LMFDB - Embedded newform 27.8.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [27,8,Mod(1,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,9,0,77] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.43439568807$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.53113$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+16.5934 q^{2} +147.340 q^{4} +114.187 q^{5} +1438.40 q^{7} +320.924 q^{8} +1894.75 q^{10} +5928.74 q^{11} -11447.2 q^{13} +23868.0 q^{14} -13534.4 q^{16} -20235.6 q^{17} -6354.94 q^{19} +16824.3 q^{20} +98377.8 q^{22} +75845.6 q^{23} -65086.4 q^{25} -189948. q^{26} +211935. q^{28} -74784.3 q^{29} -189363. q^{31} -265659. q^{32} -335778. q^{34} +164247. q^{35} -33407.2 q^{37} -105450. q^{38} +36645.3 q^{40} +141245. q^{41} -246197. q^{43} +873543. q^{44} +1.25854e6 q^{46} +335133. q^{47} +1.24547e6 q^{49} -1.08000e6 q^{50} -1.68664e6 q^{52} +1.65156e6 q^{53} +676983. q^{55} +461619. q^{56} -1.24092e6 q^{58} +2.04823e6 q^{59} -590469. q^{61} -3.14217e6 q^{62} -2.67579e6 q^{64} -1.30712e6 q^{65} +53575.5 q^{67} -2.98153e6 q^{68} +2.72541e6 q^{70} -4.95678e6 q^{71} +817542. q^{73} -554338. q^{74} -936340. q^{76} +8.52792e6 q^{77} +7.57257e6 q^{79} -1.54545e6 q^{80} +2.34373e6 q^{82} -1.01891e6 q^{83} -2.31064e6 q^{85} -4.08524e6 q^{86} +1.90267e6 q^{88} +1.37281e6 q^{89} -1.64658e7 q^{91} +1.11751e7 q^{92} +5.56099e6 q^{94} -725650. q^{95} -1.06023e7 q^{97} +2.06665e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{2} + 77 q^{4} + 180 q^{5} + 700 q^{7} + 1827 q^{8} + 1395 q^{10} + 10890 q^{11} - 5480 q^{13} + 29475 q^{14} - 15967 q^{16} + 16416 q^{17} + 16024 q^{19} + 12195 q^{20} + 60705 q^{22} + 24372 q^{23}+ \cdots + 22779738 q^{98}+O(q^{100}) $ 2 * q + 9 * q^2 + 77 * q^4 + 180 * q^5 + 700 * q^7 + 1827 * q^8 + 1395 * q^10 + 10890 * q^11 - 5480 * q^13 + 29475 * q^14 - 15967 * q^16 + 16416 * q^17 + 16024 * q^19 + 12195 * q^20 + 60705 * q^22 + 24372 * q^23 - 138880 * q^25 - 235260 * q^26 + 263875 * q^28 - 143280 * q^29 - 38708 * q^31 - 439965 * q^32 - 614088 * q^34 + 115650 * q^35 + 455620 * q^37 - 275382 * q^38 + 135765 * q^40 + 731880 * q^41 - 1088840 * q^43 + 524565 * q^44 + 1649394 * q^46 + 1561500 * q^47 + 967164 * q^49 - 519660 * q^50 - 2106380 * q^52 + 2610468 * q^53 + 1003500 * q^55 - 650475 * q^56 - 720810 * q^58 + 1731960 * q^59 - 620192 * q^61 - 4286151 * q^62 - 1040839 * q^64 - 914400 * q^65 + 346600 * q^67 - 5559624 * q^68 + 3094425 * q^70 - 4242240 * q^71 - 3145190 * q^73 - 4267710 * q^74 - 2510486 * q^76 + 4864500 * q^77 + 10110616 * q^79 - 1705545 * q^80 - 2141190 * q^82 + 644202 * q^83 + 101520 * q^85 + 2313270 * q^86 + 9374715 * q^88 + 6021000 * q^89 - 20872000 * q^91 + 14795802 * q^92 - 3751290 * q^94 + 747180 * q^95 + 4098670 * q^97 + 22779738 * 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 $

$2$	16.5934	1.46666	0.733331	−	0.679872i	$-0.237965\pi$
$2$	16.5934	1.46666	0.733331	+	0.679872i	$0.237965\pi$
$3$	0	0
$3$	0	0
$4$	147.340	1.15110
$5$	114.187	0.408527	0.204264	−	0.978916i	$-0.434520\pi$
$5$	114.187	0.408527	0.204264	+	0.978916i	$0.434520\pi$
$6$	0	0
$7$	1438.40	1.58503	0.792516	−	0.609851i	$-0.208771\pi$
$7$	1438.40	1.58503	0.792516	+	0.609851i	$0.208771\pi$
$8$	320.924	0.221609
$9$	0	0
$10$	1894.75	0.599171
$11$	5928.74	1.34304	0.671518	−	0.740988i	$-0.265643\pi$
$11$	5928.74	1.34304	0.671518	+	0.740988i	$0.265643\pi$
$12$	0	0
$13$	−11447.2	−1.44510	−0.722552	−	0.691317i	$-0.757031\pi$
$13$	−11447.2	−1.44510	−0.722552	+	0.691317i	$0.757031\pi$
$14$	23868.0	2.32471
$15$	0	0
$16$	−13534.4	−0.826072
$17$	−20235.6	−0.998955	−0.499477	−	0.866327i	$-0.666475\pi$
$17$	−20235.6	−0.998955	−0.499477	+	0.866327i	$0.666475\pi$
$18$	0	0
$19$	−6354.94	−0.212556	−0.106278	−	0.994336i	$-0.533893\pi$
$19$	−6354.94	−0.212556	−0.106278	+	0.994336i	$0.533893\pi$
$20$	16824.3	0.470254
$21$	0	0
$22$	98377.8	1.96978
$23$	75845.6	1.29982	0.649910	−	0.760012i	$-0.274807\pi$
$23$	75845.6	1.29982	0.649910	+	0.760012i	$0.274807\pi$
$24$	0	0
$25$	−65086.4	−0.833106
$26$	−189948.	−2.11948
$27$	0	0
$28$	211935.	1.82453
$29$	−74784.3	−0.569400	−0.284700	−	0.958617i	$-0.591894\pi$
$29$	−74784.3	−0.569400	−0.284700	+	0.958617i	$0.591894\pi$
$30$	0	0
$31$	−189363.	−1.14164	−0.570820	−	0.821076i	$-0.693374\pi$
$31$	−189363.	−1.14164	−0.570820	+	0.821076i	$0.693374\pi$
$32$	−265659.	−1.43318
$33$	0	0
$34$	−335778.	−1.46513
$35$	164247.	0.647528
$36$	0	0
$37$	−33407.2	−0.108426	−0.0542130	−	0.998529i	$-0.517265\pi$
$37$	−33407.2	−0.108426	−0.0542130	+	0.998529i	$0.517265\pi$
$38$	−105450.	−0.311748
$39$	0	0
$40$	36645.3	0.0905332
$41$	141245.	0.320058	0.160029	−	0.987112i	$-0.448841\pi$
$41$	141245.	0.320058	0.160029	+	0.987112i	$0.448841\pi$
$42$	0	0
$43$	−246197.	−0.472219	−0.236109	−	0.971726i	$-0.575872\pi$
$43$	−246197.	−0.472219	−0.236109	+	0.971726i	$0.575872\pi$
$44$	873543.	1.54597
$45$	0	0
$46$	1.25854e6	1.90640
$47$	335133.	0.470841	0.235421	−	0.971894i	$-0.424353\pi$
$47$	335133.	0.470841	0.235421	+	0.971894i	$0.424353\pi$
$48$	0	0
$49$	1.24547e6	1.51233
$50$	−1.08000e6	−1.22188
$51$	0	0
$52$	−1.68664e6	−1.66346
$53$	1.65156e6	1.52381	0.761904	−	0.647691i	$-0.224265\pi$
$53$	1.65156e6	1.52381	0.761904	+	0.647691i	$0.224265\pi$
$54$	0	0
$55$	676983.	0.548667
$56$	461619.	0.351257
$57$	0	0
$58$	−1.24092e6	−0.835117
$59$	2.04823e6	1.29836	0.649182	−	0.760633i	$-0.275111\pi$
