LMFDB - Embedded newform 27.10.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,10,Mod(1,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$13.9059675764$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.74166$ 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)

$f(q)$	$=$	\(q+22.4499 q^{2} -8.00000 q^{4} -493.899 q^{5} -763.000 q^{7} -11674.0 q^{8} -11088.0 q^{10} -56888.2 q^{11} -73015.0 q^{13} -17129.3 q^{14} -257984. q^{16} +168240. q^{17} -598129. q^{19} +3951.19 q^{20} -1.27714e6 q^{22} +2.40039e6 q^{23} -1.70919e6 q^{25} -1.63918e6 q^{26} +6104.00 q^{28} -4.65378e6 q^{29} -1.82410e6 q^{31} +185347. q^{32} +3.77698e6 q^{34} +376845. q^{35} +1.42722e7 q^{37} -1.34280e7 q^{38} +5.76576e6 q^{40} +2.98243e7 q^{41} +7.75690e6 q^{43} +455105. q^{44} +5.38887e7 q^{46} +3.09903e7 q^{47} -3.97714e7 q^{49} -3.83712e7 q^{50} +584120. q^{52} +9.90123e6 q^{53} +2.80970e7 q^{55} +8.90724e6 q^{56} -1.04477e8 q^{58} -1.25210e8 q^{59} -1.52766e8 q^{61} -4.09509e7 q^{62} +1.36249e8 q^{64} +3.60620e7 q^{65} -3.20752e8 q^{67} -1.34592e6 q^{68} +8.46014e6 q^{70} +1.09152e8 q^{71} +6.64640e7 q^{73} +3.20410e8 q^{74} +4.78503e6 q^{76} +4.34057e7 q^{77} -1.15357e8 q^{79} +1.27418e8 q^{80} +6.69554e8 q^{82} -6.14639e8 q^{83} -8.30935e7 q^{85} +1.74142e8 q^{86} +6.64111e8 q^{88} -3.37797e8 q^{89} +5.57104e7 q^{91} -1.92031e7 q^{92} +6.95731e8 q^{94} +2.95415e8 q^{95} -2.35307e8 q^{97} -8.92867e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{4} - 1526 q^{7} - 22176 q^{10} - 146030 q^{13} - 515968 q^{16} - 1196258 q^{19} - 2554272 q^{22} - 3418378 q^{25} + 12208 q^{28} - 3648200 q^{31} + 7553952 q^{34} + 28544350 q^{37} + 11531520 q^{40}+ \cdots - 470613686 q^{97}+O(q^{100}) $ 2 * q - 16 * q^4 - 1526 * q^7 - 22176 * q^10 - 146030 * q^13 - 515968 * q^16 - 1196258 * q^19 - 2554272 * q^22 - 3418378 * q^25 + 12208 * q^28 - 3648200 * q^31 + 7553952 * q^34 + 28544350 * q^37 + 11531520 * q^40 + 15513808 * q^43 + 107777376 * q^46 - 79542876 * q^49 + 1168240 * q^52 + 56193984 * q^55 - 208954368 * q^58 - 305532986 * q^61 + 272497664 * q^64 - 641504138 * q^67 + 16920288 * q^70 + 132927970 * q^73 + 9570064 * q^76 - 230714906 * q^79 + 1339107840 * q^82 - 166186944 * q^85 + 1328221440 * q^88 + 111420890 * q^91 + 1391461344 * q^94 - 470613686 * q^97

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 $

$2$	22.4499	0.992157	0.496078	−	0.868278i	$-0.334773\pi$
$2$	22.4499	0.992157	0.496078	+	0.868278i	$0.334773\pi$
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.0156250
$5$	−493.899	−0.353405	−0.176703	−	0.984264i	$-0.556543\pi$
$5$	−493.899	−0.353405	−0.176703	+	0.984264i	$0.556543\pi$
$6$	0	0
$7$	−763.000	−0.120111	−0.0600556	−	0.998195i	$-0.519128\pi$
$7$	−763.000	−0.120111	−0.0600556	+	0.998195i	$0.519128\pi$
$8$	−11674.0	−1.00766
$9$	0	0
$10$	−11088.0	−0.350633
$11$	−56888.2	−1.17153	−0.585767	−	0.810480i	$-0.699207\pi$
$11$	−56888.2	−1.17153	−0.585767	+	0.810480i	$0.699207\pi$
$12$	0	0
$13$	−73015.0	−0.709034	−0.354517	−	0.935050i	$-0.615355\pi$
$13$	−73015.0	−0.709034	−0.354517	+	0.935050i	$0.615355\pi$
$14$	−17129.3	−0.119169
$15$	0	0
$16$	−257984.	−0.984131
$17$	168240.	0.488550	0.244275	−	0.969706i	$-0.421450\pi$
$17$	168240.	0.488550	0.244275	+	0.969706i	$0.421450\pi$
$18$	0	0
$19$	−598129.	−1.05294	−0.526470	−	0.850194i	$-0.676485\pi$
$19$	−598129.	−1.05294	−0.526470	+	0.850194i	$0.676485\pi$
$20$	3951.19	0.00552196
$21$	0	0
$22$	−1.27714e6	−1.16235
$23$	2.40039e6	1.78857	0.894287	−	0.447493i	$-0.147683\pi$
$23$	2.40039e6	1.78857	0.894287	+	0.447493i	$0.147683\pi$
$24$	0	0
$25$	−1.70919e6	−0.875105
$26$	−1.63918e6	−0.703473
$27$	0	0
$28$	6104.00	0.00187674
$29$	−4.65378e6	−1.22184	−0.610921	−	0.791691i	$-0.709201\pi$
$29$	−4.65378e6	−1.22184	−0.610921	+	0.791691i	$0.709201\pi$
$30$	0	0
$31$	−1.82410e6	−0.354749	−0.177374	−	0.984143i	$-0.556760\pi$
$31$	−1.82410e6	−0.354749	−0.177374	+	0.984143i	$0.556760\pi$
$32$	185347.	0.0312471
$33$	0	0
$34$	3.77698e6	0.484718
$35$	376845.	0.0424479
$36$	0	0
$37$	1.42722e7	1.25194	0.625968	−	0.779848i	$-0.284704\pi$
$37$	1.42722e7	1.25194	0.625968	+	0.779848i	$0.284704\pi$
$38$	−1.34280e7	−1.04468
$39$	0	0
$40$	5.76576e6	0.356112
$41$	2.98243e7	1.64833	0.824163	−	0.566353i	$-0.191646\pi$
$41$	2.98243e7	1.64833	0.824163	+	0.566353i	$0.191646\pi$
$42$	0	0
$43$	7.75690e6	0.346003	0.173002	−	0.984922i	$-0.444653\pi$
$43$	7.75690e6	0.346003	0.173002	+	0.984922i	$0.444653\pi$
$44$	455105.	0.0183052
$45$	0	0
$46$	5.38887e7	1.77455
$47$	3.09903e7	0.926372	0.463186	−	0.886261i	$-0.346706\pi$
$47$	3.09903e7	0.926372	0.463186	+	0.886261i	$0.346706\pi$
$48$	0	0
$49$	−3.97714e7	−0.985573
$50$	−3.83712e7	−0.868241
$51$	0	0
$52$	584120.	0.0110787
$53$	9.90123e6	0.172365	0.0861823	−	0.996279i	$-0.472533\pi$
$53$	9.90123e6	0.172365	0.0861823	+	0.996279i	$0.472533\pi$
$54$	0	0
$55$	2.80970e7	0.414026
$56$	8.90724e6	0.121031
$57$	0	0
$58$	−1.04477e8	−1.21226
$59$	−1.25210e8	−1.34525	−0.672627	−	0.739982i	$-0.734834\pi$
$59$	−1.25210e8	−1.34525	−0.672627	+	0.739982i	$0.734834\pi$
$60$	0	0
$61$	−1.52766e8	−1.41268	−0.706340	−	0.707873i	$-0.749655\pi$
$61$	−1.52766e8	−1.41268	−0.706340	+	0.707873i	$0.749655\pi$
$62$	−4.09509e7	−0.351966
$63$	0	0
$64$	1.36249e8	1.01513
$65$	3.60620e7	0.250576
$66$	0	0
$67$	−3.20752e8	−1.94461	−0.972306	−	0.233712i	$-0.924913\pi$
