LMFDB - Embedded newform 81.5.b.a.80.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,5,Mod(80,81)]

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

lf = mfeigenbasis(mf)

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")

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

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			80.4
Root			$-0.102534 - 0.177594i$ of defining polynomial
Character	$\chi$	$=$	81.80
Dual form			81.5.b.a.80.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.355188i q^{2} +15.8738 q^{4} +34.7338i q^{5} -31.2109 q^{7} +11.3212i q^{8} -12.3370 q^{10} +57.7314i q^{11} -73.2956 q^{13} -11.0857i q^{14} +249.960 q^{16} +386.985i q^{17} +115.791 q^{19} +551.359i q^{20} -20.5055 q^{22} +548.312i q^{23} -581.437 q^{25} -26.0337i q^{26} -495.436 q^{28} -785.291i q^{29} +544.734 q^{31} +269.922i q^{32} -137.452 q^{34} -1084.07i q^{35} +898.827 q^{37} +41.1275i q^{38} -393.228 q^{40} -2588.85i q^{41} +2000.11 q^{43} +916.419i q^{44} -194.753 q^{46} -811.345i q^{47} -1426.88 q^{49} -206.519i q^{50} -1163.48 q^{52} -2221.00i q^{53} -2005.23 q^{55} -353.344i q^{56} +278.926 q^{58} -1512.26i q^{59} -1902.56 q^{61} +193.483i q^{62} +3903.49 q^{64} -2545.83i q^{65} +4507.09 q^{67} +6142.93i q^{68} +385.049 q^{70} -3993.54i q^{71} -3436.70 q^{73} +319.252i q^{74} +1838.05 q^{76} -1801.85i q^{77} +1202.78 q^{79} +8682.07i q^{80} +919.529 q^{82} +9256.34i q^{83} -13441.4 q^{85} +710.413i q^{86} -653.588 q^{88} +8929.99i q^{89} +2287.62 q^{91} +8703.81i q^{92} +288.180 q^{94} +4021.86i q^{95} +6670.29 q^{97} -506.811i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.355188i	0.0887969i	0.999014	+	0.0443984i	$0.0141371\pi$
$2$	0.355188i	0.0887969i	−0.999014	+	0.0443984i	$0.985863\pi$
$3$	0	0
$3$	0	0
$4$	15.8738	0.992115
$5$	34.7338i	1.38935i	0.719323	+	0.694676i	$0.244452\pi$
$5$	34.7338i	1.38935i	−0.719323	+	0.694676i	$0.755548\pi$
$6$	0	0
$7$	−31.2109	−0.636956	−0.318478	−	0.947930i	$-0.603172\pi$
$7$	−31.2109	−0.636956	−0.318478	+	0.947930i	$0.603172\pi$
$8$	11.3212i	0.176894i
$9$	0	0
$10$	−12.3370	−0.123370
$11$	57.7314i	0.477119i	0.971128	+	0.238559i	$0.0766752\pi$
$11$	57.7314i	0.477119i	−0.971128	+	0.238559i	$0.923325\pi$
$12$	0	0
$13$	−73.2956	−0.433702	−0.216851	−	0.976205i	$-0.569578\pi$
$13$	−73.2956	−0.433702	−0.216851	+	0.976205i	$0.569578\pi$
$14$	− 11.0857i	− 0.0565597i
$15$	0	0
$16$	249.960	0.976408
$17$	386.985i	1.33905i	0.742791	+	0.669524i	$0.233502\pi$
$17$	386.985i	1.33905i	−0.742791	+	0.669524i	$0.766498\pi$
$18$	0	0
$19$	115.791	0.320750	0.160375	−	0.987056i	$-0.448730\pi$
$19$	115.791	0.320750	0.160375	+	0.987056i	$0.448730\pi$
$20$	551.359i	1.37840i
$21$	0	0
$22$	−20.5055	−0.0423667
$23$	548.312i	1.03651i	0.855227	+	0.518253i	$0.173417\pi$
$23$	548.312i	1.03651i	−0.855227	+	0.518253i	$0.826583\pi$
$24$	0	0
$25$	−581.437	−0.930299
$26$	− 26.0337i	− 0.0385113i
$27$	0	0
$28$	−495.436	−0.631934
$29$	− 785.291i	− 0.933758i	−0.884321	−	0.466879i	$-0.845378\pi$
$29$	− 785.291i	− 0.933758i	0.884321	−	0.466879i	$-0.154622\pi$
$30$	0	0
$31$	544.734	0.566840	0.283420	−	0.958996i	$-0.408531\pi$
$31$	544.734	0.566840	0.283420	+	0.958996i	$0.408531\pi$
$32$	269.922i	0.263596i
$33$	0	0
$34$	−137.452	−0.118903
$35$	− 1084.07i	− 0.884957i
$36$	0	0
$37$	898.827	0.656557	0.328279	−	0.944581i	$-0.393532\pi$
$37$	898.827	0.656557	0.328279	+	0.944581i	$0.393532\pi$
$38$	41.1275i	0.0284816i
$39$	0	0
$40$	−393.228	−0.245768
$41$	− 2588.85i	− 1.54007i	−0.638003	−	0.770034i	$-0.720239\pi$
$41$	− 2588.85i	− 1.54007i	0.638003	−	0.770034i	$-0.279761\pi$
$42$	0	0
$43$	2000.11	1.08172	0.540862	−	0.841111i	$-0.318098\pi$
$43$	2000.11	1.08172	0.540862	+	0.841111i	$0.318098\pi$
$44$	916.419i	0.473357i
$45$	0	0
$46$	−194.753	−0.0920385
$47$	− 811.345i	− 0.367291i	−0.982993	−	0.183645i	$-0.941210\pi$
$47$	− 811.345i	− 0.367291i	0.982993	−	0.183645i	$-0.0587898\pi$
$48$	0	0
$49$	−1426.88	−0.594287
$50$	− 206.519i	− 0.0826077i
$51$	0	0
$52$	−1163.48	−0.430282
$53$	− 2221.00i	− 0.790672i	−0.918537	−	0.395336i	$-0.870628\pi$
$53$	− 2221.00i	− 0.790672i	0.918537	−	0.395336i	$-0.129372\pi$
$54$	0	0
$55$	−2005.23	−0.662886
$56$	− 353.344i	− 0.112674i
$57$	0	0
$58$	278.926	0.0829148
$59$	− 1512.26i	− 0.434432i	−0.976124	−	0.217216i	$-0.930302\pi$
$59$	− 1512.26i	− 0.434432i	0.976124	−	0.217216i	$-0.0696976\pi$
$60$	0	0
$61$	−1902.56	−0.511304	−0.255652	−	0.966769i	$-0.582290\pi$
$61$	−1902.56	−0.511304	−0.255652	+	0.966769i	$0.582290\pi$
$62$	193.483i	0.0503337i
$63$	0	0
$64$	3903.49	0.953001
$65$	− 2545.83i	− 0.602564i
$66$	0	0
$67$	4507.09	1.00403	0.502015	−	0.864859i	$-0.332592\pi$
$67$	4507.09	1.00403	0.502015	+	0.864859i	$0.332592\pi$
$68$	6142.93i	1.32849i
$69$	0	0
$70$	385.049	0.0785814
$71$	− 3993.54i	− 0.792213i	−0.918205	−	0.396106i	$-0.870361\pi$
$71$	− 3993.54i	− 0.792213i	0.918205	−	0.396106i	$-0.129639\pi$
$72$	0	0
$73$	−3436.70	−0.644905	−0.322452	−	0.946586i	$-0.604507\pi$
$73$	−3436.70	−0.644905	−0.322452	+	0.946586i	$0.604507\pi$
$74$	319.252i	0.0583002i
$75$	0	0
$76$	1838.05	0.318221
$77$	− 1801.85i	− 0.303904i
$78$	0	0
