LMFDB - Embedded newform 16.24.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,24,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 24, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	16.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:	$53.6326459752$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{144169}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 36042 $ x^2 - x - 36042
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-189.348$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+48964.9 q^{3} -3.19930e7 q^{5} +5.17135e9 q^{7} -9.17456e10 q^{9} +O(q^{10})$	$q+48964.9 q^{3} -3.19930e7 q^{5} +5.17135e9 q^{7} -9.17456e10 q^{9} -6.04602e11 q^{11} +7.96710e12 q^{13} -1.56653e12 q^{15} +1.98385e13 q^{17} -6.27346e14 q^{19} +2.53215e14 q^{21} +4.55685e15 q^{23} -1.08974e16 q^{25} -9.10202e15 q^{27} +4.14107e16 q^{29} -1.35683e15 q^{31} -2.96043e16 q^{33} -1.65447e17 q^{35} +3.41258e17 q^{37} +3.90108e17 q^{39} -3.69518e18 q^{41} +1.96955e18 q^{43} +2.93522e18 q^{45} -2.44381e19 q^{47} -6.25860e17 q^{49} +9.71391e17 q^{51} -6.39437e19 q^{53} +1.93431e19 q^{55} -3.07179e19 q^{57} -2.81892e20 q^{59} -4.67790e20 q^{61} -4.74449e20 q^{63} -2.54892e20 q^{65} -2.77406e20 q^{67} +2.23126e20 q^{69} -2.29069e21 q^{71} -4.56908e21 q^{73} -5.33588e20 q^{75} -3.12661e21 q^{77} +3.99005e21 q^{79} +8.19155e21 q^{81} -1.45920e22 q^{83} -6.34694e20 q^{85} +2.02767e21 q^{87} +1.80991e21 q^{89} +4.12007e22 q^{91} -6.64369e19 q^{93} +2.00707e22 q^{95} +8.25561e22 q^{97} +5.54696e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 339480 q^{3} + 73069020 q^{5} + 1359184400 q^{7} - 34999394166 q^{9}+O(q^{10}) $ 2 * q - 339480 * q^3 + 73069020 * q^5 + 1359184400 * q^7 - 34999394166 * q^9	$ 2 q - 339480 q^{3} + 73069020 q^{5} + 1359184400 q^{7} - 34999394166 q^{9} - 856801968264 q^{11} + 4376109322060 q^{13} - 42377338985040 q^{15} + 254028147597540 q^{17} - 4260600979960 q^{19} + 17\!\cdots\!44 q^{21}+ \cdots + 41\!\cdots\!12 q^{99}+O(q^{100}) $ 2 * q - 339480 * q^3 + 73069020 * q^5 + 1359184400 * q^7 - 34999394166 * q^9 - 856801968264 * q^11 + 4376109322060 * q^13 - 42377338985040 * q^15 + 254028147597540 * q^17 - 4260600979960 * q^19 + 1734031637722944 * q^21 + 8144713079008560 * q^23 - 11780274628800850 * q^25 + 5424634982716560 * q^27 + 20818433601623340 * q^29 - 137714017177000384 * q^31 + 68361366766001760 * q^33 - 565961271250425120 * q^35 - 897721264408967780 * q^37 + 1785011473665029232 * q^39 - 2294435477168314956 * q^41 + 1750760768619855800 * q^43 + 8897092690294206540 * q^45 - 15759744217656780960 * q^47 - 13461981704376200814 * q^49 - 89998362845078292144 * q^51 - 140287253401646796420 * q^53 - 7153550955060182640 * q^55 - 272752401448627175520 * q^57 - 280872989971340771880 * q^59 - 180452892516502223636 * q^61 - 690775113933935014320 * q^63 - 632168834809440380760 * q^65 - 1754233163431557625240 * q^67 - 1170560672172404223552 * q^69 - 3055033510194143328624 * q^71 - 8063408253877606149260 * q^73 - 190631089350520885800 * q^75 - 2165184764357449665600 * q^77 - 6244916814559639980640 * q^79 - 2793528580929833975598 * q^81 - 6875994082418498976120 * q^83 + 23969743087870314902520 * q^85 + 10026640653837674384880 * q^87 + 6395093086173070004820 * q^89 + 54890178162704560146016 * q^91 + 52900811441357852079360 * q^93 + 85533361066700858502000 * q^95 - 31147288846254030500540 * q^97 + 41158245132312135981912 * q^99

Display 100 coefficients 10 coefficients

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	0
$2$	0	0
$3$	48964.9	0.159584	0.0797921	−	0.996812i	$-0.474574\pi$
$3$	48964.9	0.159584	0.0797921	+	0.996812i	$0.474574\pi$
$4$	0	0
$5$	−3.19930e7	−0.293022	−0.146511	−	0.989209i	$-0.546804\pi$
$5$	−3.19930e7	−0.293022	−0.146511	+	0.989209i	$0.546804\pi$
$6$	0	0
$7$	5.17135e9	0.988500	0.494250	−	0.869320i	$-0.335443\pi$
$7$	5.17135e9	0.988500	0.494250	+	0.869320i	$0.335443\pi$
$8$	0	0
$9$	−9.17456e10	−0.974533
$10$	0	0
$11$	−6.04602e11	−0.638931	−0.319466	−	0.947598i	$-0.603503\pi$
$11$	−6.04602e11	−0.638931	−0.319466	+	0.947598i	$0.603503\pi$
$12$	0	0
$13$	7.96710e12	1.23297	0.616484	−	0.787367i	$-0.288556\pi$
$13$	7.96710e12	1.23297	0.616484	+	0.787367i	$0.288556\pi$
$14$	0	0
$15$	−1.56653e12	−0.0467617
$16$	0	0
$17$	1.98385e13	0.140393	0.0701967	−	0.997533i	$-0.477637\pi$
$17$	1.98385e13	0.140393	0.0701967	+	0.997533i	$0.477637\pi$
$18$	0	0
$19$	−6.27346e14	−1.23550	−0.617748	−	0.786377i	$-0.711955\pi$
$19$	−6.27346e14	−1.23550	−0.617748	+	0.786377i	$0.711955\pi$
$20$	0	0
$21$	2.53215e14	0.157749
$22$	0	0
$23$	4.55685e15	0.997229	0.498614	−	0.866824i	$-0.333842\pi$
$23$	4.55685e15	0.997229	0.498614	+	0.866824i	$0.333842\pi$
$24$	0	0
$25$	−1.08974e16	−0.914138
$26$	0	0
$27$	−9.10202e15	−0.315104
$28$	0	0
$29$	4.14107e16	0.630283	0.315142	−	0.949045i	$-0.397948\pi$
$29$	4.14107e16	0.630283	0.315142	+	0.949045i	$0.397948\pi$
$30$	0	0
$31$	−1.35683e15	−0.00959104	−0.00479552	−	0.999989i	$-0.501526\pi$
$31$	−1.35683e15	−0.00959104	−0.00479552	+	0.999989i	$0.501526\pi$
$32$	0	0
$33$	−2.96043e16	−0.101963
$34$	0	0
$35$	−1.65447e17	−0.289652
$36$	0	0
$37$	3.41258e17	0.315328	0.157664	−	0.987493i	$-0.449604\pi$
$37$	3.41258e17	0.315328	0.157664	+	0.987493i	$0.449604\pi$
$38$	0	0
$39$	3.90108e17	0.196762
$40$	0	0
$41$	−3.69518e18	−1.04863	−0.524313	−	0.851525i	$-0.675678\pi$
$41$	−3.69518e18	−1.04863	−0.524313	+	0.851525i	$0.675678\pi$
$42$	0	0
$43$	1.96955e18	0.323207	0.161603	−	0.986856i	$-0.448333\pi$
$43$	1.96955e18	0.323207	0.161603	+	0.986856i	$0.448333\pi$
$44$	0	0
$45$	2.93522e18	0.285559
$46$	0	0