$67$	−3.20752e8	−1.94461	−0.972306	+	0.233712i	$0.924913\pi$
$68$	−1.34592e6	−0.00763359
$69$	0	0
$70$	8.46014e6	0.0421150
$71$	1.09152e8	0.509763	0.254881	−	0.966972i	$-0.417964\pi$
$71$	1.09152e8	0.509763	0.254881	+	0.966972i	$0.417964\pi$
$72$	0	0
$73$	6.64640e7	0.273926	0.136963	−	0.990576i	$-0.456266\pi$
$73$	6.64640e7	0.273926	0.136963	+	0.990576i	$0.456266\pi$
$74$	3.20410e8	1.24212
$75$	0	0
$76$	4.78503e6	0.0164522
$77$	4.34057e7	0.140714
$78$	0	0
$79$	−1.15357e8	−0.333214	−0.166607	−	0.986023i	$-0.553281\pi$
$79$	−1.15357e8	−0.333214	−0.166607	+	0.986023i	$0.553281\pi$
$80$	1.27418e8	0.347797
$81$	0	0
$82$	6.69554e8	1.63540
$83$	−6.14639e8	−1.42157	−0.710785	−	0.703409i	$-0.751660\pi$
$83$	−6.14639e8	−1.42157	−0.710785	+	0.703409i	$0.751660\pi$
$84$	0	0
$85$	−8.30935e7	−0.172656
$86$	1.74142e8	0.343290
$87$	0	0
$88$	6.64111e8	1.18051
$89$	−3.37797e8	−0.570691	−0.285345	−	0.958425i	$-0.592108\pi$
$89$	−3.37797e8	−0.570691	−0.285345	+	0.958425i	$0.592108\pi$
$90$	0	0
$91$	5.57104e7	0.0851629
$92$	−1.92031e7	−0.0279465
$93$	0	0
$94$	6.95731e8	0.919106
$95$	2.95415e8	0.372114
$96$	0	0
$97$	−2.35307e8	−0.269874	−0.134937	−	0.990854i	$-0.543083\pi$
$97$	−2.35307e8	−0.269874	−0.134937	+	0.990854i	$0.543083\pi$
$98$	−8.92867e8	−0.977843
$99$	0	0
$100$	1.36735e7	0.0136735
$101$	−9.16170e8	−0.876052	−0.438026	−	0.898962i	$-0.644322\pi$
$101$	−9.16170e8	−0.876052	−0.438026	+	0.898962i	$0.644322\pi$
$102$	0	0
$103$	1.00799e8	0.0882451	0.0441225	−	0.999026i	$-0.485951\pi$
$103$	1.00799e8	0.0882451	0.0441225	+	0.999026i	$0.485951\pi$
$104$	8.52375e8	0.714465
$105$	0	0
$106$	2.22282e8	0.171013
$107$	1.50887e9	1.11282	0.556411	−	0.830907i	$-0.312178\pi$
$107$	1.50887e9	1.11282	0.556411	+	0.830907i	$0.312178\pi$
$108$	0	0
$109$	−2.26733e9	−1.53849	−0.769247	−	0.638951i	$-0.779369\pi$
$109$	−2.26733e9	−1.53849	−0.769247	+	0.638951i	$0.779369\pi$
$110$	6.30776e8	0.410779
$111$	0	0
$112$	1.96842e8	0.118205
$113$	1.15962e9	0.669054	0.334527	−	0.942386i	$-0.391423\pi$
$113$	1.15962e9	0.669054	0.334527	+	0.942386i	$0.391423\pi$
$114$	0	0
$115$	−1.18555e9	−0.632092
$116$	3.72303e7	0.0190913
$117$	0	0
$118$	−2.81095e9	−1.33470
$119$	−1.28367e8	−0.0586803
$120$	0	0
$121$	8.78315e8	0.372491
$122$	−3.42960e9	−1.40160
$123$	0	0
$124$	1.45928e7	0.00554295
$125$	1.80881e9	0.662672
$126$	0	0
$127$	−2.61195e9	−0.890938	−0.445469	−	0.895297i	$-0.646963\pi$
$127$	−2.61195e9	−0.890938	−0.445469	+	0.895297i	$0.646963\pi$
$128$	2.96388e9	0.975924
$129$	0	0
$130$	8.09590e8	0.248611
$131$	−3.26886e9	−0.969786	−0.484893	−	0.874574i	$-0.661141\pi$
$131$	−3.26886e9	−0.969786	−0.484893	+	0.874574i	$0.661141\pi$
$132$	0	0
$133$	4.56372e8	0.126470
$134$	−7.20087e9	−1.92936
$135$	0	0
$136$	−1.96403e9	−0.492292
$137$	−5.66884e7	−0.0137484	−0.00687419	−	0.999976i	$-0.502188\pi$
$137$	−5.66884e7	−0.0137484	−0.00687419	+	0.999976i	$0.502188\pi$
$138$	0	0
$139$	6.79897e9	1.54482	0.772408	−	0.635127i	$-0.219052\pi$
$139$	6.79897e9	1.54482	0.772408	+	0.635127i	$0.219052\pi$
$140$	−3.01476e6	−0.000663249 0
$141$	0	0
$142$	2.45045e9	0.505764
$143$	4.15369e9	0.830657
$144$	0	0
$145$	2.29850e9	0.431805
$146$	1.49211e9	0.271778
$147$	0	0
$148$	−1.14177e8	−0.0195615
$149$	1.45949e9	0.242585	0.121292	−	0.992617i	$-0.461296\pi$
$149$	1.45949e9	0.242585	0.121292	+	0.992617i	$0.461296\pi$
$150$	0	0
$151$	4.95922e9	0.776277	0.388139	−	0.921601i	$-0.373118\pi$
$151$	4.95922e9	0.776277	0.388139	+	0.921601i	$0.373118\pi$
$152$	6.98254e9	1.06100
$153$	0	0
$154$	9.74455e8	0.139611
$155$	9.00921e8	0.125370
$156$	0	0
$157$	5.69095e9	0.747543	0.373772	−	0.927521i	$-0.378064\pi$
$157$	5.69095e9	0.747543	0.373772	+	0.927521i	$0.378064\pi$
$158$	−2.58977e9	−0.330601
$159$	0	0
$160$	−9.15425e7	−0.0110429
$161$	−1.83150e9	−0.214828
$162$	0	0
$163$	7.88882e9	0.875322	0.437661	−	0.899140i	$-0.355807\pi$
$163$	7.88882e9	0.875322	0.437661	+	0.899140i	$0.355807\pi$
$164$	−2.38594e8	−0.0257551
$165$	0	0
$166$	−1.37986e10	−1.41042
$167$	9.25008e9	0.920283	0.460142	−	0.887846i	$-0.347799\pi$
$167$	9.25008e9	0.920283	0.460142	+	0.887846i	$0.347799\pi$
$168$	0	0
$169$	−5.27331e9	−0.497271
$170$	−1.86544e9	−0.171302
$171$	0	0
$172$	−6.20552e7	−0.00540630
$173$	−1.85425e10	−1.57384	−0.786922	−	0.617052i	$-0.788327\pi$
$173$	−1.85425e10	−1.57384	−0.786922	+	0.617052i	$0.788327\pi$
$174$	0	0
$175$	1.30411e9	0.105110
$176$	1.46762e10	1.15294
$177$	0	0
$178$	−7.58352e9	−0.566215
$179$	−1.83931e10	−1.33911	−0.669555	−	0.742762i	$-0.733515\pi$
$179$	−1.83931e10	−1.33911	−0.669555	+	0.742762i	$0.733515\pi$
$180$	0	0
$181$	1.17014e9	0.0810375	0.0405187	−	0.999179i	$-0.487099\pi$
$181$	1.17014e9	0.0810375	0.0405187	+	0.999179i	$0.487099\pi$
$182$	1.25070e9	0.0844949
$183$	0	0
$184$	−2.80221e10	−1.80227
$185$	−7.04901e9	−0.442441
$186$	0	0
$187$	−9.57086e9	−0.572353
$188$	−2.47922e8	−0.0144746
$189$	0	0
$190$	6.63205e9	0.369196
$191$	−1.03749e10	−0.564073	−0.282036	−	0.959404i	$-0.591010\pi$
$191$	−1.03749e10	−0.564073	−0.282036	+	0.959404i	$0.591010\pi$
$192$	0	0
$193$	−1.17395e10	−0.609037	−0.304518	−	0.952506i	$-0.598496\pi$
$193$	−1.17395e10	−0.609037	−0.304518	+	0.952506i	$0.598496\pi$
$194$	−5.28263e9	−0.267758
$195$	0	0
$196$	3.18172e8	0.0153996
$197$	2.61163e10	1.23542	0.617708	−	0.786408i	$-0.288062\pi$
$197$	2.61163e10	1.23542	0.617708	+	0.786408i	$0.288062\pi$
$198$	0	0
