962
$521$	− 269661.i	− 0.993443i	−0.867910	−	0.496721i	$-0.834537\pi$
$521$	− 269661.i	− 0.993443i	0.867910	−	0.496721i	$-0.165463\pi$
$522$	0	0
$523$	−187282.	−0.684689	−0.342344	−	0.939575i	$-0.611221\pi$
$523$	−187282.	−0.684689	−0.342344	+	0.939575i	$0.611221\pi$
$524$	91498.3i	0.333235i
$525$	0	0
$526$	290693.	1.05066
$527$	11812.8i	0.0425336i
$528$	0	0
$529$	119878.	0.428378
$530$	326576.i	1.16261i
$531$	0	0
$532$	−77673.1	−0.274440
$533$	93981.7i	0.330818i
$534$	0	0
$535$	134711.	0.470649
$536$	371336.i	1.29252i
$537$	0	0
$538$	−29026.1	−0.100282
$539$	54611.9i	0.187979i
$540$	0	0
$541$	−25587.9	−0.0874259	−0.0437129	−	0.999044i	$-0.513919\pi$
$541$	−25587.9	−0.0874259	−0.0437129	+	0.999044i	$0.513919\pi$
$542$	564248.i	1.92075i
$543$	0	0
$544$	6891.80	0.0232881
$545$	56327.8i	0.189640i
$546$	0	0
$547$	−276849.	−0.925269	−0.462634	−	0.886549i	$-0.653096\pi$
$547$	−276849.	−0.925269	−0.462634	+	0.886549i	$0.653096\pi$
$548$	− 102979.i	− 0.342915i
$549$	0	0
$550$	398105.	1.31605
$551$	29265.9i	0.0963959i
$552$	0	0
$553$	12677.9	0.0414569
$554$	− 462695.i	− 1.50756i
$555$	0	0
$556$	−93643.9	−0.302921
$557$	461786.i	1.48844i	0.667936	+	0.744219i	$0.267178\pi$
$557$	461786.i	1.48844i	−0.667936	+	0.744219i	$0.732822\pi$
$558$	0	0
$559$	194568.	0.622656
$560$	24194.9i	0.0771521i
$561$	0	0
$562$	−977989.	−3.09643
$563$	177944.i	0.561393i	0.959797	+	0.280697i	$0.0905654\pi$
$563$	177944.i	0.561393i	−0.959797	+	0.280697i	$0.909435\pi$
$564$	0	0
$565$	−52922.0	−0.165783
$566$	549930.i	1.71662i
$567$	0	0
$568$	137872.	0.427347
$569$	− 5902.63i	− 0.0182314i	−0.999958	−	0.00911572i	$-0.997098\pi$
$569$	− 5902.63i	− 0.0182314i	0.999958	−	0.00911572i	$-0.00290166\pi$
$570$	0	0
$571$	476692.	1.46206	0.731031	−	0.682344i	$-0.239039\pi$
$571$	476692.	1.46206	0.731031	+	0.682344i	$0.239039\pi$
$572$	251149.i	0.767607i
$573$	0	0
$574$	439700.	1.33454
$575$	− 194442.i	− 0.588105i
$576$	0	0
$577$	209901.	0.630468	0.315234	−	0.949014i	$-0.397917\pi$
$577$	209901.	0.630468	0.315234	+	0.949014i	$0.397917\pi$
$578$	− 546882.i	− 1.63696i
$579$	0	0
$580$	171591.	0.510082
$581$	701385.i	2.07780i
$582$	0	0
$583$	528605.	1.55523
$584$	68038.7i	0.199494i
$585$	0	0
$586$	235255.	0.685083
$587$	− 180770.i	− 0.524627i	−0.964983	−	0.262313i	$-0.915515\pi$
$587$	− 180770.i	− 0.524627i	0.964983	−	0.262313i	$-0.0844854\pi$
$588$	0	0
$589$	82937.2	0.239066
$590$	− 257349.i	− 0.739297i
$591$	0	0
$592$	−66261.9	−0.189069