$79$	1202.78	0.192722	0.0963608	−	0.995346i	$-0.469280\pi$
$79$	1202.78	0.192722	0.0963608	+	0.995346i	$0.469280\pi$
$80$	8682.07i	1.35657i
$81$	0	0
$82$	919.529	0.136753
$83$	9256.34i	1.34364i	0.740714	+	0.671820i	$0.234487\pi$
$83$	9256.34i	1.34364i	−0.740714	+	0.671820i	$0.765513\pi$
$84$	0	0
$85$	−13441.4	−1.86041
$86$	710.413i	0.0960537i
$87$	0	0
$88$	−653.588	−0.0843993
$89$	8929.99i	1.12738i	0.825986	+	0.563691i	$0.190619\pi$
$89$	8929.99i	1.12738i	−0.825986	+	0.563691i	$0.809381\pi$
$90$	0	0
$91$	2287.62	0.276249
$92$	8703.81i	1.02833i
$93$	0	0
$94$	288.180	0.0326143
$95$	4021.86i	0.445635i
$96$	0	0
$97$	6670.29	0.708926	0.354463	−	0.935070i	$-0.384664\pi$
$97$	6670.29	0.708926	0.354463	+	0.935070i	$0.384664\pi$
$98$	− 506.811i	− 0.0527708i
$99$	0	0
$100$	−9229.64	−0.922964
$101$	− 9150.06i	− 0.896977i	−0.893789	−	0.448489i	$-0.851962\pi$
$101$	− 9150.06i	− 0.896977i	0.893789	−	0.448489i	$-0.148038\pi$
$102$	0	0
$103$	15312.4	1.44334	0.721670	−	0.692237i	$-0.243375\pi$
$103$	15312.4	1.44334	0.721670	+	0.692237i	$0.243375\pi$
$104$	− 829.793i	− 0.0767190i
$105$	0	0
$106$	788.870	0.0702092
$107$	− 6099.28i	− 0.532735i	−0.963872	−	0.266367i	$-0.914177\pi$
$107$	− 6099.28i	− 0.532735i	0.963872	−	0.266367i	$-0.0858234\pi$
$108$	0	0
$109$	15169.5	1.27679	0.638393	−	0.769710i	$-0.279599\pi$
$109$	15169.5	1.27679	0.638393	+	0.769710i	$0.279599\pi$
$110$	− 712.233i	− 0.0588622i
$111$	0	0
$112$	−7801.48	−0.621929
$113$	1373.06i	0.107531i	0.998554	+	0.0537655i	$0.0171224\pi$
$113$	1373.06i	0.107531i	−0.998554	+	0.0537655i	$0.982878\pi$
$114$	0	0
$115$	−19045.0	−1.44007
$116$	− 12465.6i	− 0.926396i
$117$	0	0
$118$	537.135	0.0385762
$119$	− 12078.1i	− 0.852914i
$120$	0	0
$121$	11308.1	0.772358
$122$	− 675.767i	− 0.0454022i
$123$	0	0
$124$	8647.01	0.562371
$125$	1513.10i	0.0968386i
$126$	0	0
$127$	−19152.4	−1.18745	−0.593726	−	0.804667i	$-0.702344\pi$
$127$	−19152.4	−1.18745	−0.593726	+	0.804667i	$0.702344\pi$
$128$	5705.22i	0.348219i
$129$	0	0
$130$	904.248	0.0535058
$131$	3020.00i	0.175980i	0.996121	+	0.0879902i	$0.0280444\pi$
$131$	3020.00i	0.175980i	−0.996121	+	0.0879902i	$0.971956\pi$
$132$	0	0
$133$	−3613.93	−0.204304
$134$	1600.86i	0.0891548i
$135$	0	0
$136$	−4381.13	−0.236869
$137$	− 10147.9i	− 0.540671i	−0.962766	−	0.270336i	$-0.912865\pi$
$137$	− 10147.9i	− 0.540671i	0.962766	−	0.270336i	$-0.0871346\pi$
$138$	0	0
$139$	−35126.4	−1.81804	−0.909021	−	0.416751i	$-0.863169\pi$
$139$	−35126.4	−1.81804	−0.909021	+	0.416751i	$0.863169\pi$
$140$	− 17208.4i	− 0.877979i
$141$	0	0
$142$	1418.46	0.0703460
$143$	− 4231.45i	− 0.206927i
$144$	0	0
$145$	27276.1	1.29732
$146$	− 1220.67i	− 0.0572656i
$147$	0	0
$148$	14267.8	0.651380
$149$	− 37382.0i	− 1.68380i	−0.539636	−	0.841899i	$-0.681438\pi$
$149$	− 37382.0i	− 1.68380i	0.539636	−	0.841899i	$-0.318562\pi$
$150$	0	0
$151$	33270.0	1.45915	0.729573	−	0.683903i	$-0.239719\pi$
$151$	33270.0	1.45915	0.729573	+	0.683903i	$0.239719\pi$
$152$	1310.89i	0.0567387i
$153$	0	0
$154$	639.993	0.0269857
$155$	18920.7i	0.787541i
$156$	0	0
$157$	8080.33	0.327816	0.163908	−	0.986476i	$-0.447590\pi$
$157$	8080.33	0.327816	0.163908	+	0.986476i	$0.447590\pi$
$158$	427.211i	0.0171131i
$159$	0	0
$160$	−9375.41	−0.366227
$161$	− 17113.3i	− 0.660209i
$162$	0	0
$163$	−25427.1	−0.957022	−0.478511	−	0.878081i	$-0.658823\pi$
$163$	−25427.1	−0.957022	−0.478511	+	0.878081i	$0.658823\pi$
$164$	− 41095.0i	− 1.52792i
$165$	0	0
$166$	−3287.74	−0.119311
$167$	35052.8i	1.25687i	0.777863	+	0.628434i	$0.216304\pi$
$167$	35052.8i	1.25687i	−0.777863	+	0.628434i	$0.783696\pi$
$168$	0	0
$169$	−23188.8	−0.811903
$170$	− 4774.24i	− 0.165198i
$171$	0	0
$172$	31749.4	1.07319
$173$	14700.5i	0.491177i	0.969374	+	0.245589i	$0.0789813\pi$
$173$	14700.5i	0.491177i	−0.969374	+	0.245589i	$0.921019\pi$
$174$	0	0
$175$	18147.2	0.592560
$176$	14430.6i	0.465862i
$177$	0	0
$178$	−3171.82	−0.100108
$179$	37052.5i	1.15641i	0.815892	+	0.578205i	$0.196247\pi$
$179$	37052.5i	1.15641i	−0.815892	+	0.578205i	$0.803753\pi$
$180$	0	0
$181$	−39664.7	−1.21073	−0.605365	−	0.795948i	$-0.706973\pi$
$181$	−39664.7	−1.21073	−0.605365	+	0.795948i	$0.706973\pi$
$182$	812.533i	0.0245300i
$183$	0	0
$184$	−6207.54	−0.183351
$185$	31219.7i	0.912189i
$186$	0	0
$187$	−22341.2	−0.638885
$188$	− 12879.2i	− 0.364395i
$189$	0	0
$190$	−1428.51	−0.0395710
$191$	57637.0i	1.57992i	0.613160	+	0.789959i	$0.289898\pi$
$191$	57637.0i	1.57992i	−0.613160	+	0.789959i	$0.710102\pi$
$192$	0	0
$193$	4179.63	0.112208	0.0561039	−	0.998425i	$-0.482132\pi$
$193$	4179.63	0.112208	0.0561039	+	0.998425i	$0.482132\pi$
$194$	2369.20i	0.0629504i
$195$	0	0
$196$	−22650.1	−0.589601
$197$	22191.5i	0.571812i	0.958258	+	0.285906i	$0.0922945\pi$
$197$	22191.5i	0.571812i	−0.958258	+	0.285906i	$0.907705\pi$
$198$	0	0
$199$	50608.7	1.27797	0.638983	−	0.769221i	$-0.279355\pi$
$199$	50608.7	1.27797	0.638983	+	0.769221i	$0.279355\pi$
$200$	− 6582.56i	− 0.164564i
$201$	0	0
$202$	3249.99	0.0796488
$203$	24509.6i	0.594763i
$204$	0	0
$205$	89920.7	2.13970
$206$	5438.77i	0.128164i
$207$	0	0
$208$	−18321.0	−0.423469
$209$	6684.77i	0.153036i
$210$	0	0
$211$	−8918.05	−0.200311	−0.100156	−	0.994972i	$-0.531934\pi$