$199$	2.31315e10	1.04560	0.522800	−	0.852455i	$-0.324887\pi$
$199$	2.31315e10	1.04560	0.522800	+	0.852455i	$0.324887\pi$
$200$	1.99530e10	0.881807
$201$	0	0
$202$	−2.05680e10	−0.869181
$203$	3.55084e9	0.146757
$204$	0	0
$205$	−1.47302e10	−0.582527
$206$	2.26294e9	0.0875530
$207$	0	0
$208$	1.88367e10	0.697782
$209$	3.40265e10	1.23355
$210$	0	0
$211$	2.57907e10	0.895759	0.447880	−	0.894094i	$-0.352179\pi$
$211$	2.57907e10	0.895759	0.447880	+	0.894094i	$0.352179\pi$
$212$	−7.92099e7	−0.00269320
$213$	0	0
$214$	3.38741e10	1.10409
$215$	−3.83113e9	−0.122279
$216$	0	0
$217$	1.39179e9	0.0426093
$218$	−5.09015e10	−1.52643
$219$	0	0
$220$	−2.24776e8	−0.00646916
$221$	−1.22840e10	−0.346398
$222$	0	0
$223$	−1.00337e9	−0.0271699	−0.0135850	−	0.999908i	$-0.504324\pi$
$223$	−1.00337e9	−0.0271699	−0.0135850	+	0.999908i	$0.504324\pi$
$224$	−1.41420e8	−0.00375313
$225$	0	0
$226$	2.60333e10	0.663806
$227$	5.27911e10	1.31961	0.659803	−	0.751438i	$-0.270640\pi$
$227$	5.27911e10	1.31961	0.659803	+	0.751438i	$0.270640\pi$
$228$	0	0
$229$	3.10909e10	0.747091	0.373546	−	0.927612i	$-0.378142\pi$
$229$	3.10909e10	0.747091	0.373546	+	0.927612i	$0.378142\pi$
$230$	−2.66156e10	−0.627134
$231$	0	0
$232$	5.43281e10	1.23120
$233$	−3.41332e10	−0.758710	−0.379355	−	0.925251i	$-0.623854\pi$
$233$	−3.41332e10	−0.758710	−0.379355	+	0.925251i	$0.623854\pi$
$234$	0	0
$235$	−1.53061e10	−0.327385
$236$	1.00168e9	0.0210196
$237$	0	0
$238$	−2.88183e9	−0.0582200
$239$	−9.02670e10	−1.78953	−0.894764	−	0.446539i	$-0.852656\pi$
$239$	−9.02670e10	−1.78953	−0.894764	+	0.446539i	$0.852656\pi$
$240$	0	0
$241$	−7.73393e10	−1.47681	−0.738403	−	0.674360i	$-0.764420\pi$
$241$	−7.73393e10	−1.47681	−0.738403	+	0.674360i	$0.764420\pi$
$242$	1.97181e10	0.369570
$243$	0	0
$244$	1.22213e9	0.0220731
$245$	1.96431e10	0.348307
$246$	0	0
$247$	4.36724e10	0.746570
$248$	2.12945e10	0.357466
$249$	0	0
$250$	4.06077e10	0.657474
$251$	4.53113e10	0.720568	0.360284	−	0.932843i	$-0.382680\pi$
$251$	4.53113e10	0.720568	0.360284	+	0.932843i	$0.382680\pi$
$252$	0	0
$253$	−1.36554e11	−2.09538
$254$	−5.86380e10	−0.883950
$255$	0	0
$256$	−3.22043e9	−0.0468635
$257$	1.35887e11	1.94303	0.971514	−	0.236984i	$-0.0761588\pi$
$257$	1.35887e11	1.94303	0.971514	+	0.236984i	$0.0761588\pi$
$258$	0	0
$259$	−1.08897e10	−0.150372
$260$	−2.88496e8	−0.00391525
$261$	0	0
$262$	−7.33858e10	−0.962179
$263$	4.81731e10	0.620875	0.310437	−	0.950594i	$-0.399525\pi$
$263$	4.81731e10	0.620875	0.310437	+	0.950594i	$0.399525\pi$
$264$	0	0
$265$	−4.89021e9	−0.0609145
$266$	1.02455e10	0.125478
$267$	0	0
$268$	2.56602e9	0.0303846
$269$	−1.33744e11	−1.55736	−0.778682	−	0.627418i	$-0.784112\pi$
$269$	−1.33744e11	−1.55736	−0.778682	+	0.627418i	$0.784112\pi$
$270$	0	0
$271$	−7.03388e10	−0.792197	−0.396098	−	0.918208i	$-0.629636\pi$
$271$	−7.03388e10	−0.792197	−0.396098	+	0.918208i	$0.629636\pi$
$272$	−4.34032e10	−0.480797
$273$	0	0
$274$	−1.27265e9	−0.0136406
$275$	9.72326e10	1.02521
$276$	0	0
$277$	−1.34327e11	−1.37090	−0.685449	−	0.728121i	$-0.740394\pi$
$277$	−1.34327e11	−1.37090	−0.685449	+	0.728121i	$0.740394\pi$
$278$	1.52637e11	1.53270
$279$	0	0
$280$	−4.39927e9	−0.0427730
$281$	−1.15851e11	−1.10847	−0.554233	−	0.832362i	$-0.686988\pi$
$281$	−1.15851e11	−1.10847	−0.554233	+	0.832362i	$0.686988\pi$
$282$	0	0
$283$	1.40288e11	1.30011	0.650055	−	0.759887i	$-0.274746\pi$
$283$	1.40288e11	1.30011	0.650055	+	0.759887i	$0.274746\pi$
$284$	−8.73214e8	−0.00796504
$285$	0	0
$286$	9.32501e10	0.824142
$287$	−2.27559e10	−0.197982
$288$	0	0
$289$	−9.02832e10	−0.761319
$290$	5.16012e10	0.428419
$291$	0	0
$292$	−5.31712e8	−0.00428010
$293$	−1.59942e11	−1.26782	−0.633910	−	0.773407i	$-0.718551\pi$
$293$	−1.59942e11	−1.26782	−0.633910	+	0.773407i	$0.718551\pi$
$294$	0	0
$295$	6.18409e10	0.475420
$296$	−1.66613e11	−1.26153
$297$	0	0
$298$	3.27655e10	0.240682
$299$	−1.75265e11	−1.26816
$300$	0	0
$301$	−5.91852e9	−0.0415589
$302$	1.11334e11	0.770189
$303$	0	0
$304$	1.54308e11	1.03623
$305$	7.54512e10	0.499249
$306$	0	0
$307$	1.98532e11	1.27558	0.637791	−	0.770210i	$-0.279849\pi$
$307$	1.98532e11	1.27558	0.637791	+	0.770210i	$0.279849\pi$
$308$	−3.47245e8	−0.00219866
$309$	0	0
$310$	2.02256e10	0.124387
$311$	−2.24811e10	−0.136269	−0.0681344	−	0.997676i	$-0.521705\pi$
$311$	−2.24811e10	−0.136269	−0.0681344	+	0.997676i	$0.521705\pi$
$312$	0	0
$313$	−5.95978e10	−0.350978	−0.175489	−	0.984481i	$-0.556151\pi$
$313$	−5.95978e10	−0.350978	−0.175489	+	0.984481i	$0.556151\pi$
$314$	1.27761e11	0.741680
$315$	0	0
$316$	9.22860e8	0.00520647
$317$	3.02644e11	1.68332	0.841658	−	0.540012i	$-0.181580\pi$
$317$	3.02644e11	1.68332	0.841658	+	0.540012i	$0.181580\pi$
$318$	0	0
$319$	2.64745e11	1.43143
$320$	−6.72931e10	−0.358753
$321$	0	0
$322$	−4.11171e10	−0.213143
$323$	−1.00629e11	−0.514414
$324$	0	0
$325$	1.24796e11	0.620479
$326$	1.77104e11	0.868456
$327$	0	0
$328$	−3.48168e11	−1.66095
$329$	−2.36456e10	−0.111268
$330$	0	0
$331$	4.50482e10	0.206277	0.103139	−	0.994667i	$-0.467111\pi$
$331$	4.50482e10	0.206277	0.103139	+	0.994667i	$0.467111\pi$
$332$	4.91711e9	0.0222120
$333$	0	0
$334$	2.07664e11	0.913065
$335$	1.58419e11	0.687236
$336$	0	0
$337$	−1.02480e11	−0.432818	−0.216409	−	0.976303i	$-0.569435\pi$
$337$	−1.02480e11	−0.432818	−0.216409	+	0.976303i	$0.569435\pi$
$338$	−1.18385e11	−0.493371
$339$	0	0
$340$	6.64748e8	0.00269775
$341$	1.03770e11	0.415600
$342$	0	0