$593$	468097.i	1.33115i	0.746332	+	0.665574i	$0.231813\pi$
$593$	468097.i	1.33115i	−0.746332	+	0.665574i	$0.768187\pi$
$594$	0	0
$595$	4839.60	0.0136702
$596$	240019.i	0.675698i
$597$	0	0
$598$	195532.	0.546782
$599$	− 644967.i	− 1.79756i	−0.438399	−	0.898780i	$-0.644454\pi$
$599$	− 644967.i	− 1.79756i	0.438399	−	0.898780i	$-0.355546\pi$
$600$	0	0
$601$	255033.	0.706070	0.353035	−	0.935610i	$-0.385150\pi$
$601$	255033.	0.706070	0.353035	+	0.935610i	$0.385150\pi$
$602$	− 910302.i	− 2.51184i
$603$	0	0
$604$	267072.	0.732073
$605$	11509.8i	0.0314453i
$606$	0	0
$607$	−352702.	−0.957261	−0.478631	−	0.878016i	$-0.658867\pi$
$607$	−352702.	−0.957261	−0.478631	+	0.878016i	$0.658867\pi$
$608$	− 48386.9i	− 0.130894i
$609$	0	0
$610$	1159.42	0.00311589
$611$	59690.9i	0.159892i
$612$	0	0
$613$	−37013.8	−0.0985015	−0.0492508	−	0.998786i	$-0.515683\pi$
$613$	−37013.8	−0.0985015	−0.0492508	+	0.998786i	$0.515683\pi$
$614$	− 46223.3i	− 0.122610i
$615$	0	0
$616$	477038.	1.25716
$617$	570187.i	1.49778i	0.662696	+	0.748888i	$0.269412\pi$
$617$	570187.i	1.49778i	−0.662696	+	0.748888i	$0.730588\pi$
$618$	0	0
$619$	−440814.	−1.15047	−0.575233	−	0.817990i	$-0.695089\pi$
$619$	−440814.	−1.15047	−0.575233	+	0.817990i	$0.695089\pi$
$620$	− 486276.i	− 1.26503i
$621$	0	0
$622$	−561296.	−1.45081
$623$	30681.2i	0.0790489i
$624$	0	0
$625$	149578.	0.382920
$626$	− 481232.i	− 1.22802i
$627$	0	0
$628$	−152102.	−0.385669
$629$	13254.1i	0.0335003i
$630$	0	0
$631$	−80926.3	−0.203250	−0.101625	−	0.994823i	$-0.532404\pi$
$631$	−80926.3	−0.203250	−0.101625	+	0.994823i	$0.532404\pi$
$632$	− 17052.1i	− 0.0426917i
$633$	0	0
$634$	322643.	0.802682
$635$	− 159321.i	− 0.395118i
$636$	0	0
$637$	−32604.4	−0.0803522
$638$	− 442727.i	− 1.08766i
$639$	0	0
$640$	−331317.	−0.808879
$641$	− 152324.i	− 0.370726i	−0.982670	−	0.185363i	$-0.940654\pi$
$641$	− 152324.i	− 0.370726i	0.982670	−	0.185363i	$-0.0593461\pi$
$642$	0	0
$643$	116874.	0.282681	0.141340	−	0.989961i	$-0.454859\pi$
$643$	116874.	0.282681	0.141340	+	0.989961i	$0.454859\pi$
$644$	− 573902.i	− 1.38378i
$645$	0	0
$646$	2734.63	0.00655290
$647$	− 481407.i	− 1.15002i	−0.818148	−	0.575008i	$-0.804999\pi$
$647$	− 481407.i	− 1.15002i	0.818148	−	0.575008i	$-0.195001\pi$
$648$	0	0
$649$	−416553.	−0.988965
$650$	237677.i	0.562548i
$651$	0	0
$652$	58006.6	0.136453
$653$	− 460831.i	− 1.08073i	−0.841432	−	0.540363i	$-0.818287\pi$
$653$	− 460831.i	− 1.08073i	0.841432	−	0.540363i	$-0.181713\pi$
$654$	0	0
$655$	−40026.5	−0.0932964