$47$	−2.44381e19	−1.44192	−0.720962	−	0.692975i	$-0.756300\pi$
$47$	−2.44381e19	−1.44192	−0.720962	+	0.692975i	$0.756300\pi$
$48$	0	0
$49$	−6.25860e17	−0.0228677
$50$	0	0
$51$	9.71391e17	0.0224046
$52$	0	0
$53$	−6.39437e19	−0.947600	−0.473800	−	0.880632i	$-0.657118\pi$
$53$	−6.39437e19	−0.947600	−0.473800	+	0.880632i	$0.657118\pi$
$54$	0	0
$55$	1.93431e19	0.187221
$56$	0	0
$57$	−3.07179e19	−0.197166
$58$	0	0
$59$	−2.81892e20	−1.21698	−0.608491	−	0.793561i	$-0.708225\pi$
$59$	−2.81892e20	−1.21698	−0.608491	+	0.793561i	$0.708225\pi$
$60$	0	0
$61$	−4.67790e20	−1.37644	−0.688221	−	0.725501i	$-0.741608\pi$
$61$	−4.67790e20	−1.37644	−0.688221	+	0.725501i	$0.741608\pi$
$62$	0	0
$63$	−4.74449e20	−0.963326
$64$	0	0
$65$	−2.54892e20	−0.361287
$66$	0	0
$67$	−2.77406e20	−0.277495	−0.138747	−	0.990328i	$-0.544308\pi$
$67$	−2.77406e20	−0.277495	−0.138747	+	0.990328i	$0.544308\pi$
$68$	0	0
$69$	2.23126e20	0.159142
$70$	0	0
$71$	−2.29069e21	−1.17624	−0.588119	−	0.808775i	$-0.700131\pi$
$71$	−2.29069e21	−1.17624	−0.588119	+	0.808775i	$0.700131\pi$
$72$	0	0
$73$	−4.56908e21	−1.70457	−0.852286	−	0.523075i	$-0.824785\pi$
$73$	−4.56908e21	−1.70457	−0.852286	+	0.523075i	$0.824785\pi$
$74$	0	0
$75$	−5.33588e20	−0.145882
$76$	0	0
$77$	−3.12661e21	−0.631583
$78$	0	0
$79$	3.99005e21	0.600160	0.300080	−	0.953914i	$-0.402987\pi$
$79$	3.99005e21	0.600160	0.300080	+	0.953914i	$0.402987\pi$
$80$	0	0
$81$	8.19155e21	0.924247
$82$	0	0
$83$	−1.45920e22	−1.24370	−0.621850	−	0.783136i	$-0.713619\pi$
$83$	−1.45920e22	−1.24370	−0.621850	+	0.783136i	$0.713619\pi$
$84$	0	0
$85$	−6.34694e20	−0.0411384
$86$	0	0
$87$	2.02767e21	0.100583
$88$	0	0
$89$	1.80991e21	0.0691310	0.0345655	−	0.999402i	$-0.488995\pi$
$89$	1.80991e21	0.0691310	0.0345655	+	0.999402i	$0.488995\pi$
$90$	0	0
$91$	4.12007e22	1.21879
$92$	0	0
$93$	−6.64369e19	−0.00153058
$94$	0	0
$95$	2.00707e22	0.362027
$96$	0	0
$97$	8.25561e22	1.17186	0.585928	−	0.810363i	$-0.300730\pi$
$97$	8.25561e22	1.17186	0.585928	+	0.810363i	$0.300730\pi$
$98$	0	0
$99$	5.54696e22	0.622659
$100$	0	0
$101$	4.84234e22	0.431877	0.215938	−	0.976407i	$-0.430719\pi$
$101$	4.84234e22	0.431877	0.215938	+	0.976407i	$0.430719\pi$
$102$	0	0
$103$	3.23687e22	0.230408	0.115204	−	0.993342i	$-0.463248\pi$
$103$	3.23687e22	0.230408	0.115204	+	0.993342i	$0.463248\pi$
$104$	0	0
$105$	−8.10110e21	−0.0462239
$106$	0	0
$107$	5.92757e22	0.272247	0.136124	−	0.990692i	$-0.456536\pi$
$107$	5.92757e22	0.272247	0.136124	+	0.990692i	$0.456536\pi$
$108$	0	0
$109$	2.86887e23	1.06490	0.532448	−	0.846463i	$-0.321272\pi$
$109$	2.86887e23	1.06490	0.532448	+	0.846463i	$0.321272\pi$
$110$	0	0
$111$	1.67096e22	0.0503214
$112$	0	0
$113$	−4.87844e22	−0.119641	−0.0598204	−	0.998209i	$-0.519053\pi$
$113$	−4.87844e22	−0.119641	−0.0598204	+	0.998209i	$0.519053\pi$
$114$	0	0