$343$	6.11354e10	0.238490
$344$	−9.05539e10	−0.348654
$345$	0	0
$346$	−4.16279e11	−1.56150
$347$	−1.90947e11	−0.707017	−0.353509	−	0.935431i	$-0.615012\pi$
$347$	−1.90947e11	−0.707017	−0.353509	+	0.935431i	$0.615012\pi$
$348$	0	0
$349$	−2.17959e11	−0.786431	−0.393216	−	0.919446i	$-0.628637\pi$
$349$	−2.17959e11	−0.786431	−0.393216	+	0.919446i	$0.628637\pi$
$350$	2.92772e10	0.104285
$351$	0	0
$352$	−1.05440e10	−0.0366071
$353$	−3.13329e11	−1.07403	−0.537013	−	0.843574i	$-0.680447\pi$
$353$	−3.13329e11	−1.07403	−0.537013	+	0.843574i	$0.680447\pi$
$354$	0	0
$355$	−5.39099e10	−0.180153
$356$	2.70238e9	0.00891704
$357$	0	0
$358$	−4.12924e11	−1.32861
$359$	2.37983e11	0.756173	0.378087	−	0.925770i	$-0.376582\pi$
$359$	2.37983e11	0.756173	0.378087	+	0.925770i	$0.376582\pi$
$360$	0	0
$361$	3.50706e10	0.108683
$362$	2.62697e10	0.0804019
$363$	0	0
$364$	−4.45684e8	−0.00133067
$365$	−3.28265e10	−0.0968069
$366$	0	0
$367$	3.25345e11	0.936153	0.468077	−	0.883688i	$-0.344947\pi$
$367$	3.25345e11	0.936153	0.468077	+	0.883688i	$0.344947\pi$
$368$	−6.19263e11	−1.76019
$369$	0	0
$370$	−1.58250e11	−0.438971
$371$	−7.55464e9	−0.0207029
$372$	0	0
$373$	−5.13899e11	−1.37464	−0.687319	−	0.726356i	$-0.741212\pi$
$373$	−5.13899e11	−1.37464	−0.687319	+	0.726356i	$0.741212\pi$
$374$	−2.14865e11	−0.567863
$375$	0	0
$376$	−3.61780e11	−0.933467
$377$	3.39796e11	0.866328
$378$	0	0
$379$	6.49235e11	1.61631	0.808156	−	0.588968i	$-0.200466\pi$
$379$	6.49235e11	1.61631	0.808156	+	0.588968i	$0.200466\pi$
$380$	−2.36332e9	−0.00581429
$381$	0	0
$382$	−2.32917e11	−0.559649
$383$	5.68193e11	1.34928	0.674638	−	0.738148i	$-0.264300\pi$
$383$	5.68193e11	1.34928	0.674638	+	0.738148i	$0.264300\pi$
$384$	0	0
$385$	−2.14380e10	−0.0497292
$386$	−2.63552e11	−0.604260
$387$	0	0
$388$	1.88245e9	0.00421679
$389$	−1.85758e11	−0.411315	−0.205657	−	0.978624i	$-0.565933\pi$
$389$	−1.85758e11	−0.411315	−0.205657	+	0.978624i	$0.565933\pi$
$390$	0	0
$391$	4.03842e11	0.873808
$392$	4.64291e11	0.993122
$393$	0	0
$394$	5.86309e11	1.22573
$395$	5.69749e10	0.117760
$396$	0	0
$397$	−3.00165e11	−0.606461	−0.303230	−	0.952917i	$-0.598065\pi$
$397$	−3.00165e11	−0.606461	−0.303230	+	0.952917i	$0.598065\pi$
$398$	5.19302e11	1.03740
$399$	0	0
$400$	4.40943e11	0.861218
$401$	6.13010e11	1.18391	0.591954	−	0.805972i	$-0.298357\pi$
$401$	6.13010e11	1.18391	0.591954	+	0.805972i	$0.298357\pi$
$402$	0	0
$403$	1.33187e11	0.251529
$404$	7.32936e9	0.0136883
$405$	0	0
$406$	7.97161e10	0.145606
$407$	−8.11918e11	−1.46669
$408$	0	0
$409$	1.78447e11	0.315322	0.157661	−	0.987493i	$-0.449605\pi$
$409$	1.78447e11	0.315322	0.157661	+	0.987493i	$0.449605\pi$
$410$	−3.30692e11	−0.577958
$411$	0	0
$412$	−8.06395e8	−0.00137883
$413$	9.55350e10	0.161580
$414$	0	0
$415$	3.03569e11	0.502391
$416$	−1.35331e10	−0.0221553
$417$	0	0
$418$	7.63892e11	1.22388
$419$	1.13405e11	0.179751	0.0898753	−	0.995953i	$-0.471353\pi$
$419$	1.13405e11	0.179751	0.0898753	+	0.995953i	$0.471353\pi$
$420$	0	0
$421$	1.70461e11	0.264457	0.132229	−	0.991219i	$-0.457787\pi$
$421$	1.70461e11	0.264457	0.132229	+	0.991219i	$0.457787\pi$
$422$	5.78999e11	0.888734
$423$	0	0
$424$	−1.15587e11	−0.173685
$425$	−2.87554e11	−0.427532
$426$	0	0
$427$	1.16561e11	0.169679
$428$	−1.20710e10	−0.0173878
$429$	0	0
$430$	−8.60086e10	−0.121320
$431$	−6.06017e11	−0.845936	−0.422968	−	0.906145i	$-0.639012\pi$
$431$	−6.06017e11	−0.845936	−0.422968	+	0.906145i	$0.639012\pi$
$432$	0	0
$433$	−8.94608e11	−1.22303	−0.611515	−	0.791233i	$-0.709440\pi$
$433$	−8.94608e11	−1.22303	−0.611515	+	0.791233i	$0.709440\pi$
$434$	3.12456e10	0.0422751
$435$	0	0
$436$	1.81387e10	0.0240390
$437$	−1.43574e12	−1.88326
$438$	0	0
$439$	8.16164e11	1.04879	0.524393	−	0.851476i	$-0.324292\pi$
$439$	8.16164e11	1.04879	0.524393	+	0.851476i	$0.324292\pi$
$440$	−3.28003e11	−0.417197
$441$	0	0
$442$	−2.75776e11	−0.343681
$443$	2.55777e11	0.315533	0.157767	−	0.987476i	$-0.449571\pi$
$443$	2.55777e11	0.315533	0.157767	+	0.987476i	$0.449571\pi$
$444$	0	0
$445$	1.66838e11	0.201685
$446$	−2.25256e10	−0.0269568
$447$	0	0
$448$	−1.03958e11	−0.121929
$449$	−2.65080e11	−0.307800	−0.153900	−	0.988086i	$-0.549183\pi$
$449$	−2.65080e11	−0.307800	−0.153900	+	0.988086i	$0.549183\pi$
$450$	0	0
$451$	−1.69665e12	−1.93107
$452$	−9.27693e9	−0.0104540
$453$	0	0
$454$	1.18516e12	1.30926
$455$	−2.75153e10	−0.0300970
$456$	0	0
$457$	−1.25565e12	−1.34662	−0.673312	−	0.739358i	$-0.735129\pi$
$457$	−1.25565e12	−1.34662	−0.673312	+	0.739358i	$0.735129\pi$
$458$	6.97989e11	0.741232
$459$	0	0
$460$	9.48441e9	0.00987643
$461$	1.49720e12	1.54393	0.771963	−	0.635667i	$-0.219275\pi$
$461$	1.49720e12	1.54393	0.771963	+	0.635667i	$0.219275\pi$
$462$	0	0
$463$	−1.88248e12	−1.90377	−0.951886	−	0.306453i	$-0.900858\pi$
$463$	−1.88248e12	−1.90377	−0.951886	+	0.306453i	$0.900858\pi$
$464$	1.20060e12	1.20245
$465$	0	0
$466$	−7.66289e11	−0.752759
$467$	1.43018e11	0.139145	0.0695723	−	0.997577i	$-0.477837\pi$
$467$	1.43018e11	0.139145	0.0695723	+	0.997577i	$0.477837\pi$
$468$	0	0
$469$	2.44734e11	0.233570
$470$	−3.43621e11	−0.324817
$471$	0	0
$472$	1.46170e12	1.35556
$473$	−4.41276e11	−0.405355
$474$	0	0
$475$	1.02232e12	0.921433
$476$	1.02694e9	0.000916879 0
$477$	0	0
$478$	−2.02649e12	−1.77549
$479$	3.78756e11	0.328738	0.164369	−	0.986399i	$-0.447441\pi$
$479$	3.78756e11	0.328738	0.164369	+	0.986399i	$0.447441\pi$