$656$	− 48551.9i	− 0.112823i
$657$	0	0
$658$	279268.	0.645015
$659$	165076.i	0.380114i	0.981773	+	0.190057i	$0.0608674\pi$
$659$	165076.i	0.380114i	−0.981773	+	0.190057i	$0.939133\pi$
$660$	0	0
$661$	−281756.	−0.644868	−0.322434	−	0.946592i	$-0.604501\pi$
$661$	−281756.	−0.644868	−0.322434	+	0.946592i	$0.604501\pi$
$662$	537378.i	1.22621i
$663$	0	0
$664$	943382.	2.13969
$665$	− 33978.6i	− 0.0768355i
$666$	0	0
$667$	−216236.	−0.486046
$668$	− 997148.i	− 2.23464i
$669$	0	0
$670$	−400123.	−0.891341
$671$	− 1876.67i	− 0.00416816i
$672$	0	0
$673$	−756775.	−1.67085	−0.835423	−	0.549608i	$-0.814777\pi$
$673$	−756775.	−1.67085	−0.835423	+	0.549608i	$0.814777\pi$
$674$	307855.i	0.677683i
$675$	0	0
$676$	619357.	1.35534
$677$	− 68188.8i	− 0.148777i	−0.997229	−	0.0743885i	$-0.976300\pi$
$677$	− 68188.8i	− 0.148777i	0.997229	−	0.0743885i	$-0.0237005\pi$
$678$	0	0
$679$	−805667.	−1.74749
$680$	− 6509.39i	− 0.0140774i
$681$	0	0
$682$	−1.25465e6	−2.69746
$683$	− 31351.1i	− 0.0672066i	−0.999435	−	0.0336033i	$-0.989302\pi$
$683$	− 31351.1i	− 0.0672066i	0.999435	−	0.0336033i	$-0.0106983\pi$
$684$	0	0
$685$	45048.7	0.0960065
$686$	− 685576.i	− 1.45682i
$687$	0	0
$688$	−100516.	−0.212353
$689$	315588.i	0.664786i
$690$	0	0
$691$	374884.	0.785129	0.392565	−	0.919724i	$-0.371588\pi$
$691$	374884.	0.785129	0.392565	+	0.919724i	$0.371588\pi$
$692$	1.30866e6i	2.73285i
$693$	0	0
$694$	−290405.	−0.602955
$695$	− 40965.1i	− 0.0848095i
$696$	0	0
$697$	−9711.63	−0.0199906
$698$	1.22180e6i	2.50778i
$699$	0	0
$700$	697601.	1.42368
$701$	− 97448.5i	− 0.198308i	−0.995072	−	0.0991538i	$-0.968386\pi$
$701$	− 97448.5i	− 0.198308i	0.995072	−	0.0991538i	$-0.0316136\pi$
$702$	0	0
$703$	93056.2	0.188293
$704$	809056.i	1.63242i
$705$	0	0
$706$	−1.11192e6	−2.23082
$707$	− 589238.i	− 1.17883i
$708$	0	0
$709$	−292525.	−0.581931	−0.290965	−	0.956734i	$-0.593976\pi$
$709$	−292525.	−0.581931	−0.290965	+	0.956734i	$0.593976\pi$
$710$	148561.i	0.294704i
$711$	0	0
$712$	41267.0	0.0814034
$713$	612796.i	1.20542i
$714$	0	0
$715$	−109867.	−0.214908
$716$	− 1.28530e6i	− 2.50713i
$717$	0	0
$718$	−456426.	−0.885362
$719$	− 655579.i	− 1.26814i	−0.773276	−	0.634070i	$-0.781383\pi$
$719$	− 655579.i	− 1.26814i	0.773276	−	0.634070i	$-0.218617\pi$
$720$	0	0
$721$	189807.	0.365125
$722$	834729.i	1.60129i
$723$	0	0
$724$	1.23275e6	2.35179
$725$	− 262844.i	− 0.500061i
$726$	0	0
$727$	259261.	0.490533	0.245266	−	0.969456i	$-0.421125\pi$
$727$	259261.	0.490533	0.245266	+	0.969456i	$0.421125\pi$