$211$	−8918.05	−0.200311	−0.100156	+	0.994972i	$0.531934\pi$
$212$	− 35255.7i	− 0.784437i
$213$	0	0
$214$	2166.39	0.0473052
$215$	69471.4i	1.50290i
$216$	0	0
$217$	−17001.6	−0.361052
$218$	5388.02i	0.113375i
$219$	0	0
$220$	−31830.7	−0.657659
$221$	− 28364.3i	− 0.580747i
$222$	0	0
$223$	−8497.53	−0.170877	−0.0854384	−	0.996343i	$-0.527229\pi$
$223$	−8497.53	−0.170877	−0.0854384	+	0.996343i	$0.527229\pi$
$224$	− 8424.49i	− 0.167899i
$225$	0	0
$226$	−487.695	−0.00954842
$227$	19102.8i	0.370719i	0.982671	+	0.185360i	$0.0593450\pi$
$227$	19102.8i	0.370719i	−0.982671	+	0.185360i	$0.940655\pi$
$228$	0	0
$229$	−10723.0	−0.204478	−0.102239	−	0.994760i	$-0.532601\pi$
$229$	−10723.0	−0.204478	−0.102239	+	0.994760i	$0.532601\pi$
$230$	− 6764.53i	− 0.127874i
$231$	0	0
$232$	8890.43	0.165176
$233$	− 102200.i	− 1.88251i	−0.337694	−	0.941256i	$-0.609647\pi$
$233$	− 102200.i	− 1.88251i	0.337694	−	0.941256i	$-0.390353\pi$
$234$	0	0
$235$	28181.1	0.510296
$236$	− 24005.3i	− 0.431006i
$237$	0	0
$238$	4290.00	0.0757362
$239$	− 15520.2i	− 0.271707i	−0.990729	−	0.135854i	$-0.956622\pi$
$239$	− 15520.2i	− 0.271707i	0.990729	−	0.135854i	$-0.0433777\pi$
$240$	0	0
$241$	−71957.8	−1.23892	−0.619461	−	0.785028i	$-0.712649\pi$
$241$	−71957.8	−1.23892	−0.619461	+	0.785028i	$0.712649\pi$
$242$	4016.49i	0.0685830i
$243$	0	0
$244$	−30201.0	−0.507272
$245$	− 49561.1i	− 0.825674i
$246$	0	0
$247$	−8486.96	−0.139110
$248$	6167.03i	0.100270i
$249$	0	0
$250$	−537.435	−0.00859897
$251$	41487.8i	0.658526i	0.944238	+	0.329263i	$0.106800\pi$
$251$	41487.8i	0.658526i	−0.944238	+	0.329263i	$0.893200\pi$
$252$	0	0
$253$	−31654.8	−0.494537
$254$	− 6802.70i	− 0.105442i
$255$	0	0
$256$	60429.5	0.922080
$257$	72529.3i	1.09811i	0.835785	+	0.549057i	$0.185013\pi$
$257$	72529.3i	1.09811i	−0.835785	+	0.549057i	$0.814987\pi$
$258$	0	0
$259$	−28053.1	−0.418198
$260$	− 40412.2i	− 0.597813i
$261$	0	0
$262$	−1072.67	−0.0156265
$263$	− 85890.2i	− 1.24174i	−0.783912	−	0.620872i	$-0.786779\pi$
$263$	− 85890.2i	− 1.24174i	0.783912	−	0.620872i	$-0.213221\pi$
$264$	0	0
$265$	77143.7	1.09852
$266$	− 1283.62i	− 0.0181416i
$267$	0	0
$268$	71544.9	0.996114
$269$	− 88967.6i	− 1.22950i	−0.788724	−	0.614748i	$-0.789258\pi$
$269$	− 88967.6i	− 1.22950i	0.788724	−	0.614748i	$-0.210742\pi$
$270$	0	0
$271$	96541.6	1.31455	0.657273	−	0.753652i	$-0.271710\pi$
$271$	96541.6	1.31455	0.657273	+	0.753652i	$0.271710\pi$
$272$	96730.8i	1.30746i
$273$	0	0
$274$	3604.39	0.0480099
$275$	− 33567.2i	− 0.443863i
$276$	0	0
$277$	−23541.6	−0.306815	−0.153407	−	0.988163i	$-0.549025\pi$
$277$	−23541.6	−0.306815	−0.153407	+	0.988163i	$0.549025\pi$
$278$	− 12476.5i	− 0.161436i
$279$	0	0
$280$	12273.0	0.156543
$281$	− 58409.9i	− 0.739731i	−0.929085	−	0.369865i	$-0.879404\pi$
$281$	− 58409.9i	− 0.739731i	0.929085	−	0.369865i	$-0.120596\pi$
$282$	0	0
$283$	−76117.3	−0.950408	−0.475204	−	0.879876i	$-0.657626\pi$
$283$	−76117.3	−0.950408	−0.475204	+	0.879876i	$0.657626\pi$
$284$	− 63392.9i	− 0.785966i
$285$	0	0
$286$	1502.96	0.0183745
$287$	80800.3i	0.980956i
$288$	0	0
$289$	−66236.1	−0.793047
$290$	9688.14i	0.115198i
$291$	0	0
$292$	−54553.6	−0.639820
$293$	− 139911.i	− 1.62973i	−0.579649	−	0.814866i	$-0.696810\pi$
$293$	− 139911.i	− 1.62973i	0.579649	−	0.814866i	$-0.303190\pi$
$294$	0	0
$295$	52526.4	0.603579
$296$	10175.8i	0.116141i
$297$	0	0
$298$	13277.6	0.149516
$299$	− 40188.8i	− 0.449534i
$300$	0	0
$301$	−62425.1	−0.689011
$302$	11817.1i	0.129568i
$303$	0	0
$304$	28943.1	0.313183
$305$	− 66083.2i	− 0.710381i
$306$	0	0
$307$	−81796.8	−0.867880	−0.433940	−	0.900942i	$-0.642877\pi$
$307$	−81796.8	−0.867880	−0.433940	+	0.900942i	$0.642877\pi$
$308$	− 28602.2i	− 0.301508i
$309$	0	0
$310$	−6720.39	−0.0699312
$311$	1702.71i	0.0176043i	0.999961	+	0.00880216i	$0.00280185\pi$
$311$	1702.71i	0.0176043i	−0.999961	+	0.00880216i	$0.997198\pi$
$312$	0	0
$313$	−69960.1	−0.714105	−0.357052	−	0.934084i	$-0.616218\pi$
$313$	−69960.1	−0.714105	−0.357052	+	0.934084i	$0.616218\pi$
$314$	2870.03i	0.0291090i
$315$	0	0
$316$	19092.7	0.191202
$317$	116629.i	1.16062i	0.814397	+	0.580309i	$0.197068\pi$
$317$	116629.i	1.16062i	−0.814397	+	0.580309i	$0.802932\pi$
$318$	0	0
$319$	45335.9	0.445514
$320$	135583.i	1.32405i
$321$	0	0
$322$	6078.42	0.0586245
$323$	44809.3i	0.429500i
$324$	0	0
$325$	42616.8	0.403472
$326$	− 9031.40i	− 0.0849806i
$327$	0	0
$328$	29308.9	0.272428
$329$	25322.8i	0.233948i
$330$	0	0
$331$	100836.	0.920366	0.460183	−	0.887824i	$-0.347784\pi$
$331$	100836.	0.920366	0.460183	+	0.887824i	$0.347784\pi$
$332$	146934.i	1.33305i
$333$	0	0
$334$	−12450.3	−0.111606
$335$	156549.i	1.39495i
$336$	0	0
$337$	40094.5	0.353041	0.176520	−	0.984297i	$-0.443516\pi$
$337$	40094.5	0.353041	0.176520	+	0.984297i	$0.443516\pi$
$338$	− 8236.36i	− 0.0720945i
$339$	0	0
$340$	−213367.	−1.84574
$341$	31448.2i	0.270450i
$342$	0	0
$343$	119471.	1.01549
$344$	22643.6i	0.191350i
$345$	0	0
$346$	−5221.42	−0.0436150
$347$	226362.i	1.87994i	0.341254	+	0.939971i	$0.389148\pi$
$347$	226362.i	1.87994i	−0.341254	+	0.939971i	$0.610852\pi$
$348$	0	0
$349$	79598.3	0.653511	0.326755	−	0.945109i	$-0.394045\pi$