$480$	0	0
$481$	−1.04208e12	−0.887666
$482$	−1.73626e12	−1.46522
$483$	0	0
$484$	−7.02652e9	−0.00582018
$485$	1.16218e11	0.0953750
$486$	0	0
$487$	9.66599e9	0.00778692	0.00389346	−	0.999992i	$-0.498761\pi$
$487$	9.66599e9	0.00778692	0.00389346	+	0.999992i	$0.498761\pi$
$488$	1.78339e12	1.42350
$489$	0	0
$490$	4.40986e11	0.345575
$491$	−6.90819e11	−0.536411	−0.268205	−	0.963362i	$-0.586431\pi$
$491$	−6.90819e11	−0.536411	−0.268205	+	0.963362i	$0.586431\pi$
$492$	0	0
$493$	−7.82952e11	−0.596931
$494$	9.80443e11	0.740715
$495$	0	0
$496$	4.70589e11	0.349119
$497$	−8.32828e10	−0.0612282
$498$	0	0
$499$	7.97646e11	0.575914	0.287957	−	0.957643i	$-0.407024\pi$
$499$	7.97646e11	0.575914	0.287957	+	0.957643i	$0.407024\pi$
$500$	−1.44705e10	−0.0103542
$501$	0	0
$502$	1.01724e12	0.714916
$503$	4.71233e11	0.328231	0.164116	−	0.986441i	$-0.447523\pi$
$503$	4.71233e11	0.328231	0.164116	+	0.986441i	$0.447523\pi$
$504$	0	0
$505$	4.52495e11	0.309601
$506$	−3.06563e12	−2.07894
$507$	0	0
$508$	2.08956e10	0.0139209
$509$	2.09988e12	1.38664	0.693320	−	0.720630i	$-0.256147\pi$
$509$	2.09988e12	1.38664	0.693320	+	0.720630i	$0.256147\pi$
$510$	0	0
$511$	−5.07120e10	−0.0329016
$512$	−1.58981e12	−1.02242
$513$	0	0
$514$	3.05066e12	1.92779
$515$	−4.97847e10	−0.0311863
$516$	0	0
$517$	−1.76298e12	−1.08528
$518$	−2.44472e11	−0.149192
$519$	0	0
$520$	−4.20987e11	−0.252495
$521$	−2.49095e12	−1.48114	−0.740569	−	0.671980i	$-0.765444\pi$
$521$	−2.49095e12	−1.48114	−0.740569	+	0.671980i	$0.765444\pi$
$522$	0	0
$523$	9.27462e11	0.542049	0.271024	−	0.962572i	$-0.412638\pi$
$523$	9.27462e11	0.542049	0.271024	+	0.962572i	$0.412638\pi$
$524$	2.61509e10	0.0151529
$525$	0	0
$526$	1.08148e12	0.616005
$527$	−3.06886e11	−0.173312
$528$	0	0
$529$	3.96073e12	2.19900
$530$	−1.09785e11	−0.0604368
$531$	0	0
$532$	−3.65098e9	−0.00197609
$533$	−2.17762e12	−1.16872
$534$	0	0
$535$	−7.45230e11	−0.393277
$536$	3.74445e12	1.95951
$537$	0	0
$538$	−3.00255e12	−1.54515
$539$	2.26252e12	1.15463
$540$	0	0
$541$	1.99506e12	1.00131	0.500655	−	0.865647i	$-0.333093\pi$
$541$	1.99506e12	1.00131	0.500655	+	0.865647i	$0.333093\pi$
$542$	−1.57910e12	−0.785984
$543$	0	0
$544$	3.11827e10	0.0152658
$545$	1.11983e12	0.543712
$546$	0	0
$547$	−1.49828e11	−0.0715567	−0.0357784	−	0.999360i	$-0.511391\pi$
$547$	−1.49828e11	−0.0715567	−0.0357784	+	0.999360i	$0.511391\pi$
$548$	4.53507e8	0.000214818 0
$549$	0	0
$550$	2.18287e12	1.01717
$551$	2.78356e12	1.28653
$552$	0	0
$553$	8.80177e10	0.0400228
$554$	−3.01564e12	−1.36015
$555$	0	0
$556$	−5.43918e10	−0.0241378
$557$	6.75741e11	0.297462	0.148731	−	0.988878i	$-0.452481\pi$
$557$	6.75741e11	0.297462	0.148731	+	0.988878i	$0.452481\pi$
$558$	0	0
$559$	−5.66370e11	−0.245328
$560$	−9.72199e10	−0.0417743
$561$	0	0
$562$	−2.60086e12	−1.09977
$563$	1.25366e12	0.525885	0.262943	−	0.964811i	$-0.415307\pi$
$563$	1.25366e12	0.525885	0.262943	+	0.964811i	$0.415307\pi$
$564$	0	0
$565$	−5.72733e11	−0.236447
$566$	3.14945e12	1.28991
$567$	0	0
$568$	−1.27423e12	−0.513667
$569$	3.15164e12	1.26047	0.630233	−	0.776406i	$-0.282959\pi$
$569$	3.15164e12	1.26047	0.630233	+	0.776406i	$0.282959\pi$
$570$	0	0
$571$	−4.23310e12	−1.66647	−0.833233	−	0.552923i	$-0.813513\pi$
$571$	−4.23310e12	−1.66647	−0.833233	+	0.552923i	$0.813513\pi$
$572$	−3.32295e10	−0.0129790
$573$	0	0
$574$	−5.10870e11	−0.196429
$575$	−4.10273e12	−1.56519
$576$	0	0
$577$	4.23367e12	1.59011	0.795053	−	0.606540i	$-0.207443\pi$
$577$	4.23367e12	1.59011	0.795053	+	0.606540i	$0.207443\pi$
$578$	−2.02685e12	−0.755348
$579$	0	0
$580$	−1.83880e10	−0.00674696
$581$	4.68969e11	0.170747
$582$	0	0
$583$	−5.63263e11	−0.201931
$584$	−7.75899e11	−0.276024
$585$	0	0
$586$	−3.59068e12	−1.25788
$587$	−1.39753e12	−0.485837	−0.242919	−	0.970047i	$-0.578105\pi$
$587$	−1.39753e12	−0.485837	−0.242919	+	0.970047i	$0.578105\pi$
$588$	0	0
$589$	1.09105e12	0.373529
$590$	1.38833e12	0.471691
$591$	0	0
$592$	−3.68199e12	−1.23207
$593$	6.32131e11	0.209923	0.104962	−	0.994476i	$-0.466528\pi$
$593$	6.32131e11	0.209923	0.104962	+	0.994476i	$0.466528\pi$
$594$	0	0
$595$	6.34003e10	0.0207379
$596$	−1.16759e10	−0.00379039
$597$	0	0
$598$	−3.93468e12	−1.25821
$599$	−5.15452e12	−1.63594	−0.817970	−	0.575261i	$-0.804900\pi$
$599$	−5.15452e12	−1.63594	−0.817970	+	0.575261i	$0.804900\pi$
$600$	0	0
$601$	−7.37044e11	−0.230440	−0.115220	−	0.993340i	$-0.536757\pi$
$601$	−7.37044e11	−0.230440	−0.115220	+	0.993340i	$0.536757\pi$
$602$	−1.32870e11	−0.0412329
$603$	0	0
$604$	−3.96737e10	−0.0121293
$605$	−4.33799e11	−0.131640
$606$	0	0
$607$	1.87389e12	0.560267	0.280134	−	0.959961i	$-0.409621\pi$
$607$	1.87389e12	0.560267	0.280134	+	0.959961i	$0.409621\pi$
$608$	−1.10861e11	−0.0329013
$609$	0	0
$610$	1.69387e12	0.495333
$611$	−2.26276e12	−0.656829
$612$	0	0
$613$	−1.62540e12	−0.464931	−0.232465	−	0.972605i	$-0.574679\pi$
$613$	−1.62540e12	−0.464931	−0.232465	+	0.972605i	$0.574679\pi$
$614$	4.45703e12	1.26558
$615$	0	0
$616$	−5.06716e11	−0.141792
$617$	−4.48555e12	−1.24604	−0.623021	−	0.782205i	$-0.714095\pi$
$617$	−4.48555e12	−1.24604	−0.623021	+	0.782205i	$0.714095\pi$
$618$	0	0
$619$	2.64489e12	0.724102	0.362051	−	0.932158i	$-0.382077\pi$
$619$	2.64489e12	0.724102	0.362051	+	0.932158i	$0.382077\pi$
$620$	−7.20737e9	−0.00195891
$621$	0	0
$622$	−5.04700e11	−0.135200
$623$	2.57739e11	0.0685463
$624$	0	0
$625$	2.44489e12	0.640913