$728$	284801.i	0.537377i
$729$	0	0
$730$	−73313.3	−0.137574
$731$	20105.8i	0.0376258i
$732$	0	0
$733$	82508.9	0.153565	0.0767826	−	0.997048i	$-0.475535\pi$
$733$	82508.9	0.153565	0.0767826	+	0.997048i	$0.475535\pi$
$734$	1.15434e6i	2.14261i
$735$	0	0
$736$	357516.	0.659993
$737$	647650.i	1.19235i
$738$	0	0
$739$	−730583.	−1.33777	−0.668884	−	0.743367i	$-0.733228\pi$
$739$	−730583.	−1.33777	−0.668884	+	0.743367i	$0.733228\pi$
$740$	− 545606.i	− 0.996359i
$741$	0	0
$742$	1.47650e6	2.68180
$743$	− 451784.i	− 0.818377i	−0.912450	−	0.409189i	$-0.865812\pi$
$743$	− 451784.i	− 0.818377i	0.912450	−	0.409189i	$-0.134188\pi$
$744$	0	0
$745$	−104998.	−0.189176
$746$	− 440244.i	− 0.791071i
$747$	0	0
$748$	−25952.5	−0.0463849
$749$	− 609052.i	− 1.08565i
$750$	0	0
$751$	580162.	1.02865	0.514327	−	0.857594i	$-0.328042\pi$
$751$	580162.	1.02865	0.514327	+	0.857594i	$0.328042\pi$
$752$	− 30836.9i	− 0.0545299i
$753$	0	0
$754$	264317.	0.464924
$755$	116832.i	0.204960i
$756$	0	0
$757$	1.00242e6	1.74928	0.874641	−	0.484771i	$-0.161097\pi$
$757$	1.00242e6	1.74928	0.874641	+	0.484771i	$0.161097\pi$
$758$	− 288159.i	− 0.501526i
$759$	0	0
$760$	−45702.1	−0.0791241
$761$	84717.1i	0.146286i	0.997321	+	0.0731429i	$0.0233029\pi$
$761$	84717.1i	0.146286i	−0.997321	+	0.0731429i	$0.976697\pi$
$762$	0	0
$763$	254667.	0.437445
$764$	− 1.25840e6i	− 2.15592i
$765$	0	0
$766$	1.21313e6	2.06752
$767$	− 248691.i	− 0.422736i
$768$	0	0
$769$	−683525.	−1.15585	−0.577926	−	0.816089i	$-0.696138\pi$
$769$	−683525.	−1.15585	−0.577926	+	0.816089i	$0.696138\pi$
$770$	514019.i	0.866957i
$771$	0	0
$772$	732849.	1.22965
$773$	235888.i	0.394772i	0.980326	+	0.197386i	$0.0632453\pi$
$773$	235888.i	0.394772i	−0.980326	+	0.197386i	$0.936755\pi$
$774$	0	0
$775$	−744879.	−1.24017
$776$	1.08364e6i	1.79955i
$777$	0	0
$778$	−199475.	−0.329556
$779$	68184.8i	0.112360i
$780$	0	0
$781$	240464.	0.394229
$782$	20205.3i	0.0330409i
$783$	0	0
$784$	16843.8	0.0274036
$785$	− 66537.8i	− 0.107977i
$786$	0	0
$787$	−457456.	−0.738585	−0.369292	−	0.929313i	$-0.620400\pi$
$787$	−457456.	−0.738585	−0.369292	+	0.929313i	$0.620400\pi$
$788$	1.74178e6i	2.80504i
$789$	0	0
$790$	18374.0	0.0294408
$791$	239269.i	0.382414i
$792$	0	0
$793$	1120.41	0.00178169
$794$	1.54458e6i	2.45002i
$795$	0	0
$796$	696043.	1.09853
$797$	170922.i	0.269080i	0.990908	+	0.134540i	$0.0429556\pi$
$797$	170922.i	0.269080i	−0.990908	+	0.134540i	$0.957044\pi$
$798$	0	0
$799$	−6168.17	−0.00966191