$59$	2.04823e6	1.29836	0.649182	+	0.760633i	$0.275111\pi$
$60$	0	0
$61$	−590469.	−0.333076	−0.166538	−	0.986035i	$-0.553259\pi$
$61$	−590469.	−0.333076	−0.166538	+	0.986035i	$0.553259\pi$
$62$	−3.14217e6	−1.67440
$63$	0	0
$64$	−2.67579e6	−1.27591
$65$	−1.30712e6	−0.590364
$66$	0	0
$67$	53575.5	0.0217623	0.0108811	−	0.999941i	$-0.496536\pi$
$67$	53575.5	0.0217623	0.0108811	+	0.999941i	$0.496536\pi$
$68$	−2.98153e6	−1.14989
$69$	0	0
$70$	2.72541e6	0.949705
$71$	−4.95678e6	−1.64360	−0.821798	−	0.569779i	$-0.807029\pi$
$71$	−4.95678e6	−1.64360	−0.821798	+	0.569779i	$0.807029\pi$
$72$	0	0
$73$	817542.	0.245969	0.122984	−	0.992409i	$-0.460753\pi$
$73$	817542.	0.245969	0.122984	+	0.992409i	$0.460753\pi$
$74$	−554338.	−0.159024
$75$	0	0
$76$	−936340.	−0.244673
$77$	8.52792e6	2.12875
$78$	0	0
$79$	7.57257e6	1.72802	0.864009	−	0.503476i	$-0.167946\pi$
$79$	7.57257e6	1.72802	0.864009	+	0.503476i	$0.167946\pi$
$80$	−1.54545e6	−0.337473
$81$	0	0
$82$	2.34373e6	0.469417
$83$	−1.01891e6	−0.195597	−0.0977986	−	0.995206i	$-0.531180\pi$
$83$	−1.01891e6	−0.195597	−0.0977986	+	0.995206i	$0.531180\pi$
$84$	0	0
$85$	−2.31064e6	−0.408100
$86$	−4.08524e6	−0.692585
$87$	0	0
$88$	1.90267e6	0.297629
$89$	1.37281e6	0.206418	0.103209	−	0.994660i	$-0.467089\pi$
$89$	1.37281e6	0.206418	0.103209	+	0.994660i	$0.467089\pi$
$90$	0	0
$91$	−1.64658e7	−2.29054
$92$	1.11751e7	1.49622
$93$	0	0
$94$	5.56099e6	0.690565
$95$	−725650.	−0.0868350
$96$	0	0
$97$	−1.06023e7	−1.17950	−0.589750	−	0.807586i	$-0.700774\pi$
$97$	−1.06023e7	−1.17950	−0.589750	+	0.807586i	$0.700774\pi$
$98$	2.06665e7	2.21807
$99$	0	0
$100$	−9.58986e6	−0.958986
$101$	−1.04277e7	−1.00708	−0.503540	−	0.863972i	$-0.667969\pi$
$101$	−1.04277e7	−1.00708	−0.503540	+	0.863972i	$0.667969\pi$
$102$	0	0
$103$	1.53315e6	0.138246	0.0691232	−	0.997608i	$-0.477980\pi$
$103$	1.53315e6	0.138246	0.0691232	+	0.997608i	$0.477980\pi$
$104$	−3.67369e6	−0.320248
$105$	0	0
$106$	2.74050e7	2.23491
$107$	9.64859e6	0.761414	0.380707	−	0.924696i	$-0.375681\pi$
$107$	9.64859e6	0.761414	0.380707	+	0.924696i	$0.375681\pi$
$108$	0	0
$109$	5.79409e6	0.428541	0.214270	−	0.976774i	$-0.431263\pi$
$109$	5.79409e6	0.428541	0.214270	+	0.976774i	$0.431263\pi$
$110$	1.12334e7	0.804708
$111$	0	0
$112$	−1.94679e7	−1.30935
$113$	−1.44843e7	−0.944327	−0.472164	−	0.881511i	$-0.656527\pi$
$113$	−1.44843e7	−0.944327	−0.472164	+	0.881511i	$0.656527\pi$
$114$	0	0
$115$	8.66056e6	0.531011
$116$	−1.10188e7	−0.655435
$117$	0	0
$118$	3.39871e7	1.90426
$119$	−2.91070e7	−1.58338
$120$	0	0
$121$	1.56627e7	0.803746
$122$	−9.79788e6	−0.488509
$123$	0	0
$124$	−2.79008e7	−1.31414
$125$	−1.63528e7	−0.748873
$126$	0	0
$127$	−1.65775e7	−0.718136	−0.359068	−	0.933311i	$-0.616905\pi$
$127$	−1.65775e7	−0.718136	−0.359068	+	0.933311i	$0.616905\pi$
$128$	−1.03960e7	−0.438158
$129$	0	0
$130$	−2.16896e7	−0.865864
$131$	1.44778e7	0.562669	0.281335	−	0.959610i	$-0.409223\pi$
$131$	1.44778e7	0.562669	0.281335	+	0.959610i	$0.409223\pi$
$132$	0	0
$133$	−9.14098e6	−0.336909
$134$	888999.	0.0319179
$135$	0	0
$136$	−6.49411e6	−0.221377
$137$	4.21938e7	1.40193	0.700966	−	0.713194i	$-0.252752\pi$
$137$	4.21938e7	1.40193	0.700966	+	0.713194i	$0.252752\pi$
$138$	0	0
$139$	4.79933e7	1.51575	0.757877	−	0.652398i	$-0.226237\pi$
$139$	4.79933e7	1.51575	0.757877	+	0.652398i	$0.226237\pi$
$140$	2.42002e7	0.745368
$141$	0	0
$142$	−8.22497e7	−2.41060
$143$	−6.78676e7	−1.94083
$144$	0	0
$145$	−8.53938e6	−0.232615
$146$	1.35658e7	0.360753
$147$	0	0
$148$	−4.92223e6	−0.124809
$149$	1.15668e7	0.286457	0.143229	−	0.989690i	$-0.454252\pi$
$149$	1.15668e7	0.286457	0.143229	+	0.989690i	$0.454252\pi$
$150$	0	0
$151$	1.09933e7	0.259843	0.129921	−	0.991524i	$-0.458528\pi$
$151$	1.09933e7	0.259843	0.129921	+	0.991524i	$0.458528\pi$
$152$	−2.03945e6	−0.0471044
$153$	0	0
$154$	1.41507e8	3.12216
$155$	−2.16227e7	−0.466390
$156$	0	0
$157$	2.48018e7	0.511487	0.255744	−	0.966745i	$-0.417680\pi$
$157$	2.48018e7	0.511487	0.255744	+	0.966745i	$0.417680\pi$
$158$	1.25655e8	2.53442
$159$	0	0
$160$	−3.03348e7	−0.585492
$161$	1.09097e8	2.06025
$162$	0	0
$163$	3.04106e6	0.0550008	0.0275004	−	0.999622i	$-0.491245\pi$
$163$	3.04106e6	0.0550008	0.0275004	+	0.999622i	$0.491245\pi$
$164$	2.08111e7	0.368418
$165$	0	0
$166$	−1.69072e7	−0.286875
$167$	−1.06849e8	−1.77526	−0.887629	−	0.460559i	$-0.847649\pi$
$167$	−1.06849e8	−1.77526	−0.887629	+	0.460559i	$0.847649\pi$
$168$	0	0
$169$	6.82907e7	1.08832
$170$	−3.83414e7	−0.598545
$171$	0	0
$172$	−3.62748e7	−0.543570
$173$	9.94936e7	1.46094	0.730472	−	0.682942i	$-0.239300\pi$
$173$	9.94936e7	1.46094	0.730472	+	0.682942i	$0.239300\pi$
$174$	0	0
$175$	−9.36206e7	−1.32050
$176$	−8.02417e7	−1.10944
$177$	0	0
$178$	2.27796e7	0.302745
$179$	2.29173e7	0.298661	0.149330	−	0.988787i	$-0.452288\pi$
$179$	2.29173e7	0.298661	0.149330	+	0.988787i	$0.452288\pi$
$180$	0	0
$181$	−1.35824e8	−1.70256	−0.851280	−	0.524712i	$-0.824173\pi$
$181$	−1.35824e8	−1.70256	−0.851280	+	0.524712i	$0.824173\pi$
$182$	−2.73223e8	−3.35944
$183$	0	0
$184$	2.43407e7	0.288052
$185$	−3.81465e6	−0.0442949
$186$	0	0
$187$	−1.19972e8	−1.34163
$188$	4.93786e7	0.541984
$189$	0	0
$190$	−1.20410e7	−0.127358
$191$	1.11919e8	1.16221	0.581107	−	0.813827i	$-0.302620\pi$
$191$	1.11919e8	1.16221	0.581107	+	0.813827i	$0.302620\pi$
$192$	0	0
$193$	1.02898e8	1.03029	0.515143	−	0.857104i	$-0.327739\pi$
$193$	1.02898e8	1.03029	0.515143	+	0.857104i	$0.327739\pi$