$349$	79598.3	0.653511	0.326755	+	0.945109i	$0.394045\pi$
$350$	6445.64i	0.0526175i
$351$	0	0
$352$	−15583.0	−0.125766
$353$	− 139919.i	− 1.12286i	−0.827523	−	0.561431i	$-0.810251\pi$
$353$	− 139919.i	− 1.12286i	0.827523	−	0.561431i	$-0.189749\pi$
$354$	0	0
$355$	138711.	1.10066
$356$	141753.i	1.11849i
$357$	0	0
$358$	−13160.6	−0.102686
$359$	− 211616.i	− 1.64195i	−0.570963	−	0.820976i	$-0.693430\pi$
$359$	− 211616.i	− 1.64195i	0.570963	−	0.820976i	$-0.306570\pi$
$360$	0	0
$361$	−116913.	−0.897119
$362$	− 14088.4i	− 0.107509i
$363$	0	0
$364$	36313.3	0.274071
$365$	− 119370.i	− 0.896000i
$366$	0	0
$367$	−31993.6	−0.237537	−0.118769	−	0.992922i	$-0.537895\pi$
$367$	−31993.6	−0.237537	−0.118769	+	0.992922i	$0.537895\pi$
$368$	137056.i	1.01205i
$369$	0	0
$370$	−11088.8	−0.0809995
$371$	69319.2i	0.503623i
$372$	0	0
$373$	−123148.	−0.885133	−0.442567	−	0.896736i	$-0.645932\pi$
$373$	−123148.	−0.885133	−0.442567	+	0.896736i	$0.645932\pi$
$374$	− 7935.30i	− 0.0567310i
$375$	0	0
$376$	9185.40	0.0649714
$377$	57558.3i	0.404972i
$378$	0	0
$379$	−116524.	−0.811218	−0.405609	−	0.914047i	$-0.632941\pi$
$379$	−116524.	−0.811218	−0.405609	+	0.914047i	$0.632941\pi$
$380$	63842.4i	0.442122i
$381$	0	0
$382$	−20471.9	−0.140292
$383$	− 181112.i	− 1.23467i	−0.786701	−	0.617334i	$-0.788213\pi$
$383$	− 181112.i	− 1.23467i	0.786701	−	0.617334i	$-0.211787\pi$
$384$	0	0
$385$	62585.0	0.422229
$386$	1484.55i	0.00996371i
$387$	0	0
$388$	105883.	0.703336
$389$	− 50827.4i	− 0.335892i	−0.985796	−	0.167946i	$-0.946287\pi$
$389$	− 50827.4i	− 0.335892i	0.985796	−	0.167946i	$-0.0537134\pi$
$390$	0	0
$391$	−212188.	−1.38793
$392$	− 16154.0i	− 0.105126i
$393$	0	0
$394$	−7882.13	−0.0507751
$395$	41777.0i	0.267758i
$396$	0	0
$397$	228710.	1.45112	0.725561	−	0.688158i	$-0.241580\pi$
$397$	228710.	1.45112	0.725561	+	0.688158i	$0.241580\pi$
$398$	17975.6i	0.113479i
$399$	0	0
$400$	−145336.	−0.908351
$401$	− 188930.i	− 1.17493i	−0.809250	−	0.587465i	$-0.800126\pi$
$401$	− 188930.i	− 1.17493i	0.809250	−	0.587465i	$-0.199874\pi$
$402$	0	0
$403$	−39926.5	−0.245840
$404$	− 145247.i	− 0.889904i
$405$	0	0
$406$	−8705.50	−0.0528131
$407$	51890.5i	0.313256i
$408$	0	0
$409$	277427.	1.65845	0.829223	−	0.558918i	$-0.188783\pi$
$409$	277427.	1.65845	0.829223	+	0.558918i	$0.188783\pi$
$410$	31938.7i	0.189998i
$411$	0	0
$412$	243067.	1.43196
$413$	47198.8i	0.276714i
$414$	0	0
$415$	−321508.	−1.86679
$416$	− 19784.1i	− 0.114322i
$417$	0	0
$418$	−2374.35	−0.0135891
$419$	102422.i	0.583397i	0.956510	+	0.291699i	$0.0942205\pi$
$419$	102422.i	0.583397i	−0.956510	+	0.291699i	$0.905780\pi$
$420$	0	0
$421$	47135.8	0.265942	0.132971	−	0.991120i	$-0.457548\pi$
$421$	47135.8	0.265942	0.132971	+	0.991120i	$0.457548\pi$
$422$	− 3167.58i	− 0.0177870i
$423$	0	0
$424$	25144.3	0.139865
$425$	− 225007.i	− 1.24571i
$426$	0	0
$427$	59380.6	0.325678
$428$	− 96819.0i	− 0.528534i
$429$	0	0
$430$	−24675.4	−0.133452
$431$	− 45556.1i	− 0.245240i	−0.992454	−	0.122620i	$-0.960870\pi$
$431$	− 45556.1i	− 0.245240i	0.992454	−	0.122620i	$-0.0391297\pi$
$432$	0	0
$433$	209599.	1.11793	0.558965	−	0.829192i	$-0.311199\pi$
$433$	209599.	1.11793	0.558965	+	0.829192i	$0.311199\pi$
$434$	− 6038.76i	− 0.0320603i
$435$	0	0
$436$	240798.	1.26672
$437$	63489.5i	0.332460i
$438$	0	0
$439$	−183684.	−0.953110	−0.476555	−	0.879145i	$-0.658115\pi$
$439$	−183684.	−0.953110	−0.476555	+	0.879145i	$0.658115\pi$
$440$	− 22701.6i	− 0.117260i
$441$	0	0
$442$	10074.6	0.0515685
$443$	− 115943.i	− 0.590795i	−0.955374	−	0.295398i	$-0.904548\pi$
$443$	− 115943.i	− 0.590795i	0.955374	−	0.295398i	$-0.0954521\pi$
$444$	0	0
$445$	−310172.	−1.56633
$446$	− 3018.22i	− 0.0151733i
$447$	0	0
$448$	−121831.	−0.607020
$449$	− 328940.i	− 1.63164i	−0.578305	−	0.815820i	$-0.696286\pi$
$449$	− 328940.i	− 1.63164i	0.578305	−	0.815820i	$-0.303714\pi$
$450$	0	0
$451$	149458.	0.734795
$452$	21795.8i	0.106683i
$453$	0	0
$454$	−6785.07	−0.0329187
$455$	79457.6i	0.383807i
$456$	0	0
$457$	−212737.	−1.01862	−0.509308	−	0.860584i	$-0.670099\pi$
$457$	−212737.	−1.01862	−0.509308	+	0.860584i	$0.670099\pi$
$458$	− 3808.68i	− 0.0181570i
$459$	0	0
$460$	−302317.	−1.42872
$461$	− 48448.1i	− 0.227968i	−0.993483	−	0.113984i	$-0.963639\pi$
$461$	− 48448.1i	− 0.227968i	0.993483	−	0.113984i	$-0.0363613\pi$
$462$	0	0
$463$	169101.	0.788832	0.394416	−	0.918932i	$-0.370947\pi$
$463$	169101.	0.788832	0.394416	+	0.918932i	$0.370947\pi$
$464$	− 196292.i	− 0.911729i
$465$	0	0
$466$	36300.1	0.167161
$467$	54646.4i	0.250569i	0.992121	+	0.125285i	$0.0399844\pi$
$467$	54646.4i	0.250569i	−0.992121	+	0.125285i	$0.960016\pi$
$468$	0	0
$469$	−140670.	−0.639524
$470$	10009.6i	0.0453127i
$471$	0	0
$472$	17120.6	0.0768482
$473$	115469.i	0.516111i
$474$	0	0
$475$	−67325.1	−0.298394
$476$	− 191726.i	− 0.846189i
$477$	0	0
$478$	5512.58	0.0241268
$479$	− 138577.i	− 0.603978i	−0.953311	−	0.301989i	$-0.902349\pi$
$479$	− 138577.i	− 0.603978i	0.953311	−	0.301989i	$-0.0976506\pi$
$480$	0	0
$481$	−65880.0	−0.284750
$482$	− 25558.5i	− 0.110012i
$483$	0	0
$484$	179503.	0.766268
$485$	231684.i	0.984948i
$486$	0	0