$626$	−1.33797e12	−0.348226
$627$	0	0
$628$	−4.55276e10	−0.0116804
$629$	2.40115e12	0.611633
$630$	0	0
$631$	−2.53561e12	−0.636723	−0.318361	−	0.947969i	$-0.603133\pi$
$631$	−2.53561e12	−0.636723	−0.318361	+	0.947969i	$0.603133\pi$
$632$	1.34668e12	0.335767
$633$	0	0
$634$	6.79434e12	1.67011
$635$	1.29004e12	0.314862
$636$	0	0
$637$	2.90391e12	0.698805
$638$	5.94351e12	1.42020
$639$	0	0
$640$	−1.46386e12	−0.344897
$641$	−7.11969e11	−0.166571	−0.0832856	−	0.996526i	$-0.526541\pi$
$641$	−7.11969e11	−0.166571	−0.0832856	+	0.996526i	$0.526541\pi$
$642$	0	0
$643$	−5.97690e11	−0.137888	−0.0689441	−	0.997621i	$-0.521963\pi$
$643$	−5.97690e11	−0.137888	−0.0689441	+	0.997621i	$0.521963\pi$
$644$	1.46520e10	0.00335668
$645$	0	0
$646$	−2.25912e12	−0.510379
$647$	3.90064e12	0.875119	0.437559	−	0.899189i	$-0.355843\pi$
$647$	3.90064e12	0.875119	0.437559	+	0.899189i	$0.355843\pi$
$648$	0	0
$649$	7.12295e12	1.57601
$650$	2.80167e12	0.615612
$651$	0	0
$652$	−6.31106e10	−0.0136769
$653$	2.73627e11	0.0588912	0.0294456	−	0.999566i	$-0.490626\pi$
$653$	2.73627e11	0.0588912	0.0294456	+	0.999566i	$0.490626\pi$
$654$	0	0
$655$	1.61449e12	0.342727
$656$	−7.69419e12	−1.62217
$657$	0	0
$658$	−5.30843e11	−0.110395
$659$	−1.93239e12	−0.399126	−0.199563	−	0.979885i	$-0.563952\pi$
$659$	−1.93239e12	−0.399126	−0.199563	+	0.979885i	$0.563952\pi$
$660$	0	0
$661$	8.99819e12	1.83336	0.916682	−	0.399617i	$-0.130857\pi$
$661$	8.99819e12	1.83336	0.916682	+	0.399617i	$0.130857\pi$
$662$	1.01133e12	0.204660
$663$	0	0
$664$	7.17528e12	1.43246
$665$	−2.25402e11	−0.0446951
$666$	0	0
$667$	−1.11709e13	−2.18536
$668$	−7.40007e10	−0.0143794
$669$	0	0
$670$	3.55650e12	0.681846
$671$	8.69060e12	1.65500
$672$	0	0
$673$	−3.80149e12	−0.714309	−0.357155	−	0.934045i	$-0.616253\pi$
$673$	−3.80149e12	−0.714309	−0.357155	+	0.934045i	$0.616253\pi$
$674$	−2.30068e12	−0.429424
$675$	0	0
$676$	4.21865e10	0.00776986
$677$	−2.51015e12	−0.459252	−0.229626	−	0.973279i	$-0.573750\pi$
$677$	−2.51015e12	−0.459252	−0.229626	+	0.973279i	$0.573750\pi$
$678$	0	0
$679$	1.79539e11	0.0324149
$680$	9.70031e11	0.173978
$681$	0	0
$682$	2.32962e12	0.412341
$683$	−1.00278e12	−0.176325	−0.0881623	−	0.996106i	$-0.528099\pi$
$683$	−1.00278e12	−0.176325	−0.0881623	+	0.996106i	$0.528099\pi$
$684$	0	0
$685$	2.79983e10	0.00485875
$686$	1.37249e12	0.236619
$687$	0	0
$688$	−2.00116e12	−0.340513
$689$	−7.22939e11	−0.122212
$690$	0	0
$691$	1.14248e12	0.190632	0.0953161	−	0.995447i	$-0.469614\pi$
$691$	1.14248e12	0.190632	0.0953161	+	0.995447i	$0.469614\pi$
$692$	1.48340e11	0.0245913
$693$	0	0
$694$	−4.28675e12	−0.701472
$695$	−3.35801e12	−0.545946
$696$	0	0
$697$	5.01764e12	0.805289
$698$	−4.89317e12	−0.780263
$699$	0	0
$700$	−1.04329e10	−0.00164234
$701$	−5.29555e12	−0.828285	−0.414143	−	0.910212i	$-0.635919\pi$
$701$	−5.29555e12	−0.828285	−0.414143	+	0.910212i	$0.635919\pi$
$702$	0	0
$703$	−8.53660e12	−1.31821
$704$	−7.75095e12	−1.18926
$705$	0	0
$706$	−7.03422e12	−1.06560
$707$	6.99037e11	0.105224
$708$	0	0
$709$	−1.01497e13	−1.50849	−0.754246	−	0.656592i	$-0.771998\pi$
$709$	−1.01497e13	−1.50849	−0.754246	+	0.656592i	$0.771998\pi$
$710$	−1.21027e12	−0.178740
$711$	0	0
$712$	3.94343e12	0.575062
$713$	−4.37856e12	−0.634495
$714$	0	0
$715$	−2.05150e12	−0.293559
$716$	1.47145e11	0.0209236
$717$	0	0
$718$	5.34271e12	0.750242
$719$	5.75277e12	0.802780	0.401390	−	0.915907i	$-0.368527\pi$
$719$	5.75277e12	0.802780	0.401390	+	0.915907i	$0.368527\pi$
$720$	0	0
$721$	−7.69100e10	−0.0105992
$722$	7.87333e11	0.107830
$723$	0	0
$724$	−9.36116e9	−0.00126621
$725$	7.95420e12	1.06924
$726$	0	0
$727$	−1.10169e13	−1.46270	−0.731351	−	0.682001i	$-0.761110\pi$
$727$	−1.10169e13	−1.46270	−0.731351	+	0.682001i	$0.761110\pi$
$728$	−6.50362e11	−0.0858152
$729$	0	0
$730$	−7.36953e11	−0.0960476
$731$	1.30502e12	0.169040
$732$	0	0
$733$	−1.75672e12	−0.224768	−0.112384	−	0.993665i	$-0.535849\pi$
$733$	−1.75672e12	−0.224768	−0.112384	+	0.993665i	$0.535849\pi$
$734$	7.30398e12	0.928811
$735$	0	0
$736$	4.44905e11	0.0558878
$737$	1.82470e13	2.27818
$738$	0	0
$739$	−4.52901e12	−0.558603	−0.279301	−	0.960204i	$-0.590103\pi$
$739$	−4.52901e12	−0.558603	−0.279301	+	0.960204i	$0.590103\pi$
$740$	5.63921e10	0.00691314
$741$	0	0
$742$	−1.69601e11	−0.0205405
$743$	−2.25595e12	−0.271569	−0.135784	−	0.990738i	$-0.543355\pi$
$743$	−2.25595e12	−0.271569	−0.135784	+	0.990738i	$0.543355\pi$
$744$	0	0
$745$	−7.20841e11	−0.0857307
$746$	−1.15370e13	−1.36386
$747$	0	0
$748$	7.65669e10	0.00894301
$749$	−1.15127e12	−0.133662
$750$	0	0
$751$	−9.29242e12	−1.06598	−0.532990	−	0.846122i	$-0.678932\pi$
$751$	−9.29242e12	−1.06598	−0.532990	+	0.846122i	$0.678932\pi$
$752$	−7.99500e12	−0.911671
$753$	0	0
$754$	7.62840e12	0.859533
$755$	−2.44935e12	−0.274340
$756$	0	0
$757$	7.28505e12	0.806308	0.403154	−	0.915132i	$-0.367914\pi$
$757$	7.28505e12	0.806308	0.403154	+	0.915132i	$0.367914\pi$
$758$	1.45753e13	1.60364
$759$	0	0
$760$	−3.44867e12	−0.374965
$761$	2.78674e12	0.301207	0.150604	−	0.988594i	$-0.451878\pi$
$761$	2.78674e12	0.301207	0.150604	+	0.988594i	$0.451878\pi$
$762$	0	0
$763$	1.72998e12	0.184790
$764$	8.29995e10	0.00881364
$765$	0	0
$766$	1.27559e13	1.33869
$767$	9.14219e12	0.953830
$768$	0	0
$769$	4.63000e12	0.477433	0.238716	−	0.971089i	$-0.423273\pi$
$769$	4.63000e12	0.477433	0.238716	+	0.971089i	$0.423273\pi$
$770$	−4.81282e11	−0.0493391