$115$	−1.45787e23	−0.292210
$116$	0	0
$117$	−7.30947e23	−1.20157
$118$	0	0
$119$	1.02592e23	0.138779
$120$	0	0
$121$	−5.29886e23	−0.591767
$122$	0	0
$123$	−1.80934e23	−0.167344
$124$	0	0
$125$	7.30026e23	0.560884
$126$	0	0
$127$	−1.36871e24	−0.876131	−0.438065	−	0.898943i	$-0.644336\pi$
$127$	−1.36871e24	−0.876131	−0.438065	+	0.898943i	$0.644336\pi$
$128$	0	0
$129$	9.64389e22	0.0515787
$130$	0	0
$131$	−2.63272e24	−1.17974	−0.589869	−	0.807499i	$-0.700820\pi$
$131$	−2.63272e24	−1.17974	−0.589869	+	0.807499i	$0.700820\pi$
$132$	0	0
$133$	−3.24423e24	−1.22129
$134$	0	0
$135$	2.91201e23	0.0923325
$136$	0	0
$137$	−1.21837e24	−0.326207	−0.163103	−	0.986609i	$-0.552150\pi$
$137$	−1.21837e24	−0.326207	−0.163103	+	0.986609i	$0.552150\pi$
$138$	0	0
$139$	3.72223e24	0.843593	0.421797	−	0.906690i	$-0.361400\pi$
$139$	3.72223e24	0.843593	0.421797	+	0.906690i	$0.361400\pi$
$140$	0	0
$141$	−1.19661e24	−0.230108
$142$	0	0
$143$	−4.81693e24	−0.787782
$144$	0	0
$145$	−1.32485e24	−0.184687
$146$	0	0
$147$	−3.06452e22	−0.00364932
$148$	0	0
$149$	1.53044e25	1.56018	0.780088	−	0.625670i	$-0.215174\pi$
$149$	1.53044e25	1.56018	0.780088	+	0.625670i	$0.215174\pi$
$150$	0	0
$151$	1.25232e25	1.09516	0.547582	−	0.836752i	$-0.315548\pi$
$151$	1.25232e25	1.09516	0.547582	+	0.836752i	$0.315548\pi$
$152$	0	0
$153$	−1.82010e24	−0.136818
$154$	0	0
$155$	4.34090e22	0.00281038
$156$	0	0
$157$	1.92166e23	0.0107357	0.00536785	−	0.999986i	$-0.498291\pi$
$157$	1.92166e23	0.0107357	0.00536785	+	0.999986i	$0.498291\pi$
$158$	0	0
$159$	−3.13099e24	−0.151222
$160$	0	0
$161$	2.35651e25	0.985761
$162$	0	0
$163$	2.27639e25	0.826206	0.413103	−	0.910684i	$-0.364445\pi$
$163$	2.27639e25	0.826206	0.413103	+	0.910684i	$0.364445\pi$
$164$	0	0
$165$	9.47130e23	0.0298775
$166$	0	0
$167$	1.17984e25	0.324029	0.162015	−	0.986788i	$-0.448201\pi$
$167$	1.17984e25	0.324029	0.162015	+	0.986788i	$0.448201\pi$
$168$	0	0
$169$	2.17208e25	0.520211
$170$	0	0
$171$	5.75563e25	1.20403
$172$	0	0
$173$	−4.75865e25	−0.870871	−0.435435	−	0.900220i	$-0.643406\pi$
$173$	−4.75865e25	−0.870871	−0.435435	+	0.900220i	$0.643406\pi$
$174$	0	0
$175$	−5.63542e25	−0.903626
$176$	0	0
$177$	−1.38028e25	−0.194211
$178$	0	0
$179$	−9.84804e25	−1.21770	−0.608850	−	0.793286i	$-0.708369\pi$
$179$	−9.84804e25	−1.21770	−0.608850	+	0.793286i	$0.708369\pi$
$180$	0	0
$181$	4.76456e25	0.518465	0.259233	−	0.965815i	$-0.416530\pi$
$181$	4.76456e25	0.518465	0.259233	+	0.965815i	$0.416530\pi$
$182$	0	0
$183$	−2.29053e25	−0.219659
$184$	0	0
$185$	−1.09179e25	−0.0923980
$186$	0	0
$187$	−1.19944e25	−0.0897017
$188$	0	0
$189$	−4.70697e25	−0.311481
$190$	0	0
$191$	8.54117e25	0.500765	0.250382	−	0.968147i	$-0.419444\pi$
$191$	8.54117e25	0.500765	0.250382	+	0.968147i	$0.419444\pi$
$192$	0	0
$193$	−2.66797e26	−1.38762	−0.693812	−	0.720156i	$-0.744070\pi$