$194$	−1.75928e8	−1.72993
$195$	0	0
$196$	1.83507e8	1.74083
$197$	3.77994e7	0.352252	0.176126	−	0.984368i	$-0.443643\pi$
$197$	3.77994e7	0.352252	0.176126	+	0.984368i	$0.443643\pi$
$198$	0	0
$199$	−49088.1	−0.000441560 0	−0.000220780	−	1.00000i	$-0.500070\pi$
$199$	−49088.1	−0.000441560 0	−0.000220780		1.00000i	$0.500070\pi$
$200$	−2.08878e7	−0.184624
$201$	0	0
$202$	−1.73031e8	−1.47704
$203$	−1.07570e8	−0.902517
$204$	0	0
$205$	1.61283e7	0.130752
$206$	2.54401e7	0.202761
$207$	0	0
$208$	1.54931e8	1.19376
$209$	−3.76768e7	−0.285471
$210$	0	0
$211$	1.88572e8	1.38194	0.690969	−	0.722885i	$-0.257184\pi$
$211$	1.88572e8	1.38194	0.690969	+	0.722885i	$0.257184\pi$
$212$	2.43342e8	1.75405
$213$	0	0
$214$	1.60103e8	1.11674
$215$	−2.81125e7	−0.192914
$216$	0	0
$217$	−2.72380e8	−1.80953
$218$	9.61435e7	0.628525
$219$	0	0
$220$	9.97470e7	0.631569
$221$	2.31642e8	1.44359
$222$	0	0
$223$	1.05676e8	0.638127	0.319064	−	0.947733i	$-0.396632\pi$
$223$	1.05676e8	0.638127	0.319064	+	0.947733i	$0.396632\pi$
$224$	−3.82125e8	−2.27163
$225$	0	0
$226$	−2.40343e8	−1.38501
$227$	−4.59251e7	−0.260591	−0.130295	−	0.991475i	$-0.541593\pi$
$227$	−4.59251e7	−0.260591	−0.130295	+	0.991475i	$0.541593\pi$
$228$	0	0
$229$	−1.22895e8	−0.676254	−0.338127	−	0.941100i	$-0.609793\pi$
$229$	−1.22895e8	−0.676254	−0.338127	+	0.941100i	$0.609793\pi$
$230$	1.43708e8	0.778814
$231$	0	0
$232$	−2.40001e7	−0.126184
$233$	1.68276e8	0.871519	0.435760	−	0.900063i	$-0.356480\pi$
$233$	1.68276e8	0.871519	0.435760	+	0.900063i	$0.356480\pi$
$234$	0	0
$235$	3.82677e7	0.192351
$236$	3.01787e8	1.49454
$237$	0	0
$238$	−4.82985e8	−2.32228
$239$	8.02527e7	0.380248	0.190124	−	0.981760i	$-0.439111\pi$
$239$	8.02527e7	0.380248	0.190124	+	0.981760i	$0.439111\pi$
$240$	0	0
$241$	1.49328e8	0.687198	0.343599	−	0.939116i	$-0.388354\pi$
$241$	1.49328e8	0.687198	0.343599	+	0.939116i	$0.388354\pi$
$242$	2.59898e8	1.17882
$243$	0	0
$244$	−8.70000e7	−0.383402
$245$	1.42216e8	0.617826
$246$	0	0
$247$	7.27466e7	0.307166
$248$	−6.07711e7	−0.252997
$249$	0	0
$250$	−2.71349e8	−1.09834
$251$	−4.54290e8	−1.81332	−0.906661	−	0.421861i	$-0.861377\pi$
$251$	−4.54290e8	−1.81332	−0.906661	+	0.421861i	$0.861377\pi$
$252$	0	0
$253$	4.49668e8	1.74570
$254$	−2.75077e8	−1.05326
$255$	0	0
$256$	1.69996e8	0.633284
$257$	−2.92144e7	−0.107357	−0.0536786	−	0.998558i	$-0.517095\pi$
$257$	−2.92144e7	−0.107357	−0.0536786	+	0.998558i	$0.517095\pi$
$258$	0	0
$259$	−4.80530e7	−0.171859
$260$	−1.92592e8	−0.679566
$261$	0	0
$262$	2.40236e8	0.825246
$263$	−4.49406e8	−1.52333	−0.761664	−	0.647972i	$-0.775617\pi$
$263$	−4.49406e8	−1.52333	−0.761664	+	0.647972i	$0.775617\pi$
$264$	0	0
$265$	1.88587e8	0.622516
$266$	−1.51680e8	−0.494131
$267$	0	0
$268$	7.89384e6	0.0250505
$269$	−1.94325e7	−0.0608689	−0.0304345	−	0.999537i	$-0.509689\pi$
$269$	−1.94325e7	−0.0608689	−0.0304345	+	0.999537i	$0.509689\pi$
$270$	0	0
$271$	−2.13963e7	−0.0653051	−0.0326525	−	0.999467i	$-0.510395\pi$
$271$	−2.13963e7	−0.0653051	−0.0326525	+	0.999467i	$0.510395\pi$
$272$	2.73877e8	0.825209
$273$	0	0
$274$	7.00139e8	2.05616
$275$	−3.85880e8	−1.11889
$276$	0	0
$277$	−1.85015e8	−0.523031	−0.261515	−	0.965199i	$-0.584222\pi$
$277$	−1.85015e8	−0.523031	−0.261515	+	0.965199i	$0.584222\pi$
$278$	7.96371e8	2.22310
$279$	0	0
$280$	5.27107e7	0.143498
$281$	−3.62351e7	−0.0974221	−0.0487111	−	0.998813i	$-0.515511\pi$
$281$	−3.62351e7	−0.0974221	−0.0487111	+	0.998813i	$0.515511\pi$
$282$	0	0
$283$	−3.96100e8	−1.03885	−0.519424	−	0.854517i	$-0.673853\pi$
$283$	−3.96100e8	−1.03885	−0.519424	+	0.854517i	$0.673853\pi$
$284$	−7.30334e8	−1.89194
$285$	0	0
$286$	−1.12615e9	−2.84654
$287$	2.03167e8	0.507303
$288$	0	0
$289$	−857331.	−0.00208932
$290$	−1.41697e8	−0.341168
$291$	0	0
$292$	1.20457e8	0.283134
$293$	3.96468e8	0.920812	0.460406	−	0.887708i	$-0.347704\pi$
$293$	3.96468e8	0.920812	0.460406	+	0.887708i	$0.347704\pi$
$294$	0	0
$295$	2.33881e8	0.530417
$296$	−1.07212e7	−0.0240282
$297$	0	0
$298$	1.91932e8	0.420136
$299$	−8.68223e8	−1.87837
$300$	0	0
$301$	−3.54131e8	−0.748482
$302$	1.82417e8	0.381101
$303$	0	0
$304$	8.60101e7	0.175587
$305$	−6.74238e7	−0.136070
$306$	0	0
$307$	−3.12263e8	−0.615936	−0.307968	−	0.951397i	$-0.599649\pi$
$307$	−3.12263e8	−0.615936	−0.307968	+	0.951397i	$0.599649\pi$
$308$	1.25651e9	2.45040
$309$	0	0
$310$	−3.58794e8	−0.684037
$311$	2.82908e8	0.533316	0.266658	−	0.963791i	$-0.414081\pi$
$311$	2.82908e8	0.533316	0.266658	+	0.963791i	$0.414081\pi$
$312$	0	0
$313$	−9.55802e8	−1.76182	−0.880912	−	0.473280i	$-0.843070\pi$
$313$	−9.55802e8	−1.76182	−0.880912	+	0.473280i	$0.843070\pi$
$314$	4.11546e8	0.750179
$315$	0	0
$316$	1.11575e9	1.98912
$317$	4.13052e8	0.728278	0.364139	−	0.931345i	$-0.381363\pi$
$317$	4.13052e8	0.728278	0.364139	+	0.931345i	$0.381363\pi$
$318$	0	0
$319$	−4.43376e8	−0.764724
$320$	−3.05540e8	−0.521246
$321$	0	0
$322$	1.81028e9	3.02170
$323$	1.28596e8	0.212334
$324$	0	0
$325$	7.45059e8	1.20392
$326$	5.04615e7	0.0806676
$327$	0	0
$328$	4.53289e7	0.0709278
$329$	4.82057e8	0.746298
$330$	0	0
$331$	2.28880e8	0.346905	0.173452	−	0.984842i	$-0.444508\pi$
$331$	2.28880e8	0.346905	0.173452	+	0.984842i	$0.444508\pi$
$332$	−1.50127e8	−0.225151
$333$	0	0
$334$	−1.77298e9	−2.60370
$335$	6.11761e6	0.00889048
$336$	0	0
$337$	9.78074e8	1.39209	0.696044	−	0.717999i	$-0.254942\pi$
$337$	9.78074e8	1.39209	0.696044	+	0.717999i	$0.254942\pi$