$487$	23464.1	0.0989340	0.0494670	−	0.998776i	$-0.484248\pi$
$487$	23464.1	0.0989340	0.0494670	+	0.998776i	$0.484248\pi$
$488$	− 21539.3i	− 0.0904464i
$489$	0	0
$490$	17603.5	0.0733172
$491$	− 74326.7i	− 0.308306i	−0.988047	−	0.154153i	$-0.950735\pi$
$491$	− 74326.7i	− 0.308306i	0.988047	−	0.154153i	$-0.0492649\pi$
$492$	0	0
$493$	303895.	1.25035
$494$	− 3014.46i	− 0.0123525i
$495$	0	0
$496$	136162.	0.553467
$497$	124642.i	0.504605i
$498$	0	0
$499$	−18972.0	−0.0761925	−0.0380963	−	0.999274i	$-0.512129\pi$
$499$	−18972.0	−0.0761925	−0.0380963	+	0.999274i	$0.512129\pi$
$500$	24018.8i	0.0960750i
$501$	0	0
$502$	−14735.9	−0.0584751
$503$	117856.i	0.465818i	0.972498	+	0.232909i	$0.0748245\pi$
$503$	117856.i	0.465818i	−0.972498	+	0.232909i	$0.925175\pi$
$504$	0	0
$505$	317816.	1.24622
$506$	− 11243.4i	− 0.0439133i
$507$	0	0
$508$	−304022.	−1.17809
$509$	− 183460.i	− 0.708120i	−0.935223	−	0.354060i	$-0.884801\pi$
$509$	− 183460.i	− 0.708120i	0.935223	−	0.354060i	$-0.115199\pi$
$510$	0	0
$511$	107262.	0.410776
$512$	112747.i	0.430097i
$513$	0	0
$514$	−25761.5	−0.0975091
$515$	531858.i	2.00531i
$516$	0	0
$517$	46840.1	0.175241
$518$	− 9964.13i	− 0.0371347i
$519$	0	0
$520$	28821.9	0.106590
$521$	409498.i	1.50861i	0.656526	+	0.754303i	$0.272025\pi$
$521$	409498.i	1.50861i	−0.656526	+	0.754303i	$0.727975\pi$
$522$	0	0
$523$	−211852.	−0.774513	−0.387256	−	0.921972i	$-0.626577\pi$
$523$	−211852.	−0.774513	−0.387256	+	0.921972i	$0.626577\pi$
$524$	47939.0i	0.174593i
$525$	0	0
$526$	30507.1	0.110263
$527$	210804.i	0.759026i
$528$	0	0
$529$	−20804.7	−0.0743448
$530$	27400.5i	0.0975453i
$531$	0	0
$532$	−57367.0	−0.202693
$533$	189751.i	0.667930i
$534$	0	0
$535$	211851.	0.740156
$536$	51025.7i	0.177607i
$537$	0	0
$538$	31600.2	0.109175
$539$	− 82375.9i	− 0.283545i
$540$	0	0
$541$	44016.5	0.150391	0.0751954	−	0.997169i	$-0.476042\pi$
$541$	44016.5	0.150391	0.0751954	+	0.997169i	$0.476042\pi$
$542$	34290.4i	0.116728i
$543$	0	0
$544$	−104456.	−0.352967
$545$	526894.i	1.77391i
$546$	0	0
$547$	−42859.1	−0.143241	−0.0716207	−	0.997432i	$-0.522817\pi$
$547$	−42859.1	−0.143241	−0.0716207	+	0.997432i	$0.522817\pi$
$548$	− 161085.i	− 0.536408i
$549$	0	0
$550$	11922.6	0.0394137
$551$	− 90929.5i	− 0.299503i
$552$	0	0
$553$	−37539.6	−0.122755
$554$	− 8361.69i	− 0.0272442i
$555$	0	0
$556$	−557591.	−1.80371
$557$	311007.i	1.00244i	0.865319	+	0.501222i	$0.167116\pi$
$557$	311007.i	1.00244i	−0.865319	+	0.501222i	$0.832884\pi$
$558$	0	0
$559$	−146599.	−0.469145
$560$	− 270975.i	− 0.864078i
$561$	0	0
$562$	20746.5	0.0656858
$563$	285768.i	0.901563i	0.892634	+	0.450782i	$0.148855\pi$
$563$	285768.i	0.901563i	−0.892634	+	0.450782i	$0.851145\pi$
$564$	0	0
$565$	−47691.7	−0.149398
$566$	− 27035.9i	− 0.0843933i
$567$	0	0
$568$	45211.7	0.140137
$569$	− 136683.i	− 0.422171i	−0.977468	−	0.211086i	$-0.932300\pi$
$569$	− 136683.i	− 0.422171i	0.977468	−	0.211086i	$-0.0676999\pi$
$570$	0	0
$571$	−28510.0	−0.0874431	−0.0437215	−	0.999044i	$-0.513921\pi$
$571$	−28510.0	−0.0874431	−0.0437215	+	0.999044i	$0.513921\pi$
$572$	− 67169.4i	− 0.205296i
$573$	0	0
$574$	−28699.3	−0.0871058
$575$	− 318809.i	− 0.964261i
$576$	0	0
$577$	−293742.	−0.882297	−0.441149	−	0.897434i	$-0.645429\pi$
$577$	−293742.	−0.882297	−0.441149	+	0.897434i	$0.645429\pi$
$578$	− 23526.2i	− 0.0704201i
$579$	0	0
$580$	432977.	1.28709
$581$	− 288898.i	− 0.855840i
$582$	0	0
$583$	128221.	0.377244
$584$	− 38907.5i	− 0.114080i
$585$	0	0
$586$	49694.6	0.144715
$587$	386055.i	1.12040i	0.828357	+	0.560200i	$0.189276\pi$
$587$	386055.i	1.12040i	−0.828357	+	0.560200i	$0.810724\pi$
$588$	0	0
$589$	63075.2	0.181814
$590$	18656.7i	0.0535959i
$591$	0	0
$592$	224671.	0.641067
$593$	− 305581.i	− 0.868995i	−0.900673	−	0.434497i	$-0.856926\pi$
$593$	− 305581.i	− 0.868995i	0.900673	−	0.434497i	$-0.143074\pi$
$594$	0	0
$595$	419519.	1.18500
$596$	− 593396.i	− 1.67052i
$597$	0	0
$598$	14274.6	0.0399172
$599$	− 264738.i	− 0.737840i	−0.929461	−	0.368920i	$-0.879728\pi$
$599$	− 264738.i	− 0.737840i	0.929461	−	0.368920i	$-0.120272\pi$
$600$	0	0
$601$	−132203.	−0.366009	−0.183005	−	0.983112i	$-0.558582\pi$
$601$	−132203.	−0.366009	−0.183005	+	0.983112i	$0.558582\pi$
$602$	− 22172.6i	− 0.0611820i
$603$	0	0
$604$	528123.	1.44764
$605$	392773.i	1.07308i
$606$	0	0
$607$	−254146.	−0.689772	−0.344886	−	0.938645i	$-0.612082\pi$
$607$	−254146.	−0.689772	−0.344886	+	0.938645i	$0.612082\pi$
$608$	31254.5i	0.0845484i
$609$	0	0
$610$	23471.9	0.0630797
$611$	59468.0i	0.159295i
$612$	0	0
$613$	−492878.	−1.31165	−0.655826	−	0.754912i	$-0.727680\pi$
$613$	−492878.	−1.31165	−0.655826	+	0.754912i	$0.727680\pi$
$614$	− 29053.2i	− 0.0770650i
$615$	0	0
$616$	20399.0	0.0537587
$617$	121278.i	0.318574i	0.987232	+	0.159287i	$0.0509195\pi$
$617$	121278.i	0.318574i	−0.987232	+	0.159287i	$0.949081\pi$
$618$	0	0
$619$	307364.	0.802180	0.401090	−	0.916039i	$-0.368631\pi$
$619$	307364.	0.802180	0.401090	+	0.916039i	$0.368631\pi$
$620$	300344.i	0.781331i
$621$	0	0
$622$	−604.780	−0.00156321
$623$	− 278713.i	− 0.718092i
$624$	0	0
$625$	−415954.	−1.06484
$626$	− 24849.0i	− 0.0634103i
$627$	0	0
$628$	128266.	0.325231
$629$	347832.i	0.879161i