$193$	−2.66797e26	−1.38762	−0.693812	+	0.720156i	$0.744070\pi$
$194$	0	0
$195$	−1.24807e25	−0.0576557
$196$	0	0
$197$	−1.00070e26	−0.411095	−0.205547	−	0.978647i	$-0.565897\pi$
$197$	−1.00070e26	−0.411095	−0.205547	+	0.978647i	$0.565897\pi$
$198$	0	0
$199$	−4.46897e26	−1.63454	−0.817272	−	0.576251i	$-0.804515\pi$
$199$	−4.46897e26	−1.63454	−0.817272	+	0.576251i	$0.804515\pi$
$200$	0	0
$201$	−1.35831e25	−0.0442838
$202$	0	0
$203$	2.14149e26	0.623035
$204$	0	0
$205$	1.18220e26	0.307271
$206$	0	0
$207$	−4.18071e26	−0.971832
$208$	0	0
$209$	3.79295e26	0.789396
$210$	0	0
$211$	7.96987e26	1.48663	0.743315	−	0.668941i	$-0.233252\pi$
$211$	7.96987e26	1.48663	0.743315	+	0.668941i	$0.233252\pi$
$212$	0	0
$213$	−1.12163e26	−0.187709
$214$	0	0
$215$	−6.30120e25	−0.0947067
$216$	0	0
$217$	−7.01664e24	−0.00948074
$218$	0	0
$219$	−2.23724e26	−0.272023
$220$	0	0
$221$	1.58056e26	0.173101
$222$	0	0
$223$	5.69004e26	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$223$	5.69004e26	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$224$	0	0
$225$	9.99787e26	0.890858
$226$	0	0
$227$	1.82165e27	1.46611	0.733056	−	0.680168i	$-0.238093\pi$
$227$	1.82165e27	1.46611	0.733056	+	0.680168i	$0.238093\pi$
$228$	0	0
$229$	1.11330e27	0.810032	0.405016	−	0.914310i	$-0.367266\pi$
$229$	1.11330e27	0.810032	0.405016	+	0.914310i	$0.367266\pi$
$230$	0	0
$231$	−1.53094e26	−0.100791
$232$	0	0
$233$	−1.63214e27	−0.973118	−0.486559	−	0.873648i	$-0.661748\pi$
$233$	−1.63214e27	−0.973118	−0.486559	+	0.873648i	$0.661748\pi$
$234$	0	0
$235$	7.81849e26	0.422515
$236$	0	0
$237$	1.95372e26	0.0957761
$238$	0	0
$239$	−2.58550e27	−1.15072	−0.575358	−	0.817902i	$-0.695137\pi$
$239$	−2.58550e27	−1.15072	−0.575358	+	0.817902i	$0.695137\pi$
$240$	0	0
$241$	1.86784e27	0.755340	0.377670	−	0.925940i	$-0.376725\pi$
$241$	1.86784e27	0.755340	0.377670	+	0.925940i	$0.376725\pi$
$242$	0	0
$243$	1.25799e27	0.462600
$244$	0	0
$245$	2.00232e25	0.00670074
$246$	0	0
$247$	−4.99813e27	−1.52333
$248$	0	0
$249$	−7.14493e26	−0.198475
$250$	0	0
$251$	4.11170e27	1.04177	0.520887	−	0.853626i	$-0.325601\pi$
$251$	4.11170e27	1.04177	0.520887	+	0.853626i	$0.325601\pi$
$252$	0	0
$253$	−2.75508e27	−0.637160
$254$	0	0
$255$	−3.10777e25	−0.00656503
$256$	0	0
$257$	−5.10426e27	−0.985603	−0.492802	−	0.870142i	$-0.664027\pi$
$257$	−5.10426e27	−0.985603	−0.492802	+	0.870142i	$0.664027\pi$
$258$	0	0
$259$	1.76476e27	0.311702
$260$	0	0
$261$	−3.79925e27	−0.614232
$262$	0	0
$263$	−9.95433e27	−1.47408	−0.737040	−	0.675849i	$-0.763777\pi$
$263$	−9.95433e27	−1.47408	−0.737040	+	0.675849i	$0.763777\pi$
$264$	0	0
$265$	2.04575e27	0.277668
$266$	0	0
$267$	8.86222e25	0.0110322
$268$	0	0
$269$	6.42524e26	0.0734071	0.0367036	−	0.999326i	$-0.488314\pi$
$269$	6.42524e26	0.0734071	0.0367036	+	0.999326i	$0.488314\pi$