$771$	0	0
$772$	9.39164e10	0.00951620
$773$	−8.49811e12	−0.856080	−0.428040	−	0.903760i	$-0.640796\pi$
$773$	−8.49811e12	−0.856080	−0.428040	+	0.903760i	$0.640796\pi$
$774$	0	0
$775$	3.11773e12	0.310442
$776$	2.74697e12	0.271941
$777$	0	0
$778$	−4.17026e12	−0.408089
$779$	−1.78388e13	−1.73559
$780$	0	0
$781$	−6.20944e12	−0.597204
$782$	9.06623e12	0.866954
$783$	0	0
$784$	1.02604e13	0.969933
$785$	−2.81075e12	−0.264186
$786$	0	0
$787$	3.57079e12	0.331801	0.165900	−	0.986143i	$-0.446947\pi$
$787$	3.57079e12	0.331801	0.165900	+	0.986143i	$0.446947\pi$
$788$	−2.08930e11	−0.0193034
$789$	0	0
$790$	1.27908e12	0.116836
$791$	−8.84787e11	−0.0803608
$792$	0	0
$793$	1.11542e13	1.00164
$794$	−6.73868e12	−0.601704
$795$	0	0
$796$	−1.85052e11	−0.0163375
$797$	−9.06883e12	−0.796139	−0.398070	−	0.917355i	$-0.630320\pi$
$797$	−9.06883e12	−0.796139	−0.398070	+	0.917355i	$0.630320\pi$
$798$	0	0
$799$	5.21381e12	0.452579
$800$	−3.16793e11	−0.0273445
$801$	0	0
$802$	1.37620e13	1.17462
$803$	−3.78101e12	−0.320914
$804$	0	0
$805$	9.04576e11	0.0759213
$806$	2.99003e12	0.249556
$807$	0	0
$808$	1.06953e13	0.882762
$809$	−1.10403e13	−0.906173	−0.453087	−	0.891467i	$-0.649677\pi$
$809$	−1.10403e13	−0.906173	−0.453087	+	0.891467i	$0.649677\pi$
$810$	0	0
$811$	−1.04507e12	−0.0848302	−0.0424151	−	0.999100i	$-0.513505\pi$
$811$	−1.04507e12	−0.0848302	−0.0424151	+	0.999100i	$0.513505\pi$
$812$	−2.84067e10	−0.00229308
$813$	0	0
$814$	−1.82275e13	−1.45518
$815$	−3.89628e12	−0.309343
$816$	0	0
$817$	−4.63963e12	−0.364321
$818$	4.00612e12	0.312849
$819$	0	0
$820$	1.17841e11	0.00910198
$821$	1.04318e13	0.801340	0.400670	−	0.916223i	$-0.368777\pi$
$821$	1.04318e13	0.801340	0.400670	+	0.916223i	$0.368777\pi$
$822$	0	0
$823$	7.03298e12	0.534367	0.267184	−	0.963646i	$-0.413907\pi$
$823$	7.03298e12	0.534367	0.267184	+	0.963646i	$0.413907\pi$
$824$	−1.17673e12	−0.0889210
$825$	0	0
$826$	2.14476e12	0.160313
$827$	1.69223e13	1.25801	0.629006	−	0.777400i	$-0.283462\pi$
$827$	1.69223e13	1.25801	0.629006	+	0.777400i	$0.283462\pi$
$828$	0	0
$829$	2.27155e12	0.167042	0.0835212	−	0.996506i	$-0.473383\pi$
$829$	2.27155e12	0.167042	0.0835212	+	0.996506i	$0.473383\pi$
$830$	6.81511e12	0.498450
$831$	0	0
$832$	−9.94821e12	−0.719764
$833$	−6.69114e12	−0.481502
$834$	0	0
$835$	−4.56860e12	−0.325233
$836$	−2.72212e11	−0.0192743
$837$	0	0
$838$	2.54594e12	0.178341
$839$	−1.79457e13	−1.25035	−0.625175	−	0.780484i	$-0.714973\pi$
$839$	−1.79457e13	−1.25035	−0.625175	+	0.780484i	$0.714973\pi$
$840$	0	0
$841$	7.15056e12	0.492899
$842$	3.82684e12	0.262383
$843$	0	0
$844$	−2.06325e11	−0.0139962
$845$	2.60448e12	0.175738
$846$	0	0
$847$	−6.70154e11	−0.0447404
$848$	−2.55436e12	−0.169629
$849$	0	0
$850$	−6.45557e12	−0.424179
$851$	3.42588e13	2.23918
$852$	0	0
$853$	−2.20102e13	−1.42349	−0.711744	−	0.702439i	$-0.752095\pi$
$853$	−2.20102e13	−1.42349	−0.711744	+	0.702439i	$0.752095\pi$
$854$	2.61678e12	0.168348
$855$	0	0
$856$	−1.76145e13	−1.12135
$857$	2.34048e13	1.48215	0.741074	−	0.671424i	$-0.234317\pi$
$857$	2.34048e13	1.48215	0.741074	+	0.671424i	$0.234317\pi$
$858$	0	0
$859$	4.26808e10	0.00267463	0.00133732	−	0.999999i	$-0.499574\pi$
$859$	4.26808e10	0.00267463	0.00133732	+	0.999999i	$0.499574\pi$
$860$	3.06490e10	0.00191062
$861$	0	0
$862$	−1.36051e13	−0.839301
$863$	−3.10119e13	−1.90318	−0.951590	−	0.307370i	$-0.900551\pi$
$863$	−3.10119e13	−1.90318	−0.951590	+	0.307370i	$0.900551\pi$
$864$	0	0
$865$	9.15814e12	0.556205
$866$	−2.00839e13	−1.21344
$867$	0	0
$868$	−1.11343e10	−0.000665770 0
$869$	6.56247e12	0.390372
$870$	0	0
$871$	2.34197e13	1.37880
$872$	2.64688e13	1.55028
$873$	0	0
$874$	−3.22324e13	−1.86849
$875$	−1.38012e12	−0.0795943
$876$	0	0
$877$	−1.64046e13	−0.936416	−0.468208	−	0.883618i	$-0.655100\pi$
$877$	−1.64046e13	−0.936416	−0.468208	+	0.883618i	$0.655100\pi$
$878$	1.83228e13	1.04056
$879$	0	0
$880$	−7.24857e12	−0.407456
$881$	−2.25741e13	−1.26246	−0.631231	−	0.775595i	$-0.717450\pi$
$881$	−2.25741e13	−1.26246	−0.631231	+	0.775595i	$0.717450\pi$
$882$	0	0
$883$	−7.75871e12	−0.429503	−0.214752	−	0.976669i	$-0.568894\pi$
$883$	−7.75871e12	−0.429503	−0.214752	+	0.976669i	$0.568894\pi$
$884$	9.82723e10	0.00541247
$885$	0	0
$886$	5.74219e12	0.313058
$887$	−4.44036e12	−0.240858	−0.120429	−	0.992722i	$-0.538427\pi$
$887$	−4.44036e12	−0.240858	−0.120429	+	0.992722i	$0.538427\pi$
$888$	0	0
$889$	1.99291e12	0.107012
$890$	3.74549e12	0.200103
$891$	0	0
$892$	8.02694e9	0.000424530 0
$893$	−1.85362e13	−0.975414
$894$	0	0
$895$	9.08433e12	0.473249
$896$	−2.26144e12	−0.117219
$897$	0	0
$898$	−5.95104e12	−0.305386
$899$	8.48897e12	0.433447
$900$	0	0
$901$	1.66578e12	0.0842087
$902$	−3.80897e13	−1.91592
$903$	0	0
$904$	−1.35373e13	−0.674178
$905$	−5.77933e11	−0.0286391
$906$	0	0
$907$	1.71069e13	0.839342	0.419671	−	0.907676i	$-0.362145\pi$
$907$	1.71069e13	0.839342	0.419671	+	0.907676i	$0.362145\pi$
$908$	−4.22329e11	−0.0206189
$909$	0	0
$910$	−6.17717e11	−0.0298609
$911$	1.88236e13	0.905460	0.452730	−	0.891648i	$-0.350450\pi$
$911$	1.88236e13	0.905460	0.452730	+	0.891648i	$0.350450\pi$
$912$	0	0
$913$	3.49657e13	1.66542
$914$	−2.81893e13	−1.33606
$915$	0	0
$916$	−2.48727e11	−0.0116733
$917$	2.49414e12	0.116482
$918$	0	0
$919$	−4.20992e13	−1.94695	−0.973473	−	0.228800i	$-0.926520\pi$
$919$	−4.20992e13	−1.94695	−0.973473	+	0.228800i	$0.926520\pi$
$920$	1.38401e13	0.636933