$338$	1.13317e9	1.59620
$339$	0	0
$340$	−3.40451e8	−0.469763
$341$	−1.12268e9	−1.53326
$342$	0	0
$343$	6.06895e8	0.812053
$344$	−7.90106e7	−0.104648
$345$	0	0
$346$	1.65094e9	2.14271
$347$	−1.42099e8	−0.182573	−0.0912867	−	0.995825i	$-0.529098\pi$
$347$	−1.42099e8	−0.182573	−0.0912867	+	0.995825i	$0.529098\pi$
$348$	0	0
$349$	−4.63334e8	−0.583452	−0.291726	−	0.956502i	$-0.594230\pi$
$349$	−4.63334e8	−0.583452	−0.291726	+	0.956502i	$0.594230\pi$
$350$	−1.55348e9	−1.93673
$351$	0	0
$352$	−1.57502e9	−1.92481
$353$	−2.20636e8	−0.266971	−0.133486	−	0.991051i	$-0.542617\pi$
$353$	−2.20636e8	−0.266971	−0.133486	+	0.991051i	$0.542617\pi$
$354$	0	0
$355$	−5.65998e8	−0.671454
$356$	2.02271e8	0.237607
$357$	0	0
$358$	3.80276e8	0.438035
$359$	2.57110e8	0.293284	0.146642	−	0.989190i	$-0.453153\pi$
$359$	2.57110e8	0.293284	0.146642	+	0.989190i	$0.453153\pi$
$360$	0	0
$361$	−8.53486e8	−0.954820
$362$	−2.25378e9	−2.49708
$363$	0	0
$364$	−2.42607e9	−2.63663
$365$	9.33525e7	0.100485
$366$	0	0
$367$	7.22719e8	0.763200	0.381600	−	0.924328i	$-0.375373\pi$
$367$	7.22719e8	0.763200	0.381600	+	0.924328i	$0.375373\pi$
$368$	−1.02652e9	−1.07374
$369$	0	0
$370$	−6.32980e7	−0.0649657
$371$	2.37562e9	2.41528
$372$	0	0
$373$	−5.89159e8	−0.587829	−0.293915	−	0.955832i	$-0.594958\pi$
$373$	−5.89159e8	−0.587829	−0.293915	+	0.955832i	$0.594958\pi$
$374$	−1.99074e9	−1.96772
$375$	0	0
$376$	1.07552e8	0.104343
$377$	8.56073e8	0.822842
$378$	0	0
$379$	9.02541e8	0.851589	0.425794	−	0.904820i	$-0.359995\pi$
$379$	9.02541e8	0.851589	0.425794	+	0.904820i	$0.359995\pi$
$380$	−1.06918e8	−0.0999556
$381$	0	0
$382$	1.85711e9	1.70458
$383$	5.59375e8	0.508754	0.254377	−	0.967105i	$-0.418130\pi$
$383$	5.59375e8	0.508754	0.254377	+	0.967105i	$0.418130\pi$
$384$	0	0
$385$	9.73776e8	0.869654
$386$	1.70743e9	1.51108
$387$	0	0
$388$	−1.56215e9	−1.35772
$389$	−1.34140e9	−1.15541	−0.577705	−	0.816246i	$-0.696052\pi$
$389$	−1.34140e9	−1.15541	−0.577705	+	0.816246i	$0.696052\pi$
$390$	0	0
$391$	−1.53478e9	−1.29846
$392$	3.99700e8	0.335145
$393$	0	0
$394$	6.27221e8	0.516635
$395$	8.64687e8	0.705942
$396$	0	0
$397$	2.24763e9	1.80285	0.901423	−	0.432939i	$-0.142523\pi$
$397$	2.24763e9	1.80285	0.901423	+	0.432939i	$0.142523\pi$
$398$	−814537.	−0.000647620 0
$399$	0	0
$400$	8.80903e8	0.688205
$401$	−2.40503e9	−1.86258	−0.931289	−	0.364280i	$-0.881315\pi$
$401$	−2.40503e9	−1.86258	−0.931289	+	0.364280i	$0.881315\pi$
$402$	0	0
$403$	2.16768e9	1.64979
$404$	−1.53642e9	−1.15925
$405$	0	0
$406$	−1.78495e9	−1.32369
$407$	−1.98062e8	−0.145620
$408$	0	0
$409$	7.63994e8	0.552152	0.276076	−	0.961136i	$-0.410966\pi$
$409$	7.63994e8	0.552152	0.276076	+	0.961136i	$0.410966\pi$
$410$	2.67623e8	0.191770
$411$	0	0
$412$	2.25895e8	0.159135
$413$	2.94618e9	2.05795
$414$	0	0
$415$	−1.16346e8	−0.0799068
$416$	3.04106e9	2.07109
$417$	0	0
$418$	−6.25185e8	−0.418689
$419$	2.88611e8	0.191675	0.0958373	−	0.995397i	$-0.469447\pi$
$419$	2.88611e8	0.191675	0.0958373	+	0.995397i	$0.469447\pi$
$420$	0	0
$421$	−1.55888e9	−1.01818	−0.509090	−	0.860713i	$-0.670018\pi$
$421$	−1.55888e9	−1.01818	−0.509090	+	0.860713i	$0.670018\pi$
$422$	3.12905e9	2.02684
$423$	0	0
$424$	5.30027e8	0.337689
$425$	1.31706e9	0.832235
$426$	0	0
$427$	−8.49334e8	−0.527935
$428$	1.42163e9	0.876461
$429$	0	0
$430$	−4.66481e8	−0.282940
$431$	−9.87526e8	−0.594125	−0.297063	−	0.954858i	$-0.596007\pi$
$431$	−9.87526e8	−0.594125	−0.297063	+	0.954858i	$0.596007\pi$
$432$	0	0
$433$	−2.07048e9	−1.22564	−0.612820	−	0.790223i	$-0.709965\pi$
$433$	−2.07048e9	−1.22564	−0.612820	+	0.790223i	$0.709965\pi$
$434$	−4.51971e9	−2.65398
$435$	0	0
$436$	8.53703e8	0.493292
$437$	−4.81994e8	−0.276285
$438$	0	0
$439$	−3.10766e9	−1.75311	−0.876553	−	0.481306i	$-0.840163\pi$
$439$	−3.10766e9	−1.75311	−0.876553	+	0.481306i	$0.840163\pi$
$440$	2.17260e8	0.121589
$441$	0	0
$442$	3.84373e9	2.11726
$443$	1.77729e9	0.971280	0.485640	−	0.874159i	$-0.338587\pi$
$443$	1.77729e9	0.971280	0.485640	+	0.874159i	$0.338587\pi$
$444$	0	0
$445$	1.56757e8	0.0843271
$446$	1.75352e9	0.935917
$447$	0	0
$448$	−3.84887e9	−2.02237
$449$	−3.38966e9	−1.76723	−0.883616	−	0.468212i	$-0.844898\pi$
$449$	−3.38966e9	−1.76723	−0.883616	+	0.468212i	$0.844898\pi$
$450$	0	0
$451$	8.37404e8	0.429850
$452$	−2.13412e9	−1.08701
$453$	0	0
$454$	−7.62052e8	−0.382199
$455$	−1.88017e9	−0.935746
$456$	0	0
$457$	−2.09644e9	−1.02749	−0.513744	−	0.857944i	$-0.671742\pi$
$457$	−2.09644e9	−1.02749	−0.513744	+	0.857944i	$0.671742\pi$
$458$	−2.03924e9	−0.991836
$459$	0	0
$460$	1.27605e9	0.611246
$461$	1.78459e9	0.848371	0.424185	−	0.905575i	$-0.360560\pi$
$461$	1.78459e9	0.848371	0.424185	+	0.905575i	$0.360560\pi$
$462$	0	0
$463$	2.29800e9	1.07601	0.538005	−	0.842942i	$-0.319178\pi$
$463$	2.29800e9	1.07601	0.538005	+	0.842942i	$0.319178\pi$
$464$	1.01216e9	0.470365
$465$	0	0
$466$	2.79227e9	1.27822
$467$	−8.59979e8	−0.390732	−0.195366	−	0.980730i	$-0.562589\pi$
$467$	−8.59979e8	−0.390732	−0.195366	+	0.980730i	$0.562589\pi$
$468$	0	0
$469$	7.70632e7	0.0344939
$470$	6.34991e8	0.282114
$471$	0	0
$472$	6.57326e8	0.287729
$473$	−1.45964e9	−0.634207
$474$	0	0
$475$	4.13620e8	0.177082
$476$	−4.28865e9	−1.82262
$477$	0	0
$478$	1.33166e9	0.557695
$479$	3.06607e7	0.0127470	0.00637349	−	0.999980i	$-0.497971\pi$
$479$	3.06607e7	0.0127470	0.00637349	+	0.999980i	$0.497971\pi$
$480$	0	0
$481$	3.82420e8	0.156687
$482$	2.47786e9	1.00789