$270$	0	0
$271$	−1.19246e28	−1.25111	−0.625557	−	0.780178i	$-0.715128\pi$
$271$	−1.19246e28	−1.25111	−0.625557	+	0.780178i	$0.715128\pi$
$272$	0	0
$273$	2.01739e27	0.194500
$274$	0	0
$275$	6.58858e27	0.584071
$276$	0	0
$277$	−5.29510e27	−0.431874	−0.215937	−	0.976407i	$-0.569281\pi$
$277$	−5.29510e27	−0.431874	−0.215937	+	0.976407i	$0.569281\pi$
$278$	0	0
$279$	1.24483e26	0.00934678
$280$	0	0
$281$	9.67433e27	0.669111	0.334556	−	0.942376i	$-0.391414\pi$
$281$	9.67433e27	0.669111	0.334556	+	0.942376i	$0.391414\pi$
$282$	0	0
$283$	2.62436e28	1.67294	0.836468	−	0.548016i	$-0.184617\pi$
$283$	2.62436e28	1.67294	0.836468	+	0.548016i	$0.184617\pi$
$284$	0	0
$285$	9.82759e26	0.0577738
$286$	0	0
$287$	−1.91091e28	−1.03657
$288$	0	0
$289$	−1.95740e28	−0.980290
$290$	0	0
$291$	4.04235e27	0.187010
$292$	0	0
$293$	3.20151e28	1.36892	0.684458	−	0.729052i	$-0.260039\pi$
$293$	3.20151e28	1.36892	0.684458	+	0.729052i	$0.260039\pi$
$294$	0	0
$295$	9.01856e27	0.356602
$296$	0	0
$297$	5.50310e27	0.201330
$298$	0	0
$299$	3.63049e28	1.22955
$300$	0	0
$301$	1.01853e28	0.319490
$302$	0	0
$303$	2.37105e27	0.0689207
$304$	0	0
$305$	1.49660e28	0.403328
$306$	0	0
$307$	1.15407e28	0.288496	0.144248	−	0.989542i	$-0.453924\pi$
$307$	1.15407e28	0.288496	0.144248	+	0.989542i	$0.453924\pi$
$308$	0	0
$309$	1.58493e27	0.0367695
$310$	0	0
$311$	−3.62817e28	−0.781525	−0.390763	−	0.920491i	$-0.627789\pi$
$311$	−3.62817e28	−0.781525	−0.390763	+	0.920491i	$0.627789\pi$
$312$	0	0
$313$	2.56117e28	0.512481	0.256241	−	0.966613i	$-0.417516\pi$
$313$	2.56117e28	0.512481	0.256241	+	0.966613i	$0.417516\pi$
$314$	0	0
$315$	1.51791e28	0.282276
$316$	0	0
$317$	−2.91424e27	−0.0503899	−0.0251950	−	0.999683i	$-0.508021\pi$
$317$	−2.91424e27	−0.0503899	−0.0251950	+	0.999683i	$0.508021\pi$
$318$	0	0
$319$	−2.50370e28	−0.402708
$320$	0	0
$321$	2.90243e27	0.0434464
$322$	0	0
$323$	−1.24456e28	−0.173455
$324$	0	0
$325$	−8.68205e28	−1.12710
$326$	0	0
$327$	1.40474e28	0.169940
$328$	0	0
$329$	−1.26378e29	−1.42534
$330$	0	0
$331$	−4.00537e28	−0.421328	−0.210664	−	0.977558i	$-0.567563\pi$
$331$	−4.00537e28	−0.421328	−0.210664	+	0.977558i	$0.567563\pi$
$332$	0	0
$333$	−3.13089e28	−0.307298
$334$	0	0
$335$	8.87504e27	0.0813121
$336$	0	0
$337$	1.31131e29	1.12192	0.560959	−	0.827843i	$-0.310432\pi$
$337$	1.31131e29	1.12192	0.560959	+	0.827843i	$0.310432\pi$
$338$	0	0
$339$	−2.38872e27	−0.0190928
$340$	0	0
$341$	8.20342e26	0.00612801
$342$	0	0
$343$	−1.44770e29	−1.01110
$344$	0	0
$345$	−7.13846e27	−0.0466321
$346$	0	0
$347$	−1.14051e29	−0.697126	−0.348563	−	0.937285i	$-0.613330\pi$
$347$	−1.14051e29	−0.697126	−0.348563	+	0.937285i	$0.613330\pi$
$348$	0	0
$349$	7.28518e28	0.416820	0.208410	−	0.978042i	$-0.433171\pi$
$349$	7.28518e28	0.416820	0.208410	+	0.978042i	$0.433171\pi$
$350$	0	0