$630$	0	0
$631$	−254196.	−0.638425	−0.319212	−	0.947683i	$-0.603418\pi$
$631$	−254196.	−0.638425	−0.319212	+	0.947683i	$0.603418\pi$
$632$	13616.9i	0.0340912i
$633$	0	0
$634$	−41425.3	−0.103059
$635$	− 665236.i	− 1.64979i
$636$	0	0
$637$	104584.	0.257743
$638$	16102.8i	0.0395602i
$639$	0	0
$640$	−198164.	−0.483799
$641$	365987.i	0.890736i	0.895348	+	0.445368i	$0.146927\pi$
$641$	365987.i	0.890736i	−0.895348	+	0.445368i	$0.853073\pi$
$642$	0	0
$643$	230783.	0.558189	0.279094	−	0.960264i	$-0.409966\pi$
$643$	230783.	0.558189	0.279094	+	0.960264i	$0.409966\pi$
$644$	− 271653.i	− 0.655003i
$645$	0	0
$646$	−15915.7	−0.0381383
$647$	− 278596.i	− 0.665529i	−0.943010	−	0.332764i	$-0.892019\pi$
$647$	− 278596.i	− 0.665529i	0.943010	−	0.332764i	$-0.107981\pi$
$648$	0	0
$649$	87304.7	0.207276
$650$	15136.9i	0.0358271i
$651$	0	0
$652$	−403626.	−0.949476
$653$	78506.8i	0.184111i	0.995754	+	0.0920557i	$0.0293438\pi$
$653$	78506.8i	0.184111i	−0.995754	+	0.0920557i	$0.970656\pi$
$654$	0	0
$655$	−104896.	−0.244499
$656$	− 647111.i	− 1.50373i
$657$	0	0
$658$	−8994.34	−0.0207739
$659$	− 857273.i	− 1.97401i	−0.160703	−	0.987003i	$-0.551376\pi$
$659$	− 857273.i	− 1.97401i	0.160703	−	0.987003i	$-0.448624\pi$
$660$	0	0
$661$	22344.0	0.0511396	0.0255698	−	0.999673i	$-0.491860\pi$
$661$	22344.0	0.0511396	0.0255698	+	0.999673i	$0.491860\pi$
$662$	35815.8i	0.0817257i
$663$	0	0
$664$	−104793.	−0.237681
$665$	− 125526.i	− 0.283850i
$666$	0	0
$667$	430584.	0.967846
$668$	556423.i	1.24696i
$669$	0	0
$670$	−55604.1	−0.123867
$671$	− 109838.i	− 0.243953i
$672$	0	0
$673$	−663478.	−1.46486	−0.732430	−	0.680842i	$-0.761614\pi$
$673$	−663478.	−1.46486	−0.732430	+	0.680842i	$0.761614\pi$
$674$	14241.1i	0.0313489i
$675$	0	0
$676$	−368095.	−0.805501
$677$	− 228121.i	− 0.497722i	−0.968539	−	0.248861i	$-0.919944\pi$
$677$	− 228121.i	− 0.497722i	0.968539	−	0.248861i	$-0.0800563\pi$
$678$	0	0
$679$	−208185.	−0.451555
$680$	− 152173.i	− 0.329094i
$681$	0	0
$682$	−11170.0	−0.0240151
$683$	− 53606.6i	− 0.114915i	−0.998348	−	0.0574575i	$-0.981701\pi$
$683$	− 53606.6i	− 0.114915i	0.998348	−	0.0574575i	$-0.0182994\pi$
$684$	0	0
$685$	352474.	0.751182
$686$	42434.8i	0.0901724i
$687$	0	0
$688$	499948.	1.05620
$689$	162789.i	0.342915i
$690$	0	0
$691$	757326.	1.58609	0.793043	−	0.609166i	$-0.208496\pi$
$691$	757326.	1.58609	0.793043	+	0.609166i	$0.208496\pi$
$692$	233353.i	0.487305i
$693$	0	0
$694$	−80400.9	−0.166933
$695$	− 1.22007e6i	− 2.52590i
$696$	0	0
$697$	1.00185e6	2.06222
$698$	28272.3i	0.0580297i
$699$	0	0
$700$	288065.	0.587888
$701$	− 506359.i	− 1.03044i	−0.857058	−	0.515220i	$-0.827710\pi$
$701$	− 506359.i	− 1.03044i	0.857058	−	0.515220i	$-0.172290\pi$
$702$	0	0
$703$	104076.	0.210591
$704$	225354.i	0.454695i
$705$	0	0
$706$	49697.4	0.0997067
$707$	285581.i	0.571335i
$708$	0	0
$709$	−651243.	−1.29554	−0.647770	−	0.761836i	$-0.724298\pi$
$709$	−651243.	−1.29554	−0.647770	+	0.761836i	$0.724298\pi$
$710$	49268.4i	0.0977354i
$711$	0	0
$712$	−101098.	−0.199427
$713$	298684.i	0.587533i
$714$	0	0
$715$	146974.	0.287495
$716$	588166.i	1.14729i
$717$	0	0
$718$	75163.5	0.145800
$719$	69724.6i	0.134874i	0.997724	+	0.0674370i	$0.0214822\pi$
$719$	69724.6i	0.134874i	−0.997724	+	0.0674370i	$0.978518\pi$
$720$	0	0
$721$	−477913.	−0.919345
$722$	− 41526.2i	− 0.0796614i
$723$	0	0
$724$	−629631.	−1.20118
$725$	456597.i	0.868675i
$726$	0	0
$727$	−747728.	−1.41473	−0.707367	−	0.706846i	$-0.750117\pi$
$727$	−747728.	−1.41473	−0.707367	+	0.706846i	$0.750117\pi$
$728$	25898.6i	0.0488667i
$729$	0	0
$730$	42398.6	0.0795620
$731$	774011.i	1.44848i
$732$	0	0
$733$	65738.5	0.122352	0.0611761	−	0.998127i	$-0.480515\pi$
$733$	65738.5	0.122352	0.0611761	+	0.998127i	$0.480515\pi$
$734$	− 11363.7i	− 0.0210925i
$735$	0	0
$736$	−148001.	−0.273218
$737$	260201.i	0.479042i
$738$	0	0
$739$	977921.	1.79067	0.895334	−	0.445395i	$-0.146937\pi$
$739$	977921.	1.79067	0.895334	+	0.445395i	$0.146937\pi$
$740$	495576.i	0.904996i
$741$	0	0
$742$	−24621.3	−0.0447202
$743$	− 504874.i	− 0.914545i	−0.889327	−	0.457272i	$-0.848826\pi$
$743$	− 504874.i	− 0.914545i	0.889327	−	0.457272i	$-0.151174\pi$
$744$	0	0
$745$	1.29842e6	2.33939
$746$	− 43740.5i	− 0.0785971i
$747$	0	0
$748$	−354640.	−0.633847
$749$	190364.i	0.339329i
$750$	0	0
$751$	−532791.	−0.944663	−0.472332	−	0.881421i	$-0.656588\pi$
$751$	−532791.	−0.944663	−0.472332	+	0.881421i	$0.656588\pi$
$752$	− 202804.i	− 0.358626i
$753$	0	0
$754$	−20444.0	−0.0359603
$755$	1.15559e6i	2.02727i
$756$	0	0
$757$	−293571.	−0.512297	−0.256148	−	0.966637i	$-0.582454\pi$
$757$	−293571.	−0.512297	−0.256148	+	0.966637i	$0.582454\pi$
$758$	− 41388.0i	− 0.0720337i
$759$	0	0
$760$	−45532.2	−0.0788301
$761$	25691.2i	0.0443624i	0.999754	+	0.0221812i	$0.00706108\pi$
$761$	25691.2i	0.0443624i	−0.999754	+	0.0221812i	$0.992939\pi$
$762$	0	0
$763$	−473453.	−0.813257
$764$	914920.i	1.56746i
$765$	0	0
$766$	64328.9	0.109635
$767$	110842.i	0.188414i
$768$	0	0
$769$	−647144.	−1.09433	−0.547165	−	0.837025i	$-0.684293\pi$
$769$	−647144.	−1.09433	−0.547165	+	0.837025i	$0.684293\pi$
$770$	22229.4i	0.0374927i
$771$	0	0
$772$	66346.8	0.111323