$483$	0	0
$484$	2.30775e9	0.925190
$485$	−1.21064e9	−0.481858
$486$	0	0
$487$	−5.78717e8	−0.227047	−0.113523	−	0.993535i	$-0.536214\pi$
$487$	−5.78717e8	−0.227047	−0.113523	+	0.993535i	$0.536214\pi$
$488$	−1.89496e8	−0.0738125
$489$	0	0
$490$	2.35984e9	0.906142
$491$	3.29719e9	1.25707	0.628534	−	0.777782i	$-0.283655\pi$
$491$	3.29719e9	1.25707	0.628534	+	0.777782i	$0.283655\pi$
$492$	0	0
$493$	1.51331e9	0.568805
$494$	1.20711e9	0.450509
$495$	0	0
$496$	2.56291e9	0.943076
$497$	−7.12985e9	−2.60515
$498$	0	0
$499$	−4.27181e9	−1.53907	−0.769537	−	0.638602i	$-0.779513\pi$
$499$	−4.27181e9	−1.53907	−0.769537	+	0.638602i	$0.779513\pi$
$500$	−2.40944e9	−0.862026
$501$	0	0
$502$	−7.53820e9	−2.65953
$503$	1.85857e9	0.651165	0.325582	−	0.945514i	$-0.394440\pi$
$503$	1.85857e9	0.651165	0.325582	+	0.945514i	$0.394440\pi$
$504$	0	0
$505$	−1.19071e9	−0.411419
$506$	7.46152e9	2.56036
$507$	0	0
$508$	−2.44254e9	−0.826645
$509$	−2.12609e9	−0.714610	−0.357305	−	0.933988i	$-0.616304\pi$
$509$	−2.12609e9	−0.714610	−0.357305	+	0.933988i	$0.616304\pi$
$510$	0	0
$511$	1.17596e9	0.389868
$512$	4.15150e9	1.36697
$513$	0	0
$514$	−4.84766e8	−0.157457
$515$	1.75065e8	0.0564774
$516$	0	0
$517$	1.98691e9	0.632357
$518$	−7.97362e8	−0.252059
$519$	0	0
$520$	−4.19487e8	−0.130830
$521$	−5.97837e9	−1.85204	−0.926021	−	0.377472i	$-0.876793\pi$
$521$	−5.97837e9	−1.85204	−0.926021	+	0.377472i	$0.876793\pi$
$522$	0	0
$523$	5.69436e8	0.174056	0.0870280	−	0.996206i	$-0.472263\pi$
$523$	5.69436e8	0.174056	0.0870280	+	0.996206i	$0.472263\pi$
$524$	2.13317e9	0.647687
$525$	0	0
$526$	−7.45717e9	−2.23421
$527$	3.83188e9	1.14045
$528$	0	0
$529$	2.34773e9	0.689530
$530$	3.12929e9	0.913021
$531$	0	0
$532$	−1.34684e9	−0.387815
$533$	−1.61686e9	−0.462518
$534$	0	0
$535$	1.10174e9	0.311058
$536$	1.71937e7	0.00482271
$537$	0	0
$538$	−3.22451e8	−0.0892742
$539$	7.38403e9	2.03111
$540$	0	0
$541$	−1.88830e9	−0.512722	−0.256361	−	0.966581i	$-0.582524\pi$
$541$	−1.88830e9	−0.512722	−0.256361	+	0.966581i	$0.582524\pi$
$542$	−3.55038e8	−0.0957805
$543$	0	0
$544$	5.37579e9	1.43168
$545$	6.61608e8	0.175070
$546$	0	0
$547$	5.88629e8	0.153775	0.0768875	−	0.997040i	$-0.475502\pi$
$547$	5.88629e8	0.153775	0.0768875	+	0.997040i	$0.475502\pi$
$548$	6.21686e9	1.61376
$549$	0	0
$550$	−6.40305e9	−1.64103
$551$	4.75250e8	0.121030
$552$	0	0
$553$	1.08924e10	2.73897
$554$	−3.07002e9	−0.767109
$555$	0	0
$556$	7.07135e9	1.74478
$557$	5.59795e9	1.37257	0.686287	−	0.727331i	$-0.259239\pi$
$557$	5.59795e9	1.37257	0.686287	+	0.727331i	$0.259239\pi$
$558$	0	0
$559$	2.81828e9	0.682405
$560$	−2.22298e9	−0.534905
$561$	0	0
$562$	−6.01263e8	−0.142885
$563$	−3.06186e9	−0.723113	−0.361557	−	0.932350i	$-0.617755\pi$
$563$	−3.06186e9	−0.723113	−0.361557	+	0.932350i	$0.617755\pi$
$564$	0	0
$565$	−1.65391e9	−0.385783
$566$	−6.57263e9	−1.52364
$567$	0	0
$568$	−1.59075e9	−0.364236
$569$	1.74572e9	0.397267	0.198633	−	0.980074i	$-0.436350\pi$
$569$	1.74572e9	0.397267	0.198633	+	0.980074i	$0.436350\pi$
$570$	0	0
$571$	−5.28295e9	−1.18755	−0.593773	−	0.804633i	$-0.702362\pi$
$571$	−5.28295e9	−1.18755	−0.593773	+	0.804633i	$0.702362\pi$
$572$	−9.99965e9	−2.23408
$573$	0	0
$574$	3.37123e9	0.744042
$575$	−4.93651e9	−1.08289
$576$	0	0
$577$	−6.38776e9	−1.38431	−0.692155	−	0.721749i	$-0.743338\pi$
$577$	−6.38776e9	−1.38431	−0.692155	+	0.721749i	$0.743338\pi$
$578$	−1.42260e7	−0.00306433
$579$	0	0
$580$	−1.25820e9	−0.267763
$581$	−1.46561e9	−0.310028
$582$	0	0
$583$	9.79169e9	2.04653
$584$	2.62369e8	0.0545089
$585$	0	0
$586$	6.57874e9	1.35052
$587$	−5.93488e9	−1.21110	−0.605548	−	0.795809i	$-0.707046\pi$
$587$	−5.93488e9	−1.21110	−0.605548	+	0.795809i	$0.707046\pi$
$588$	0	0
$589$	1.20339e9	0.242663
$590$	3.88087e9	0.777942
$591$	0	0
$592$	4.52145e8	0.0895677
$593$	−2.18286e9	−0.429867	−0.214933	−	0.976629i	$-0.568953\pi$
$593$	−2.18286e9	−0.429867	−0.214933	+	0.976629i	$0.568953\pi$
$594$	0	0
$595$	−3.32364e9	−0.646852
$596$	1.70425e9	0.329740
$597$	0	0
$598$	−1.44068e10	−2.75494
$599$	8.51890e9	1.61953	0.809766	−	0.586752i	$-0.199594\pi$
$599$	8.51890e9	1.61953	0.809766	+	0.586752i	$0.199594\pi$
$600$	0	0
$601$	−6.59540e9	−1.23931	−0.619656	−	0.784874i	$-0.712728\pi$
$601$	−6.59540e9	−1.23931	−0.619656	+	0.784874i	$0.712728\pi$
$602$	−5.87623e9	−1.09777
$603$	0	0
$604$	1.61976e9	0.299104
$605$	1.78848e9	0.328352
$606$	0	0
$607$	−6.61480e9	−1.20048	−0.600242	−	0.799819i	$-0.704929\pi$
$607$	−6.61480e9	−1.20048	−0.600242	+	0.799819i	$0.704929\pi$
$608$	1.68825e9	0.304631
$609$	0	0
$610$	−1.11879e9	−0.199569
$611$	−3.83635e9	−0.680414
$612$	0	0
$613$	−6.77351e9	−1.18769	−0.593843	−	0.804581i	$-0.702390\pi$
$613$	−6.77351e9	−1.18769	−0.593843	+	0.804581i	$0.702390\pi$
$614$	−5.18150e9	−0.903370
$615$	0	0
$616$	2.73682e9	0.471751
$617$	1.09658e10	1.87950	0.939751	−	0.341860i	$-0.111057\pi$
$617$	1.09658e10	1.87950	0.939751	+	0.341860i	$0.111057\pi$
$618$	0	0
$619$	5.57764e9	0.945221	0.472610	−	0.881271i	$-0.343312\pi$
$619$	5.57764e9	0.945221	0.472610	+	0.881271i	$0.343312\pi$
$620$	−3.18590e9	−0.536861
$621$	0	0
$622$	4.69441e9	0.782194
$623$	1.97466e9	0.327178
$624$	0	0
$625$	3.21759e9	0.527171
$626$	−1.58600e10	−2.58400
$627$	0	0
$628$	3.65431e9	0.588772
$629$	6.76015e8	0.108313
$630$	0	0
$631$	−6.36080e9	−1.00788	−0.503940	−	0.863739i	$-0.668117\pi$
$631$	−6.36080e9	−1.00788	−0.503940	+	0.863739i	$0.668117\pi$
$632$	2.43022e9	0.382944