$921$	0	0
$922$	3.36121e13	1.53182
$923$	−7.96971e12	−0.361439
$924$	0	0
$925$	−2.43938e13	−1.09558
$926$	−4.22615e13	−1.88884
$927$	0	0
$928$	−8.62564e11	−0.0381791
$929$	1.23689e13	0.544831	0.272415	−	0.962180i	$-0.412177\pi$
$929$	1.23689e13	0.544831	0.272415	+	0.962180i	$0.412177\pi$
$930$	0	0
$931$	2.37885e13	1.03775
$932$	2.73066e11	0.0118548
$933$	0	0
$934$	3.21076e12	0.138053
$935$	4.72703e12	0.202272
$936$	0	0
$937$	2.10110e13	0.890467	0.445234	−	0.895414i	$-0.353121\pi$
$937$	2.10110e13	0.890467	0.445234	+	0.895414i	$0.353121\pi$
$938$	5.49426e12	0.231738
$939$	0	0
$940$	1.22449e11	0.00511539
$941$	1.49975e13	0.623543	0.311771	−	0.950157i	$-0.399078\pi$
$941$	1.49975e13	0.623543	0.311771	+	0.950157i	$0.399078\pi$
$942$	0	0
$943$	7.15900e13	2.94815
$944$	3.23021e13	1.32391
$945$	0	0
$946$	−9.90662e12	−0.402175
$947$	1.30832e12	0.0528615	0.0264307	−	0.999651i	$-0.491586\pi$
$947$	1.30832e12	0.0528615	0.0264307	+	0.999651i	$0.491586\pi$
$948$	0	0
$949$	−4.85287e12	−0.194223
$950$	2.29509e13	0.914206
$951$	0	0
$952$	1.49855e12	0.0591297
$953$	4.18691e13	1.64428	0.822140	−	0.569285i	$-0.192780\pi$
$953$	4.18691e13	1.64428	0.822140	+	0.569285i	$0.192780\pi$
$954$	0	0
$955$	5.12417e12	0.199346
$956$	7.22136e11	0.0279614
$957$	0	0
$958$	8.50306e12	0.326160
$959$	4.32532e10	0.00165133
$960$	0	0
$961$	−2.31123e13	−0.874153
$962$	−2.33947e13	−0.880703
$963$	0	0
$964$	6.18714e11	0.0230751
$965$	5.79815e12	0.215237
$966$	0	0
$967$	−1.10706e12	−0.0407148	−0.0203574	−	0.999793i	$-0.506480\pi$
$967$	−1.10706e12	−0.0407148	−0.0203574	+	0.999793i	$0.506480\pi$
$968$	−1.02534e13	−0.375344
$969$	0	0
$970$	2.60908e12	0.0946270
$971$	−2.21624e13	−0.800075	−0.400038	−	0.916499i	$-0.631003\pi$
$971$	−2.21624e13	−0.800075	−0.400038	+	0.916499i	$0.631003\pi$
$972$	0	0
$973$	−5.18762e12	−0.185550
$974$	2.17001e11	0.00772585
$975$	0	0
$976$	3.94113e13	1.39026
$977$	−2.19683e13	−0.771384	−0.385692	−	0.922628i	$-0.626037\pi$
$977$	−2.19683e13	−0.771384	−0.385692	+	0.922628i	$0.626037\pi$
$978$	0	0
$979$	1.92166e13	0.668583
$980$	−1.57145e11	−0.00544229
$981$	0	0
$982$	−1.55088e13	−0.532204
$983$	4.07229e13	1.39107	0.695533	−	0.718494i	$-0.255168\pi$
$983$	4.07229e13	1.39107	0.695533	+	0.718494i	$0.255168\pi$
$984$	0	0
$985$	−1.28988e13	−0.436602
$986$	−1.75772e13	−0.592249
$987$	0	0
$988$	−3.49379e11	−0.0116652
$989$	1.86196e13	0.618853
$990$	0	0
$991$	1.69138e13	0.557070	0.278535	−	0.960426i	$-0.410151\pi$
$991$	1.69138e13	0.557070	0.278535	+	0.960426i	$0.410151\pi$
$992$	−3.38091e11	−0.0110849
$993$	0	0
$994$	−1.86969e12	−0.0607480
$995$	−1.14246e13	−0.369521
$996$	0	0
$997$	−4.00661e13	−1.28425	−0.642125	−	0.766600i	$-0.721947\pi$
$997$	−4.00661e13	−1.28425	−0.642125	+	0.766600i	$0.721947\pi$
$998$	1.79071e13	0.571397
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.a.a.1.2	yes	2
3.2	odd	2	inner	27.10.a.a.1.1	✓	2
9.2	odd	6		81.10.c.f.28.2		4
9.4	even	3		81.10.c.f.55.1		4
9.5	odd	6		81.10.c.f.55.2		4
9.7	even	3		81.10.c.f.28.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.a.a.1.1	✓	2	3.2	odd	2	inner
27.10.a.a.1.2	yes	2	1.1	even	1	trivial
81.10.c.f.28.1		4	9.7	even	3
81.10.c.f.28.2		4	9.2	odd	6
81.10.c.f.55.1		4	9.4	even	3
81.10.c.f.55.2		4	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+22.4499 q^{2} -8.00000 q^{4} -493.899 q^{5} -763.000 q^{7} -11674.0 q^{8} -11088.0 q^{10} -56888.2 q^{11} -73015.0 q^{13} -17129.3 q^{14} -257984. q^{16} +168240. q^{17} -598129. q^{19} +3951.19 q^{20} -1.27714e6 q^{22} +2.40039e6 q^{23} -1.70919e6 q^{25} -1.63918e6 q^{26} +6104.00 q^{28} -4.65378e6 q^{29} -1.82410e6 q^{31} +185347. q^{32} +3.77698e6 q^{34} +376845. q^{35} +1.42722e7 q^{37} -1.34280e7 q^{38} +5.76576e6 q^{40} +2.98243e7 q^{41} +7.75690e6 q^{43} +455105. q^{44} +5.38887e7 q^{46} +3.09903e7 q^{47} -3.97714e7 q^{49} -3.83712e7 q^{50} +584120. q^{52} +9.90123e6 q^{53} +2.80970e7 q^{55} +8.90724e6 q^{56} -1.04477e8 q^{58} -1.25210e8 q^{59} -1.52766e8 q^{61} -4.09509e7 q^{62} +1.36249e8 q^{64} +3.60620e7 q^{65} -3.20752e8 q^{67} -1.34592e6 q^{68} +8.46014e6 q^{70} +1.09152e8 q^{71} +6.64640e7 q^{73} +3.20410e8 q^{74} +4.78503e6 q^{76} +4.34057e7 q^{77} -1.15357e8 q^{79} +1.27418e8 q^{80} +6.69554e8 q^{82} -6.14639e8 q^{83} -8.30935e7 q^{85} +1.74142e8 q^{86} +6.64111e8 q^{88} -3.37797e8 q^{89} +5.57104e7 q^{91} -1.92031e7 q^{92} +6.95731e8 q^{94} +2.95415e8 q^{95} -2.35307e8 q^{97} -8.92867e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{4} - 1526 q^{7} - 22176 q^{10} - 146030 q^{13} - 515968 q^{16} - 1196258 q^{19} - 2554272 q^{22} - 3418378 q^{25} + 12208 q^{28} - 3648200 q^{31} + 7553952 q^{34} + 28544350 q^{37} + 11531520 q^{40}+ \cdots - 470613686 q^{97}+O(q^{100}) \) 2 * q - 16 * q^4 - 1526 * q^7 - 22176 * q^10 - 146030 * q^13 - 515968 * q^16 - 1196258 * q^19 - 2554272 * q^22 - 3418378 * q^25 + 12208 * q^28 - 3648200 * q^31 + 7553952 * q^34 + 28544350 * q^37 + 11531520 * q^40 + 15513808 * q^43 + 107777376 * q^46 - 79542876 * q^49 + 1168240 * q^52 + 56193984 * q^55 - 208954368 * q^58 - 305532986 * q^61 + 272497664 * q^64 - 641504138 * q^67 + 16920288 * q^70 + 132927970 * q^73 + 9570064 * q^76 - 230714906 * q^79 + 1339107840 * q^82 - 166186944 * q^85 + 1328221440 * q^88 + 111420890 * q^91 + 1391461344 * q^94 - 470613686 * q^97

Introduction

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