$773$	− 610238.i	− 1.02127i	−0.859798	−	0.510634i	$-0.829411\pi$
$773$	− 610238.i	− 1.02127i	0.859798	−	0.510634i	$-0.170589\pi$
$774$	0	0
$775$	−316728.	−0.527331
$776$	75515.6i	0.125405i
$777$	0	0
$778$	18053.3	0.0298261
$779$	− 299766.i	− 0.493977i
$780$	0	0
$781$	230553.	0.377980
$782$	− 75366.6i	− 0.123244i
$783$	0	0
$784$	−356664.	−0.580266
$785$	280661.i	0.455451i
$786$	0	0
$787$	−580321.	−0.936955	−0.468477	−	0.883475i	$-0.655197\pi$
$787$	−580321.	−0.936955	−0.468477	+	0.883475i	$0.655197\pi$
$788$	352264.i	0.567303i
$789$	0	0
$790$	−14838.7	−0.0237761
$791$	− 42854.5i	− 0.0684926i
$792$	0	0
$793$	139449.	0.221753
$794$	81234.9i	0.128855i
$795$	0	0
$796$	803355.	1.26789
$797$	1.05714e6i	1.66424i	0.554598	+	0.832118i	$0.312872\pi$
$797$	1.05714e6i	1.66424i	−0.554598	+	0.832118i	$0.687128\pi$
$798$	0	0
$799$	313978.	0.491820
$800$	− 156943.i	− 0.245223i
$801$	0	0
$802$	67105.5	0.104330
$803$	− 198405.i	− 0.307696i
$804$	0	0
$805$	594409.	0.917263
$806$	− 14181.4i	− 0.0218298i
$807$	0	0
$808$	103590.	0.158670
$809$	355969.i	0.543895i	0.962312	+	0.271947i	$0.0876677\pi$
$809$	355969.i	0.543895i	−0.962312	+	0.271947i	$0.912332\pi$
$810$	0	0
$811$	−136417.	−0.207409	−0.103704	−	0.994608i	$-0.533070\pi$
$811$	−136417.	−0.207409	−0.103704	+	0.994608i	$0.533070\pi$
$812$	389061.i	0.590074i
$813$	0	0
$814$	−18430.9	−0.0278161
$815$	− 883181.i	− 1.32964i
$816$	0	0
$817$	231594.	0.346964
$818$	98538.4i	0.147265i
$819$	0	0
$820$	1.42739e6	2.12282
$821$	− 455979.i	− 0.676485i	−0.941059	−	0.338243i	$-0.890168\pi$
$821$	− 455979.i	− 0.676485i	0.941059	−	0.338243i	$-0.109832\pi$
$822$	0	0
$823$	503590.	0.743494	0.371747	−	0.928334i	$-0.378759\pi$
$823$	503590.	0.743494	0.371747	+	0.928334i	$0.378759\pi$
$824$	173355.i	0.255318i
$825$	0	0
$826$	−16764.4	−0.0245713
$827$	− 551861.i	− 0.806898i	−0.915002	−	0.403449i	$-0.867811\pi$
$827$	− 551861.i	− 0.806898i	0.915002	−	0.403449i	$-0.132189\pi$
$828$	0	0
$829$	−182308.	−0.265275	−0.132638	−	0.991165i	$-0.542345\pi$
$829$	−182308.	−0.265275	−0.132638	+	0.991165i	$0.542345\pi$
$830$	− 114196.i	− 0.165765i
$831$	0	0
$832$	−286109.	−0.413318
$833$	− 552182.i	− 0.795778i
$834$	0	0
$835$	−1.21752e6	−1.74623
$836$	106113.i	0.151829i
$837$	0	0
$838$	−36379.0	−0.0518039
$839$	− 415684.i	− 0.590526i	−0.955416	−	0.295263i	$-0.904593\pi$
$839$	− 415684.i	− 0.590526i	0.955416	−	0.295263i	$-0.0954073\pi$
$840$	0	0
$841$	90599.4	0.128095
$842$	16742.1i	0.0236148i
$843$	0	0
$844$	−141564.	−0.198732
$845$	− 805434.i	− 1.12802i
$846$	0	0
$847$	−352935.	−0.491958
$848$	− 555161.i	− 0.772018i
$849$	0	0
$850$	79919.8	0.110616
$851$	492837.i	0.680525i
$852$	0	0
$853$	150271.	0.206527	0.103263	−	0.994654i	$-0.467072\pi$
$853$	150271.	0.206527	0.103263	+	0.994654i	$0.467072\pi$
$854$	21091.3i	0.0289192i
$855$	0	0
$856$	69051.1	0.0942374
$857$	144768.i	0.197111i	0.995132	+	0.0985554i	$0.0314222\pi$
$857$	144768.i	0.197111i	−0.995132	+	0.0985554i	$0.968578\pi$
$858$	0	0
$859$	333897.	0.452508	0.226254	−	0.974068i	$-0.427352\pi$
$859$	333897.	0.452508	0.226254	+	0.974068i	$0.427352\pi$
$860$	1.10278e6i	1.49105i
$861$	0	0
$862$	16180.9	0.0217766
$863$	− 1.13280e6i	− 1.52101i	−0.649330	−	0.760507i	$-0.724951\pi$
$863$	− 1.13280e6i	− 1.52101i	0.649330	−	0.760507i	$-0.275049\pi$
$864$	0	0
$865$	−510603.	−0.682418
$866$	74447.1i	0.0992686i
$867$	0	0
$868$	−269881.	−0.358206
$869$	69437.9i	0.0919511i
$870$	0	0
$871$	−330350.	−0.435450
$872$	171737.i	0.225855i
$873$	0	0
$874$	−22550.7	−0.0295214
$875$	− 47225.2i	− 0.0616820i
$876$	0	0
$877$	496541.	0.645589	0.322795	−	0.946469i	$-0.395378\pi$
$877$	496541.	0.645589	0.322795	+	0.946469i	$0.395378\pi$
$878$	− 65242.4i	− 0.0846332i
$879$	0	0
$880$	−501228.	−0.647247
$881$	1.21533e6i	1.56583i	0.622131	+	0.782913i	$0.286267\pi$
$881$	1.21533e6i	1.56583i	−0.622131	+	0.782913i	$0.713733\pi$
$882$	0	0
$883$	−999070.	−1.28137	−0.640685	−	0.767804i	$-0.721349\pi$
$883$	−999070.	−1.28137	−0.640685	+	0.767804i	$0.721349\pi$
$884$	− 450250.i	− 0.576168i
$885$	0	0
$886$	41181.5	0.0524608
$887$	− 1.08316e6i	− 1.37672i	−0.725367	−	0.688362i	$-0.758330\pi$
$887$	− 1.08316e6i	− 1.37672i	0.725367	−	0.688362i	$-0.241670\pi$
$888$	0	0
$889$	597763.	0.756355
$890$	− 110169.i	− 0.139085i
$891$	0	0
$892$	−134888.	−0.169529
$893$	− 93946.4i	− 0.117809i
$894$	0	0
$895$	−1.28698e6	−1.60666
$896$	− 178065.i	− 0.221800i
$897$	0	0
$898$	116836.	0.144885
$899$	− 427774.i	− 0.529292i
$900$	0	0
$901$	859491.	1.05875
$902$	53085.6i	0.0652475i
$903$	0	0
$904$	−15544.7	−0.0190216
$905$	− 1.37771e6i	− 1.68213i
$906$	0	0
$907$	1.27685e6	1.55212	0.776061	−	0.630658i	$-0.217215\pi$
$907$	1.27685e6	1.55212	0.776061	+	0.630658i	$0.217215\pi$
$908$	303235.i	0.367796i
$909$	0	0
$910$	−28222.4	−0.0340809
$911$	273486.i	0.329532i	0.986333	+	0.164766i	$0.0526869\pi$
$911$	273486.i	0.329532i	−0.986333	+	0.164766i	$0.947313\pi$
$912$	0	0
$913$	−534381.	−0.641076
$914$	− 75561.6i	− 0.0904500i
$915$	0	0
$916$	−170215.	−0.202865
$917$	− 94256.7i	− 0.112092i
$918$	0	0
$919$	1.15882e6	1.37210	0.686051	−	0.727553i	$-0.259343\pi$
$919$	1.15882e6	1.37210	0.686051	+	0.727553i	$0.259343\pi$