$633$	0	0
$634$	6.85393e9	1.06814
$635$	−1.89293e9	−0.293378
$636$	0	0
$637$	−1.42571e10	−2.18547
$638$	−7.35711e9	−1.12159
$639$	0	0
$640$	−1.18709e9	−0.179000
$641$	−7.54816e8	−0.113198	−0.0565989	−	0.998397i	$-0.518026\pi$
$641$	−7.54816e8	−0.113198	−0.0565989	+	0.998397i	$0.518026\pi$
$642$	0	0
$643$	1.04241e10	1.54633	0.773164	−	0.634206i	$-0.218673\pi$
$643$	1.04241e10	1.54633	0.773164	+	0.634206i	$0.218673\pi$
$644$	1.60744e10	2.37155
$645$	0	0
$646$	2.13385e9	0.311422
$647$	8.26499e9	1.19971	0.599857	−	0.800107i	$-0.295224\pi$
$647$	8.26499e9	1.19971	0.599857	+	0.800107i	$0.295224\pi$
$648$	0	0
$649$	1.21434e10	1.74375
$650$	1.23631e10	1.76575
$651$	0	0
$652$	4.48072e8	0.0633113
$653$	9.12709e9	1.28273	0.641366	−	0.767235i	$-0.278368\pi$
$653$	9.12709e9	1.28273	0.641366	+	0.767235i	$0.278368\pi$
$654$	0	0
$655$	1.65317e9	0.229866
$656$	−1.91166e9	−0.264391
$657$	0	0
$658$	7.99895e9	1.09457
$659$	1.50605e9	0.204994	0.102497	−	0.994733i	$-0.467317\pi$
$659$	1.50605e9	0.204994	0.102497	+	0.994733i	$0.467317\pi$
$660$	0	0
$661$	9.79847e9	1.31963	0.659816	−	0.751427i	$-0.270634\pi$
$661$	9.79847e9	1.31963	0.659816	+	0.751427i	$0.270634\pi$
$662$	3.79790e9	0.508792
$663$	0	0
$664$	−3.26993e8	−0.0433461
$665$	−1.04378e9	−0.137636
$666$	0	0
$667$	−5.67206e9	−0.740117
$668$	−1.57431e10	−2.04350
$669$	0	0
$670$	1.01512e8	0.0130393
$671$	−3.50073e9	−0.447332
$672$	0	0
$673$	−9.18829e9	−1.16194	−0.580968	−	0.813927i	$-0.697326\pi$
$673$	−9.18829e9	−1.16194	−0.580968	+	0.813927i	$0.697326\pi$
$674$	1.62296e10	2.04172
$675$	0	0
$676$	1.00620e10	1.25277
$677$	1.01615e10	1.25862	0.629312	−	0.777153i	$-0.283337\pi$
$677$	1.01615e10	1.25862	0.629312	+	0.777153i	$0.283337\pi$
$678$	0	0
$679$	−1.52504e10	−1.86955
$680$	−7.41541e8	−0.0904386
$681$	0	0
$682$	−1.86291e10	−2.24878
$683$	−9.85009e9	−1.18295	−0.591477	−	0.806322i	$-0.701455\pi$
$683$	−9.85009e9	−1.18295	−0.591477	+	0.806322i	$0.701455\pi$
$684$	0	0
$685$	4.81798e9	0.572727
$686$	1.00704e10	1.19101
$687$	0	0
$688$	3.33212e9	0.390087
$689$	−1.89059e10	−2.20206
$690$	0	0
$691$	3.54327e9	0.408536	0.204268	−	0.978915i	$-0.434519\pi$
$691$	3.54327e9	0.408536	0.204268	+	0.978915i	$0.434519\pi$
$692$	1.46594e10	1.68169
$693$	0	0
$694$	−2.35790e9	−0.267773
$695$	5.48020e9	0.619226
$696$	0	0
$697$	−2.85818e9	−0.319724
$698$	−7.68828e9	−0.855727
$699$	0	0
$700$	−1.37941e10	−1.52002
$701$	1.22196e10	1.33982	0.669908	−	0.742444i	$-0.266334\pi$
$701$	1.22196e10	1.33982	0.669908	+	0.742444i	$0.266334\pi$
$702$	0	0
$703$	2.12301e8	0.0230466
$704$	−1.58640e10	−1.71360
$705$	0	0
$706$	−3.66109e9	−0.391556
$707$	−1.49992e10	−1.59625
$708$	0	0
$709$	1.01823e10	1.07297	0.536483	−	0.843911i	$-0.319753\pi$
$709$	1.01823e10	1.07297	0.536483	+	0.843911i	$0.319753\pi$
$710$	−9.39183e9	−0.984795
$711$	0	0
$712$	4.40569e8	0.0457440
$713$	−1.43623e10	−1.48392
$714$	0	0
$715$	−7.74959e9	−0.792880
$716$	3.37665e9	0.343788
$717$	0	0
$718$	4.26632e9	0.430148
$719$	2.32175e9	0.232951	0.116476	−	0.993194i	$-0.462840\pi$
$719$	2.32175e9	0.232951	0.116476	+	0.993194i	$0.462840\pi$
$720$	0	0
$721$	2.20529e9	0.219125
$722$	−1.41622e10	−1.40040
$723$	0	0
$724$	−2.00124e10	−1.95981
$725$	4.86744e9	0.474370
$726$	0	0
$727$	1.20892e10	1.16688	0.583439	−	0.812157i	$-0.301707\pi$
$727$	1.20892e10	1.16688	0.583439	+	0.812157i	$0.301707\pi$
$728$	−5.28426e9	−0.507603
$729$	0	0
$730$	1.54903e9	0.147377
$731$	4.98196e9	0.471725
$732$	0	0
$733$	1.35185e10	1.26784	0.633919	−	0.773399i	$-0.281445\pi$
$733$	1.35185e10	1.26784	0.633919	+	0.773399i	$0.281445\pi$
$734$	1.19924e10	1.11936
$735$	0	0
$736$	−2.01491e10	−1.86287
$737$	3.17635e8	0.0292275
$738$	0	0
$739$	9.64611e9	0.879218	0.439609	−	0.898189i	$-0.355117\pi$
$739$	9.64611e9	0.879218	0.439609	+	0.898189i	$0.355117\pi$
$740$	−5.62053e8	−0.0509878
$741$	0	0
$742$	3.94196e10	3.54240
$743$	−1.26934e10	−1.13532	−0.567660	−	0.823263i	$-0.692151\pi$
$743$	−1.26934e10	−1.13532	−0.567660	+	0.823263i	$0.692151\pi$
$744$	0	0
$745$	1.32077e9	0.117025
$746$	−9.77613e9	−0.862147
$747$	0	0
$748$	−1.76767e10	−1.54435
$749$	1.38786e10	1.20686
$750$	0	0
$751$	−1.83866e9	−0.158403	−0.0792013	−	0.996859i	$-0.525237\pi$
$751$	−1.83866e9	−0.158403	−0.0792013	+	0.996859i	$0.525237\pi$
$752$	−4.53581e9	−0.388949
$753$	0	0
$754$	1.42052e10	1.20683
$755$	1.25529e9	0.106153
$756$	0	0
$757$	−3.69951e9	−0.309962	−0.154981	−	0.987917i	$-0.549532\pi$
$757$	−3.69951e9	−0.309962	−0.154981	+	0.987917i	$0.549532\pi$
$758$	1.49762e10	1.24899
$759$	0	0
$760$	−2.32879e8	−0.0192434
$761$	−9.10481e9	−0.748901	−0.374451	−	0.927247i	$-0.622169\pi$
$761$	−9.10481e9	−0.748901	−0.374451	+	0.927247i	$0.622169\pi$
$762$	0	0
$763$	8.33424e9	0.679251
$764$	1.64902e10	1.33782
$765$	0	0
$766$	9.28193e9	0.746170
$767$	−2.34466e10	−1.87627
$768$	0	0
$769$	−4.87498e9	−0.386572	−0.193286	−	0.981142i	$-0.561915\pi$
$769$	−4.87498e9	−0.386572	−0.193286	+	0.981142i	$0.561915\pi$
$770$	1.61582e10	1.27549
$771$	0	0
$772$	1.51611e10	1.18596
$773$	−1.86703e9	−0.145386	−0.0726930	−	0.997354i	$-0.523159\pi$
$773$	−1.86703e9	−0.145386	−0.0726930	+	0.997354i	$0.523159\pi$
$774$	0	0
$775$	1.23249e10	0.951106
$776$	−3.40253e9	−0.261388
$777$	0	0
$778$	−2.22584e10	−1.69460
$779$	−8.97603e8	−0.0680304
$780$	0	0
$781$	−2.93874e10	−2.20741
$782$	−2.54673e10	−1.90440
$783$	0	0
$784$	−1.68566e10	−1.24929
$785$	2.83204e9	0.208956