$351$	−7.25167e28	−0.388514
$352$	0	0
$353$	−3.68425e29	−1.84902	−0.924508	−	0.381164i	$-0.875523\pi$
$353$	−3.68425e29	−1.84902	−0.924508	+	0.381164i	$0.875523\pi$
$354$	0	0
$355$	7.32860e28	0.344663
$356$	0	0
$357$	5.02340e27	0.0221469
$358$	0	0
$359$	2.45802e29	1.01625	0.508123	−	0.861284i	$-0.330339\pi$
$359$	2.45802e29	1.01625	0.508123	+	0.861284i	$0.330339\pi$
$360$	0	0
$361$	1.35734e29	0.526448
$362$	0	0
$363$	−2.59458e28	−0.0944367
$364$	0	0
$365$	1.46179e29	0.499477
$366$	0	0
$367$	4.73025e29	1.51783	0.758917	−	0.651187i	$-0.225729\pi$
$367$	4.73025e29	1.51783	0.758917	+	0.651187i	$0.225729\pi$
$368$	0	0
$369$	3.39016e29	1.02192
$370$	0	0
$371$	−3.30675e29	−0.936703
$372$	0	0
$373$	−2.90241e29	−0.772872	−0.386436	−	0.922316i	$-0.626294\pi$
$373$	−2.90241e29	−0.772872	−0.386436	+	0.922316i	$0.626294\pi$
$374$	0	0
$375$	3.57456e28	0.0895083
$376$	0	0
$377$	3.29924e29	0.777120
$378$	0	0
$379$	−2.72591e29	−0.604172	−0.302086	−	0.953281i	$-0.597683\pi$
$379$	−2.72591e29	−0.604172	−0.302086	+	0.953281i	$0.597683\pi$
$380$	0	0
$381$	−6.70187e28	−0.139817
$382$	0	0
$383$	9.91721e29	1.94806	0.974032	−	0.226408i	$-0.0726984\pi$
$383$	9.91721e29	1.94806	0.974032	+	0.226408i	$0.0726984\pi$
$384$	0	0
$385$	1.00030e29	0.185068
$386$	0	0
$387$	−1.80698e29	−0.314976
$388$	0	0
$389$	8.39840e29	1.37967	0.689836	−	0.723966i	$-0.257683\pi$
$389$	8.39840e29	1.37967	0.689836	+	0.723966i	$0.257683\pi$
$390$	0	0
$391$	9.04012e28	0.140004
$392$	0	0
$393$	−1.28911e29	−0.188268
$394$	0	0
$395$	−1.27654e29	−0.175860
$396$	0	0
$397$	9.94808e27	0.0129315	0.00646574	−	0.999979i	$-0.497942\pi$
$397$	9.94808e27	0.0129315	0.00646574	+	0.999979i	$0.497942\pi$
$398$	0	0
$399$	−1.58853e29	−0.194898
$400$	0	0
$401$	−4.81480e29	−0.557723	−0.278861	−	0.960331i	$-0.589957\pi$
$401$	−4.81480e29	−0.557723	−0.278861	+	0.960331i	$0.589957\pi$
$402$	0	0
$403$	−1.08100e28	−0.0118254
$404$	0	0
$405$	−2.62072e29	−0.270825
$406$	0	0
$407$	−2.06325e29	−0.201473
$408$	0	0
$409$	−8.22531e29	−0.759162	−0.379581	−	0.925159i	$-0.623932\pi$
$409$	−8.22531e29	−0.759162	−0.379581	+	0.925159i	$0.623932\pi$
$410$	0	0
$411$	−5.96573e28	−0.0520575
$412$	0	0
$413$	−1.45776e30	−1.20299
$414$	0	0
$415$	4.66841e29	0.364431
$416$	0	0
$417$	1.82259e29	0.134624
$418$	0	0
$419$	−2.72055e30	−1.90194	−0.950969	−	0.309287i	$-0.899910\pi$
$419$	−2.72055e30	−1.90194	−0.950969	+	0.309287i	$0.899910\pi$
$420$	0	0
$421$	1.73534e30	1.14852	0.574261	−	0.818672i	$-0.305289\pi$
$421$	1.73534e30	1.14852	0.574261	+	0.818672i	$0.305289\pi$
$422$	0	0
$423$	2.24209e30	1.40520
$424$	0	0
$425$	−2.16188e29	−0.128339
$426$	0	0
$427$	−2.41911e30	−1.36061
$428$	0	0
$429$	−2.35860e29	−0.125718
$430$	0	0
$431$	−2.30954e30	−1.16691	−0.583455	−	0.812146i	$-0.698299\pi$
$431$	−2.30954e30	−1.16691	−0.583455	+	0.812146i	$0.698299\pi$