$800$	434575.i	0.679023i
$801$	0	0
$802$	−1.53465e6	−2.38594
$803$	118667.i	0.184034i
$804$	0	0
$805$	251057.	0.387418
$806$	− 749053.i	− 1.15303i
$807$	0	0
$808$	−792541.	−1.21395
$809$	− 252280.i	− 0.385465i	−0.981251	−	0.192733i	$-0.938265\pi$
$809$	− 252280.i	− 0.385465i	0.981251	−	0.192733i	$-0.0617350\pi$
$810$	0	0
$811$	166784.	0.253579	0.126789	−	0.991930i	$-0.459533\pi$
$811$	166784.	0.253579	0.126789	+	0.991930i	$0.459533\pi$
$812$	− 775793.i	− 1.17661i
$813$	0	0
$814$	−1.40773e6	−2.12457
$815$	25375.3i	0.0382029i
$816$	0	0
$817$	141161.	0.211481
$818$	1.95124e6i	2.91611i
$819$	0	0
$820$	399780.	0.594557
$821$	− 272312.i	− 0.403999i	−0.979386	−	0.201999i	$-0.935256\pi$
$821$	− 272312.i	− 0.403999i	0.979386	−	0.201999i	$-0.0647439\pi$
$822$	0	0
$823$	964628.	1.42416	0.712082	−	0.702096i	$-0.247752\pi$
$823$	964628.	1.42416	0.712082	+	0.702096i	$0.247752\pi$
$824$	− 255295.i	− 0.376000i
$825$	0	0
$826$	−1.16352e6	−1.70535
$827$	− 430905.i	− 0.630043i	−0.949085	−	0.315021i	$-0.897988\pi$
$827$	− 430905.i	− 0.630043i	0.949085	−	0.315021i	$-0.102012\pi$
$828$	0	0
$829$	462870.	0.673520	0.336760	−	0.941591i	$-0.390669\pi$
$829$	462870.	0.673520	0.336760	+	0.941591i	$0.390669\pi$
$830$	1.01651e6i	1.47556i
$831$	0	0
$832$	−483023.	−0.697784
$833$	− 3369.19i	− 0.00485551i
$834$	0	0
$835$	436209.	0.625635
$836$	182211.i	0.260713i
$837$	0	0
$838$	241618.	0.344066
$839$	− 804698.i	− 1.14317i	−0.820545	−	0.571583i	$-0.806330\pi$
$839$	− 804698.i	− 1.14317i	0.820545	−	0.571583i	$-0.193670\pi$
$840$	0	0
$841$	414975.	0.586719
$842$	− 1.57659e6i	− 2.22380i
$843$	0	0
$844$	−1.24060e6	−1.74160
$845$	270942.i	0.379457i
$846$	0	0
$847$	52037.6	0.0725354
$848$	− 163036.i	− 0.226721i
$849$	0	0
$850$	−24560.4	−0.0339936
$851$	687563.i	0.949409i
$852$	0	0
$853$	211580.	0.290788	0.145394	−	0.989374i	$-0.453555\pi$
$853$	211580.	0.290788	0.145394	+	0.989374i	$0.453555\pi$
$854$	− 5241.94i	− 0.00718747i
$855$	0	0
$856$	−819191.	−1.11799
$857$	− 283303.i	− 0.385736i	−0.981225	−	0.192868i	$-0.938221\pi$
$857$	− 283303.i	− 0.385736i	0.981225	−	0.192868i	$-0.0617788\pi$
$858$	0	0
$859$	−30997.2	−0.0420083	−0.0210042	−	0.999779i	$-0.506686\pi$
$859$	−30997.2	−0.0420083	−0.0210042	+	0.999779i	$0.506686\pi$
$860$	− 827655.i	− 1.11906i
$861$	0	0
$862$	1.06647e6	1.43527
$863$	− 1.35611e6i	− 1.82085i	−0.413677	−	0.910424i	$-0.635756\pi$
$863$	− 1.35611e6i	− 1.82085i	0.413677	−	0.910424i	$-0.364244\pi$
$864$	0	0