$786$	0	0
$787$	−6.76549e8	−0.0494752	−0.0247376	−	0.999694i	$-0.507875\pi$
$787$	−6.76549e8	−0.0494752	−0.0247376	+	0.999694i	$0.507875\pi$
$788$	5.56939e9	0.405477
$789$	0	0
$790$	1.43481e10	1.03538
$791$	−2.08343e10	−1.49679
$792$	0	0
$793$	6.75924e9	0.481329
$794$	3.72959e10	2.64417
$795$	0	0
$796$	−7.23266e6	−0.000508279 0
$797$	2.78422e10	1.94805	0.974025	−	0.226440i	$-0.0727088\pi$
$797$	2.78422e10	1.94805	0.974025	+	0.226440i	$0.0727088\pi$
$798$	0	0
$799$	−6.78163e9	−0.470349
$800$	1.72908e10	1.19399
$801$	0	0
$802$	−3.99075e10	−2.73177
$803$	4.84699e9	0.330345
$804$	0	0
$805$	1.24574e10	0.841670
$806$	3.59692e10	2.41968
$807$	0	0
$808$	−3.34650e9	−0.223178
$809$	−5.64823e9	−0.375053	−0.187527	−	0.982260i	$-0.560047\pi$
$809$	−5.64823e9	−0.375053	−0.187527	+	0.982260i	$0.560047\pi$
$810$	0	0
$811$	−7.40101e9	−0.487212	−0.243606	−	0.969874i	$-0.578330\pi$
$811$	−7.40101e9	−0.487212	−0.243606	+	0.969874i	$0.578330\pi$
$812$	−1.58494e10	−1.03888
$813$	0	0
$814$	−3.28652e9	−0.213575
$815$	3.47249e8	0.0224693
$816$	0	0
$817$	1.56457e9	0.100373
$818$	1.26773e10	0.809820
$819$	0	0
$820$	2.37635e9	0.150509
$821$	1.52865e10	0.964066	0.482033	−	0.876153i	$-0.339899\pi$
$821$	1.52865e10	0.964066	0.482033	+	0.876153i	$0.339899\pi$
$822$	0	0
$823$	−2.97997e10	−1.86343	−0.931714	−	0.363194i	$-0.881686\pi$
$823$	−2.97997e10	−1.86343	−0.931714	+	0.363194i	$0.881686\pi$
$824$	4.92024e8	0.0306367
$825$	0	0
$826$	4.88872e10	3.01832
$827$	−6.76407e9	−0.415852	−0.207926	−	0.978145i	$-0.566671\pi$
$827$	−6.76407e9	−0.415852	−0.207926	+	0.978145i	$0.566671\pi$
$828$	0	0
$829$	−2.81440e10	−1.71571	−0.857857	−	0.513888i	$-0.828205\pi$
$829$	−2.81440e10	−1.71571	−0.857857	+	0.513888i	$0.828205\pi$
$830$	−1.93058e9	−0.117196
$831$	0	0
$832$	3.06304e10	1.84383
$833$	−2.52028e10	−1.51075
$834$	0	0
$835$	−1.22007e10	−0.725241
$836$	−5.55131e9	−0.328605
$837$	0	0
$838$	4.78904e9	0.281122
$839$	−8.02651e9	−0.469202	−0.234601	−	0.972092i	$-0.575378\pi$
$839$	−8.02651e9	−0.469202	−0.234601	+	0.972092i	$0.575378\pi$
$840$	0	0
$841$	−1.16572e10	−0.675784
$842$	−2.58671e10	−1.49333
$843$	0	0
$844$	2.77843e10	1.59075
$845$	7.79790e9	0.444610
$846$	0	0
$847$	2.25294e10	1.27396
$848$	−2.23529e10	−1.25877
$849$	0	0
$850$	2.18546e10	1.22061
$851$	−2.53378e9	−0.140934
$852$	0	0
$853$	3.45404e10	1.90549	0.952743	−	0.303779i	$-0.0982483\pi$
$853$	3.45404e10	1.90549	0.952743	+	0.303779i	$0.0982483\pi$
$854$	−1.40933e10	−0.774303
$855$	0	0
$856$	3.09647e9	0.168736
$857$	1.82602e10	0.990995	0.495498	−	0.868609i	$-0.334986\pi$
$857$	1.82602e10	0.990995	0.495498	+	0.868609i	$0.334986\pi$
$858$	0	0
$859$	−2.17949e10	−1.17322	−0.586608	−	0.809871i	$-0.699537\pi$
$859$	−2.17949e10	−1.17322	−0.586608	+	0.809871i	$0.699537\pi$
$860$	−4.14210e9	−0.222063
$861$	0	0
$862$	−1.63864e10	−0.871381
$863$	1.25884e10	0.666702	0.333351	−	0.942803i	$-0.391821\pi$
$863$	1.25884e10	0.666702	0.333351	+	0.942803i	$0.391821\pi$
$864$	0	0
$865$	1.13609e10	0.596835
$866$	−3.43562e10	−1.79760
$867$	0	0
$868$	−4.01327e10	−2.08295
$869$	4.48958e10	2.32079
$870$	0	0
$871$	−6.13291e8	−0.0314487
$872$	1.85946e9	0.0949685
$873$	0	0
$874$	−7.99792e9	−0.405216
$875$	−2.35220e10	−1.18699
$876$	0	0
$877$	1.83497e10	0.918607	0.459304	−	0.888279i	$-0.348099\pi$
$877$	1.83497e10	0.918607	0.459304	+	0.888279i	$0.348099\pi$
$878$	−5.15666e10	−2.57121
$879$	0	0
$880$	−9.16254e9	−0.453238
$881$	1.25257e10	0.617146	0.308573	−	0.951201i	$-0.400149\pi$
$881$	1.25257e10	0.617146	0.308573	+	0.951201i	$0.400149\pi$
$882$	0	0
$883$	2.79816e10	1.36776	0.683880	−	0.729594i	$-0.260291\pi$
$883$	2.79816e10	1.36776	0.683880	+	0.729594i	$0.260291\pi$
$884$	3.41303e10	1.66172
$885$	0	0
$886$	2.94912e10	1.42454
$887$	−2.74565e10	−1.32103	−0.660515	−	0.750813i	$-0.729662\pi$
$887$	−2.74565e10	−1.32103	−0.660515	+	0.750813i	$0.729662\pi$
$888$	0	0
$889$	−2.38452e10	−1.13827
$890$	2.60113e9	0.123679
$891$	0	0
$892$	1.55703e10	0.734547
$893$	−2.12975e9	−0.100080
$894$	0	0
$895$	2.61686e9	0.122011
$896$	−1.49537e10	−0.694495
$897$	0	0
$898$	−5.62459e10	−2.59193
$899$	1.41614e10	0.650049
$900$	0	0
$901$	−3.34205e10	−1.52221
$902$	1.38954e10	0.630445
$903$	0	0
$904$	−4.64836e9	−0.209271
$905$	−1.55093e10	−0.695541
$906$	0	0
$907$	3.16369e9	0.140789	0.0703944	−	0.997519i	$-0.477574\pi$
$907$	3.16369e9	0.140789	0.0703944	+	0.997519i	$0.477574\pi$
$908$	−6.76662e9	−0.299966
$909$	0	0
$910$	−3.11984e10	−1.37242
$911$	−1.72740e10	−0.756969	−0.378485	−	0.925608i	$-0.623555\pi$
$911$	−1.72740e10	−0.756969	−0.378485	+	0.925608i	$0.623555\pi$
$912$	0	0
$913$	−6.04085e9	−0.262694
$914$	−3.47871e10	−1.50698
$915$	0	0
$916$	−1.81074e10	−0.778434
$917$	2.08249e10	0.891849
$918$	0	0
$919$	9.42182e9	0.400434	0.200217	−	0.979752i	$-0.435835\pi$
$919$	9.42182e9	0.400434	0.200217	+	0.979752i	$0.435835\pi$
$920$	2.77938e9	0.117677
$921$	0	0
$922$	2.96124e10	1.24427
$923$	5.67414e10	2.37517
$924$	0	0
$925$	2.17435e9	0.0903303
$926$	3.81316e10	1.57814
$927$	0	0
$928$	1.98671e10	0.816051
$929$	−2.32592e10	−0.951788	−0.475894	−	0.879503i	$-0.657875\pi$
$929$	−2.32592e10	−0.951788	−0.475894	+	0.879503i	$0.657875\pi$
$930$	0	0
$931$	−7.91486e9	−0.321454
$932$	2.47939e10	1.00320
$933$	0	0
$934$	−1.42700e10	−0.573072
$935$	−1.36992e10	−0.548093
$936$	0	0
$937$	2.57549e10	1.02275	0.511377	−	0.859357i	$-0.329136\pi$
$937$	2.57549e10	1.02275	0.511377	+	0.859357i	$0.329136\pi$