$920$	− 215612.i	− 0.254740i
$921$	0	0
$922$	17208.1	0.0202429
$923$	292709.i	0.343584i
$924$	0	0
$925$	−522611.	−0.610795
$926$	60062.6i	0.0700459i
$927$	0	0
$928$	211967.	0.246135
$929$	852706.i	0.988025i	0.869455	+	0.494012i	$0.164470\pi$
$929$	852706.i	0.988025i	−0.869455	+	0.494012i	$0.835530\pi$
$930$	0	0
$931$	−165220.	−0.190618
$932$	− 1.62230e6i	− 1.86767i
$933$	0	0
$934$	−19409.7	−0.0222498
$935$	− 775993.i	− 0.887636i
$936$	0	0
$937$	629989.	0.717552	0.358776	−	0.933424i	$-0.383194\pi$
$937$	629989.	0.717552	0.358776	+	0.933424i	$0.383194\pi$
$938$	− 49964.3i	− 0.0567877i
$939$	0	0
$940$	447343.	0.506273
$941$	423252.i	0.477991i	0.971021	+	0.238996i	$0.0768182\pi$
$941$	423252.i	0.477991i	−0.971021	+	0.238996i	$0.923182\pi$
$942$	0	0
$943$	1.41950e6	1.59629
$944$	− 378004.i	− 0.424182i
$945$	0	0
$946$	−41013.1	−0.0458290
$947$	− 1.39444e6i	− 1.55489i	−0.628952	−	0.777444i	$-0.716516\pi$
$947$	− 1.39444e6i	− 1.55489i	0.628952	−	0.777444i	$-0.283484\pi$
$948$	0	0
$949$	251895.	0.279696
$950$	− 23913.1i	− 0.0264965i
$951$	0	0
$952$	136739.	0.150875
$953$	− 24001.0i	− 0.0264267i	−0.999913	−	0.0132134i	$-0.995794\pi$
$953$	− 24001.0i	− 0.0264267i	0.999913	−	0.0132134i	$-0.00420607\pi$
$954$	0	0
$955$	−2.00195e6	−2.19506
$956$	− 246365.i	− 0.269565i
$957$	0	0
$958$	49220.9	0.0536314
$959$	316723.i	0.344384i
$960$	0	0
$961$	−626786.	−0.678692
$962$	− 23399.8i	− 0.0252849i
$963$	0	0
$964$	−1.14225e6	−1.22915
$965$	145174.i	0.155896i
$966$	0	0
$967$	1.04406e6	1.11654	0.558268	−	0.829660i	$-0.311466\pi$
$967$	1.04406e6	1.11654	0.558268	+	0.829660i	$0.311466\pi$
$968$	128021.i	0.136625i
$969$	0	0
$970$	−82291.4	−0.0874603
$971$	605213.i	0.641904i	0.947096	+	0.320952i	$0.104003\pi$
$971$	605213.i	0.641904i	−0.947096	+	0.320952i	$0.895997\pi$
$972$	0	0
$973$	1.09632e6	1.15801
$974$	8334.15i	0.00878503i
$975$	0	0
$976$	−475565.	−0.499241
$977$	− 48105.1i	− 0.0503967i	−0.999682	−	0.0251983i	$-0.991978\pi$
$977$	− 48105.1i	− 0.0503967i	0.999682	−	0.0251983i	$-0.00802173\pi$
$978$	0	0
$979$	−515540.	−0.537895
$980$	− 786724.i	− 0.819163i
$981$	0	0
$982$	26399.9	0.0273766
$983$	− 123883.i	− 0.128205i	−0.997943	−	0.0641024i	$-0.979582\pi$
$983$	− 123883.i	− 0.128205i	0.997943	−	0.0641024i	$-0.0204184\pi$
$984$	0	0
$985$	−770794.	−0.794448
$986$	107940.i	0.111027i
$987$	0	0
$988$	−134721.	−0.138013
$989$	1.09668e6i	1.12121i
$990$	0	0
$991$	851418.	0.866953	0.433477	−	0.901165i	$-0.357287\pi$
$991$	851418.	0.866953	0.433477	+	0.901165i	$0.357287\pi$
$992$	147036.i	0.149417i
$993$	0	0
$994$	−44271.3	−0.0448073
$995$	1.75783e6i	1.77555i
$996$	0	0
$997$	−1.60305e6	−1.61271	−0.806355	−	0.591432i	$-0.798563\pi$
$997$	−1.60305e6	−1.61271	−0.806355	+	0.591432i	$0.798563\pi$
$998$	− 6738.62i	− 0.00676566i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.5.b.a.80.4		6
3.2	odd	2	inner	81.5.b.a.80.3		6
4.3	odd	2		1296.5.e.c.161.6		6
9.2	odd	6		27.5.d.a.17.2		6
9.4	even	3		27.5.d.a.8.2		6
9.5	odd	6		9.5.d.a.2.2	✓	6
9.7	even	3		9.5.d.a.5.2	yes	6
12.11	even	2		1296.5.e.c.161.1		6
36.7	odd	6		144.5.q.a.113.1		6
36.11	even	6		432.5.q.a.17.3		6
36.23	even	6		144.5.q.a.65.1		6
36.31	odd	6		432.5.q.a.305.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.d.a.2.2	✓	6	9.5	odd	6
9.5.d.a.5.2	yes	6	9.7	even	3
27.5.d.a.8.2		6	9.4	even	3
27.5.d.a.17.2		6	9.2	odd	6
81.5.b.a.80.3		6	3.2	odd	2	inner
81.5.b.a.80.4		6	1.1	even	1	trivial
144.5.q.a.65.1		6	36.23	even	6
144.5.q.a.113.1		6	36.7	odd	6
432.5.q.a.17.3		6	36.11	even	6
432.5.q.a.305.3		6	36.31	odd	6
1296.5.e.c.161.1		6	12.11	even	2
1296.5.e.c.161.6		6	4.3	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.355188i q^{2} +15.8738 q^{4} +34.7338i q^{5} -31.2109 q^{7} +11.3212i q^{8} -12.3370 q^{10} +57.7314i q^{11} -73.2956 q^{13} -11.0857i q^{14} +249.960 q^{16} +386.985i q^{17} +115.791 q^{19} +551.359i q^{20} -20.5055 q^{22} +548.312i q^{23} -581.437 q^{25} -26.0337i q^{26} -495.436 q^{28} -785.291i q^{29} +544.734 q^{31} +269.922i q^{32} -137.452 q^{34} -1084.07i q^{35} +898.827 q^{37} +41.1275i q^{38} -393.228 q^{40} -2588.85i q^{41} +2000.11 q^{43} +916.419i q^{44} -194.753 q^{46} -811.345i q^{47} -1426.88 q^{49} -206.519i q^{50} -1163.48 q^{52} -2221.00i q^{53} -2005.23 q^{55} -353.344i q^{56} +278.926 q^{58} -1512.26i q^{59} -1902.56 q^{61} +193.483i q^{62} +3903.49 q^{64} -2545.83i q^{65} +4507.09 q^{67} +6142.93i q^{68} +385.049 q^{70} -3993.54i q^{71} -3436.70 q^{73} +319.252i q^{74} +1838.05 q^{76} -1801.85i q^{77} +1202.78 q^{79} +8682.07i q^{80} +919.529 q^{82} +9256.34i q^{83} -13441.4 q^{85} +710.413i q^{86} -653.588 q^{88} +8929.99i q^{89} +2287.62 q^{91} +8703.81i q^{92} +288.180 q^{94} +4021.86i q^{95} +6670.29 q^{97} -506.811i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 30 q^{4} - 24 q^{7} - 36 q^{10} + 12 q^{13} - 30 q^{16} - 258 q^{19} + 738 q^{22} + 546 q^{25} + 1308 q^{28} - 2580 q^{31} - 1026 q^{34} + 12 q^{37} + 2628 q^{40} + 570 q^{43} - 5760 q^{46} + 3726 q^{49}+ \cdots + 57918 q^{97}+O(q^{100}) \) 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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