$865$	−572482.	−0.765120
$866$	− 1.74714e6i	− 2.32965i
$867$	0	0
$868$	−2.19853e6	−2.91806
$869$	− 29740.7i	− 0.0393833i
$870$	0	0
$871$	−386661.	−0.509675
$872$	− 342534.i	− 0.450475i
$873$	0	0
$874$	141860.	0.185711
$875$	697491.i	0.911009i
$876$	0	0
$877$	−452930.	−0.588887	−0.294443	−	0.955669i	$-0.595134\pi$
$877$	−452930.	−0.588887	−0.294443	+	0.955669i	$0.595134\pi$
$878$	241246.i	0.312948i
$879$	0	0
$880$	56758.1	0.0732930
$881$	− 1.13788e6i	− 1.46603i	−0.680211	−	0.733016i	$-0.738112\pi$
$881$	− 1.13788e6i	− 1.46603i	0.680211	−	0.733016i	$-0.261888\pi$
$882$	0	0
$883$	970367.	1.24456	0.622279	−	0.782796i	$-0.286207\pi$
$883$	970367.	1.24456	0.622279	+	0.782796i	$0.286207\pi$
$884$	− 15494.2i	− 0.0198273i
$885$	0	0
$886$	−901257.	−1.14810
$887$	− 1.29729e6i	− 1.64889i	−0.565944	−	0.824444i	$-0.691488\pi$
$887$	− 1.29729e6i	− 1.64889i	0.565944	−	0.824444i	$-0.308512\pi$
$888$	0	0
$889$	−720318.	−0.911425
$890$	44466.1i	0.0561369i
$891$	0	0
$892$	−590181.	−0.741747
$893$	43306.4i	0.0543061i
$894$	0	0
$895$	562261.	0.701926
$896$	1.49794e6i	1.86585i
$897$	0	0
$898$	1.14279e6	1.41714
$899$	828370.i	1.02496i
$900$	0	0
$901$	−32611.4	−0.0401716
$902$	− 1.03148e6i	− 1.26779i
$903$	0	0
$904$	321823.	0.393804
$905$	539275.i	0.658436i
$906$	0	0
$907$	153968.	0.187161	0.0935807	−	0.995612i	$-0.470169\pi$
$907$	153968.	0.187161	0.0935807	+	0.995612i	$0.470169\pi$
$908$	− 1.34099e6i	− 1.62650i
$909$	0	0
$910$	−306880.	−0.370583
$911$	− 1.07095e6i	− 1.29042i	−0.764006	−	0.645209i	$-0.776770\pi$
$911$	− 1.07095e6i	− 1.29042i	0.764006	−	0.645209i	$-0.223230\pi$
$912$	0	0
$913$	1.64536e6	1.97387
$914$	− 755565.i	− 0.904440i
$915$	0	0
$916$	−768220.	−0.915576
$917$	180966.i	0.215208i
$918$	0	0
$919$	747688.	0.885298	0.442649	−	0.896695i	$-0.354039\pi$
$919$	747688.	0.885298	0.442649	+	0.896695i	$0.354039\pi$
$920$	− 337678.i	− 0.398958i
$921$	0	0
$922$	2.65679e6	3.12533
$923$	143562.i	0.168514i
$924$	0	0
$925$	−835761.	−0.976784
$926$	− 475625.i	− 0.554680i
$927$	0	0
$928$	483285.	0.561187
$929$	1.48404e6i	1.71955i	0.510675	+	0.859774i	$0.329396\pi$
$929$	1.48404e6i	1.71955i	−0.510675	+	0.859774i	$0.670604\pi$
$930$	0	0
$931$	−23654.9	−0.0272911
$932$	− 1.18651e6i	− 1.36596i
$933$	0	0
$934$	2.35107e6	2.69509
$935$	− 11353.1i	− 0.0129865i
$936$	0	0
$937$	−650560.	−0.740983	−0.370491	−	0.928836i	$-0.620811\pi$
$937$	−650560.	−0.740983	−0.370491	+	0.928836i	$0.620811\pi$
$938$	1.80902e6i	2.05607i
$939$	0	0
$940$	253913.	0.287362