$938$	1.27874e9	0.0505909
$939$	0	0
$940$	5.63839e9	0.221415
$941$	−2.13825e10	−0.836557	−0.418278	−	0.908319i	$-0.637366\pi$
$941$	−2.13825e10	−0.836557	−0.418278	+	0.908319i	$0.637366\pi$
$942$	0	0
$943$	1.07128e10	0.416018
$944$	−2.77215e10	−1.07254
$945$	0	0
$946$	−2.42203e10	−0.930167
$947$	3.90679e10	1.49484	0.747420	−	0.664351i	$-0.231292\pi$
$947$	3.90679e10	1.49484	0.747420	+	0.664351i	$0.231292\pi$
$948$	0	0
$949$	−9.35860e9	−0.355450
$950$	6.86336e9	0.259719
$951$	0	0
$952$	−9.34115e9	−0.350890
$953$	−1.44692e10	−0.541527	−0.270763	−	0.962646i	$-0.587276\pi$
$953$	−1.44692e10	−0.541527	−0.270763	+	0.962646i	$0.587276\pi$
$954$	0	0
$955$	1.27796e10	0.474796
$956$	1.18245e10	0.437702
$957$	0	0
$958$	5.08764e8	0.0186955
$959$	6.06918e10	2.22211
$960$	0	0
$961$	8.34567e9	0.303340
$962$	6.34564e9	0.229807
$963$	0	0
$964$	2.20021e10	0.791032
$965$	1.17496e10	0.420900
$966$	0	0
$967$	3.68538e10	1.31066	0.655329	−	0.755343i	$-0.272530\pi$
$967$	3.68538e10	1.31066	0.655329	+	0.755343i	$0.272530\pi$
$968$	5.02655e9	0.178117
$969$	0	0
$970$	−2.00886e10	−0.706723
$971$	−1.36704e10	−0.479198	−0.239599	−	0.970872i	$-0.577016\pi$
$971$	−1.36704e10	−0.479198	−0.239599	+	0.970872i	$0.577016\pi$
$972$	0	0
$973$	6.90337e10	2.40252
$974$	−9.60288e9	−0.333001
$975$	0	0
$976$	7.99162e9	0.275144
$977$	3.51417e10	1.20557	0.602785	−	0.797904i	$-0.294058\pi$
$977$	3.51417e10	1.20557	0.602785	+	0.797904i	$0.294058\pi$
$978$	0	0
$979$	8.13905e9	0.277226
$980$	2.09541e10	0.711178
$981$	0	0
$982$	5.47116e10	1.84369
$983$	3.29903e10	1.10777	0.553885	−	0.832593i	$-0.313145\pi$
$983$	3.29903e10	1.10777	0.553885	+	0.832593i	$0.313145\pi$
$984$	0	0
$985$	4.31620e9	0.143905
$986$	2.51109e10	0.834244
$987$	0	0
$988$	1.07185e10	0.353578
$989$	−1.86730e10	−0.613799
$990$	0	0
$991$	−2.08796e10	−0.681496	−0.340748	−	0.940155i	$-0.610680\pi$
$991$	−2.08796e10	−0.681496	−0.340748	+	0.940155i	$0.610680\pi$
$992$	5.03060e10	1.63617
$993$	0	0
$994$	−1.18308e11	−3.82088
$995$	−5.60521e6	−0.000180389 0
$996$	0	0
$997$	−3.29887e10	−1.05422	−0.527111	−	0.849797i	$-0.676725\pi$
$997$	−3.29887e10	−1.05422	−0.527111	+	0.849797i	$0.676725\pi$
$998$	−7.08837e10	−2.25730
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.a.e.1.2	yes	2
3.2	odd	2		27.8.a.b.1.1	✓	2
4.3	odd	2		432.8.a.q.1.2		2
9.2	odd	6		81.8.c.h.28.2		4
9.4	even	3		81.8.c.d.55.1		4
9.5	odd	6		81.8.c.h.55.2		4
9.7	even	3		81.8.c.d.28.1		4
12.11	even	2		432.8.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.a.b.1.1	✓	2	3.2	odd	2
27.8.a.e.1.2	yes	2	1.1	even	1	trivial
81.8.c.d.28.1		4	9.7	even	3
81.8.c.d.55.1		4	9.4	even	3
81.8.c.h.28.2		4	9.2	odd	6
81.8.c.h.55.2		4	9.5	odd	6
432.8.a.j.1.1		2	12.11	even	2
432.8.a.q.1.2		2	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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+16.5934 q^{2} +147.340 q^{4} +114.187 q^{5} +1438.40 q^{7} +320.924 q^{8} +1894.75 q^{10} +5928.74 q^{11} -11447.2 q^{13} +23868.0 q^{14} -13534.4 q^{16} -20235.6 q^{17} -6354.94 q^{19} +16824.3 q^{20} +98377.8 q^{22} +75845.6 q^{23} -65086.4 q^{25} -189948. q^{26} +211935. q^{28} -74784.3 q^{29} -189363. q^{31} -265659. q^{32} -335778. q^{34} +164247. q^{35} -33407.2 q^{37} -105450. q^{38} +36645.3 q^{40} +141245. q^{41} -246197. q^{43} +873543. q^{44} +1.25854e6 q^{46} +335133. q^{47} +1.24547e6 q^{49} -1.08000e6 q^{50} -1.68664e6 q^{52} +1.65156e6 q^{53} +676983. q^{55} +461619. q^{56} -1.24092e6 q^{58} +2.04823e6 q^{59} -590469. q^{61} -3.14217e6 q^{62} -2.67579e6 q^{64} -1.30712e6 q^{65} +53575.5 q^{67} -2.98153e6 q^{68} +2.72541e6 q^{70} -4.95678e6 q^{71} +817542. q^{73} -554338. q^{74} -936340. q^{76} +8.52792e6 q^{77} +7.57257e6 q^{79} -1.54545e6 q^{80} +2.34373e6 q^{82} -1.01891e6 q^{83} -2.31064e6 q^{85} -4.08524e6 q^{86} +1.90267e6 q^{88} +1.37281e6 q^{89} -1.64658e7 q^{91} +1.11751e7 q^{92} +5.56099e6 q^{94} -725650. q^{95} -1.06023e7 q^{97} +2.06665e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{2} + 77 q^{4} + 180 q^{5} + 700 q^{7} + 1827 q^{8} + 1395 q^{10} + 10890 q^{11} - 5480 q^{13} + 29475 q^{14} - 15967 q^{16} + 16416 q^{17} + 16024 q^{19} + 12195 q^{20} + 60705 q^{22} + 24372 q^{23}+ \cdots + 22779738 q^{98}+O(q^{100}) \) 2 * q + 9 * q^2 + 77 * q^4 + 180 * q^5 + 700 * q^7 + 1827 * q^8 + 1395 * q^10 + 10890 * q^11 - 5480 * q^13 + 29475 * q^14 - 15967 * q^16 + 16416 * q^17 + 16024 * q^19 + 12195 * q^20 + 60705 * q^22 + 24372 * q^23 - 138880 * q^25 - 235260 * q^26 + 263875 * q^28 - 143280 * q^29 - 38708 * q^31 - 439965 * q^32 - 614088 * q^34 + 115650 * q^35 + 455620 * q^37 - 275382 * q^38 + 135765 * q^40 + 731880 * q^41 - 1088840 * q^43 + 524565 * q^44 + 1649394 * q^46 + 1561500 * q^47 + 967164 * q^49 - 519660 * q^50 - 2106380 * q^52 + 2610468 * q^53 + 1003500 * q^55 - 650475 * q^56 - 720810 * q^58 + 1731960 * q^59 - 620192 * q^61 - 4286151 * q^62 - 1040839 * q^64 - 914400 * q^65 + 346600 * q^67 - 5559624 * q^68 + 3094425 * q^70 - 4242240 * q^71 - 3145190 * q^73 - 4267710 * q^74 - 2510486 * q^76 + 4864500 * q^77 + 10110616 * q^79 - 1705545 * q^80 - 2141190 * q^82 + 644202 * q^83 + 101520 * q^85 + 2313270 * q^86 + 9374715 * q^88 + 6021000 * q^89 - 20872000 * q^91 + 14795802 * q^92 - 3751290 * q^94 + 747180 * q^95 + 4098670 * q^97 + 22779738 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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