$941$	406958.i	0.459590i	0.973239	+	0.229795i	$0.0738056\pi$
$941$	406958.i	0.459590i	−0.973239	+	0.229795i	$0.926194\pi$
$942$	0	0
$943$	−503795.	−0.566540
$944$	128476.i	0.144171i
$945$	0	0
$946$	−2.13545e6	−2.38620
$947$	830009.i	0.925514i	0.886485	+	0.462757i	$0.153140\pi$
$947$	830009.i	0.925514i	−0.886485	+	0.462757i	$0.846860\pi$
$948$	0	0
$949$	−70846.6	−0.0786659
$950$	172437.i	0.191066i
$951$	0	0
$952$	−29430.0	−0.0324726
$953$	927863.i	1.02164i	0.859688	+	0.510820i	$0.170658\pi$
$953$	927863.i	1.02164i	−0.859688	+	0.510820i	$0.829342\pi$
$954$	0	0
$955$	550496.	0.603597
$956$	554198.i	0.606386i
$957$	0	0
$958$	−167741.	−0.182771
$959$	− 203672.i	− 0.221460i
$960$	0	0
$961$	1.42401e6	1.54194
$962$	− 840444.i	− 0.908152i
$963$	0	0
$964$	1.56002e6	1.67872
$965$	320589.i	0.344266i
$966$	0	0
$967$	−1.11248e6	−1.18970	−0.594851	−	0.803836i	$-0.702789\pi$
$967$	−1.11248e6	−1.18970	−0.594851	+	0.803836i	$0.702789\pi$
$968$	− 69991.9i	− 0.0746959i
$969$	0	0
$970$	−1.16765e6	−1.24099
$971$	− 68160.6i	− 0.0722927i	−0.999347	−	0.0361464i	$-0.988492\pi$
$971$	− 68160.6i	− 0.0722927i	0.999347	−	0.0361464i	$-0.0115083\pi$
$972$	0	0
$973$	−185210.	−0.195631
$974$	− 1.62018e6i	− 1.70783i
$975$	0	0
$976$	−578.817	−0.000607633 0
$977$	791113.i	0.828799i	0.910095	+	0.414399i	$0.136008\pi$
$977$	791113.i	0.828799i	−0.910095	+	0.414399i	$0.863992\pi$
$978$	0	0
$979$	71974.0	0.0750949
$980$	138693.i	0.144412i
$981$	0	0
$982$	2.24263e6	2.32560
$983$	− 1.09523e6i	− 1.13344i	−0.823912	−	0.566718i	$-0.808213\pi$
$983$	− 1.09523e6i	− 1.13344i	0.823912	−	0.566718i	$-0.191787\pi$
$984$	0	0
$985$	−761950.	−0.785334
$986$	27313.3i	0.0280944i
$987$	0	0
$988$	−108784.	−0.111442
$989$	1.04300e6i	1.06633i
$990$	0	0
$991$	278492.	0.283573	0.141787	−	0.989897i	$-0.454715\pi$
$991$	278492.	0.283573	0.141787	+	0.989897i	$0.454715\pi$
$992$	− 1.36959e6i	− 1.39177i
$993$	0	0
$994$	671666.	0.679799
$995$	304488.i	0.307556i
$996$	0	0
$997$	1.33145e6	1.33947	0.669736	−	0.742599i	$-0.266407\pi$
$997$	1.33145e6	1.33947	0.669736	+	0.742599i	$0.266407\pi$
$998$	1.29633e6i	1.30154i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.5.b.a.80.1		6
3.2	odd	2	inner	81.5.b.a.80.6		6
4.3	odd	2		1296.5.e.c.161.3		6
9.2	odd	6		27.5.d.a.17.3		6
9.4	even	3		27.5.d.a.8.3		6
9.5	odd	6		9.5.d.a.2.1	✓	6
9.7	even	3		9.5.d.a.5.1	yes	6
12.11	even	2		1296.5.e.c.161.4		6
36.7	odd	6		144.5.q.a.113.2		6
36.11	even	6		432.5.q.a.17.2		6
36.23	even	6		144.5.q.a.65.2		6
36.31	odd	6		432.5.q.a.305.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.d.a.2.1	✓	6	9.5	odd	6
9.5.d.a.5.1	yes	6	9.7	even	3
27.5.d.a.8.3		6	9.4	even	3
27.5.d.a.17.3		6	9.2	odd	6
81.5.b.a.80.1		6	1.1	even	1	trivial
81.5.b.a.80.6		6	3.2	odd	2	inner
144.5.q.a.65.2		6	36.23	even	6
144.5.q.a.113.2		6	36.7	odd	6
432.5.q.a.17.2		6	36.11	even	6
432.5.q.a.305.2		6	36.31	odd	6
1296.5.e.c.161.3		6	4.3	odd	2
1296.5.e.c.161.4		6	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	81.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.55250i q^{2} -26.9353 q^{4} -11.7830i q^{5} -53.2728 q^{7} +71.6534i q^{8} -77.2081 q^{10} +124.971i q^{11} -74.6104 q^{13} +349.070i q^{14} +38.5445 q^{16} -7.70989i q^{17} -54.1307 q^{19} +317.378i q^{20} +818.874 q^{22} -399.954i q^{23} +486.161 q^{25} +488.885i q^{26} +1434.92 q^{28} -540.653i q^{29} -1532.17 q^{31} +893.891i q^{32} -50.5190 q^{34} +627.714i q^{35} -1719.10 q^{37} +354.691i q^{38} +844.292 q^{40} -1259.63i q^{41} -2607.79 q^{43} -3366.13i q^{44} -2620.70 q^{46} -800.034i q^{47} +436.996 q^{49} -3185.57i q^{50} +2009.65 q^{52} -4229.81i q^{53} +1472.54 q^{55} -3817.18i q^{56} -3542.63 q^{58} +3333.19i q^{59} -15.0169 q^{61} +10039.5i q^{62} +6473.94 q^{64} +879.134i q^{65} +5182.39 q^{67} +207.668i q^{68} +4113.10 q^{70} -1924.16i q^{71} +949.554 q^{73} +11264.4i q^{74} +1458.02 q^{76} -6657.57i q^{77} -237.980 q^{79} -454.169i q^{80} -8253.75 q^{82} -13165.9i q^{83} -90.8456 q^{85} +17087.5i q^{86} -8954.61 q^{88} -575.925i q^{89} +3974.71 q^{91} +10772.9i q^{92} -5242.22 q^{94} +637.822i q^{95} +15123.4 q^{97} -2863.41i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 30 q^{4} - 24 q^{7} - 36 q^{10} + 12 q^{13} - 30 q^{16} - 258 q^{19} + 738 q^{22} + 546 q^{25} + 1308 q^{28} - 2580 q^{31} - 1026 q^{34} + 12 q^{37} + 2628 q^{40} + 570 q^{43} - 5760 q^{46} + 3726 q^{49}+ \cdots + 57918 q^{97}+O(q^{100}) \) 6 * q - 30 * q^4 - 24 * q^7 - 36 * q^10 + 12 * q^13 - 30 * q^16 - 258 * q^19 + 738 * q^22 + 546 * q^25 + 1308 * q^28 - 2580 * q^31 - 1026 * q^34 + 12 * q^37 + 2628 * q^40 + 570 * q^43 - 5760 * q^46 + 3726 * q^49 + 480 * q^52 + 2016 * q^55 - 12924 * q^58 - 7260 * q^61 + 15450 * q^64 + 10110 * q^67 + 19368 * q^70 - 14622 * q^73 + 8094 * q^76 - 9528 * q^79 - 9702 * q^82 - 24732 * q^85 - 29574 * q^88 + 34836 * q^91 + 25416 * q^94 + 57918 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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