$432$	0	0
$433$	−4.49441e29	−0.215309	−0.107655	−	0.994188i	$-0.534334\pi$
$433$	−4.49441e29	−0.215309	−0.107655	+	0.994188i	$0.534334\pi$
$434$	0	0
$435$	−6.48713e28	−0.0294731
$436$	0	0
$437$	−2.85872e30	−1.23207
$438$	0	0
$439$	1.98640e30	0.812315	0.406157	−	0.913803i	$-0.366868\pi$
$439$	1.98640e30	0.812315	0.406157	+	0.913803i	$0.366868\pi$
$440$	0	0
$441$	5.74199e28	0.0222853
$442$	0	0
$443$	2.09074e30	0.770294	0.385147	−	0.922855i	$-0.374151\pi$
$443$	2.09074e30	0.770294	0.385147	+	0.922855i	$0.374151\pi$
$444$	0	0
$445$	−5.79046e28	−0.0202569
$446$	0	0
$447$	7.49376e29	0.248979
$448$	0	0
$449$	2.38796e30	0.753694	0.376847	−	0.926276i	$-0.377008\pi$
$449$	2.38796e30	0.753694	0.376847	+	0.926276i	$0.377008\pi$
$450$	0	0
$451$	2.23411e30	0.670000
$452$	0	0
$453$	6.13195e29	0.174771
$454$	0	0
$455$	−1.31813e30	−0.357132
$456$	0	0
$457$	1.54921e30	0.399093	0.199547	−	0.979888i	$-0.436053\pi$
$457$	1.54921e30	0.399093	0.199547	+	0.979888i	$0.436053\pi$
$458$	0	0
$459$	−1.80571e29	−0.0442386
$460$	0	0
$461$	4.06665e30	0.947711	0.473855	−	0.880603i	$-0.342862\pi$
$461$	4.06665e30	0.947711	0.473855	+	0.880603i	$0.342862\pi$
$462$	0	0
$463$	6.37975e29	0.141456	0.0707282	−	0.997496i	$-0.477468\pi$
$463$	6.37975e29	0.141456	0.0707282	+	0.997496i	$0.477468\pi$
$464$	0	0
$465$	2.12552e27	0.000448493 0
$466$	0	0
$467$	−1.15365e30	−0.231701	−0.115851	−	0.993267i	$-0.536959\pi$
$467$	−1.15365e30	−0.231701	−0.115851	+	0.993267i	$0.536959\pi$
$468$	0	0
$469$	−1.43456e30	−0.274304
$470$	0	0
$471$	9.40938e27	0.00171325
$472$	0	0
$473$	−1.19080e30	−0.206507
$474$	0	0
$475$	6.83643e30	1.12941
$476$	0	0
$477$	5.86655e30	0.923468
$478$	0	0
$479$	2.12936e30	0.319441	0.159721	−	0.987162i	$-0.448941\pi$
$479$	2.12936e30	0.319441	0.159721	+	0.987162i	$0.448941\pi$
$480$	0	0
$481$	2.71884e30	0.388790
$482$	0	0
$483$	1.15386e30	0.157312
$484$	0	0
$485$	−2.64122e30	−0.343380
$486$	0	0
$487$	−4.84200e30	−0.600401	−0.300201	−	0.953876i	$-0.597054\pi$
$487$	−4.84200e30	−0.600401	−0.300201	+	0.953876i	$0.597054\pi$
$488$	0	0
$489$	1.11463e30	0.131849
$490$	0	0
$491$	3.44338e29	0.0388640	0.0194320	−	0.999811i	$-0.493814\pi$
$491$	3.44338e29	0.0388640	0.0194320	+	0.999811i	$0.493814\pi$
$492$	0	0
$493$	8.21528e29	0.0884877
$494$	0	0
$495$	−1.77464e30	−0.182453
$496$	0	0
$497$	−1.18460e31	−1.16271
$498$	0	0
$499$	1.27549e31	1.19542	0.597711	−	0.801711i	$-0.296077\pi$
$499$	1.27549e31	1.19542	0.597711	+	0.801711i	$0.296077\pi$
$500$	0	0
$501$	5.77708e29	0.0517100
$502$	0	0
$503$	−1.67840e31	−1.43504	−0.717520	−	0.696538i	$-0.754723\pi$
$503$	−1.67840e31	−1.43504	−0.717520	+	0.696538i	$0.754723\pi$
$504$	0	0
$505$	−1.54921e30	−0.126549
$506$	0	0
$507$	1.06356e30	0.0830175
$508$	0	0
$509$	−2.11935e31	−1.58106	−0.790529	−	0.612425i	$-0.790194\pi$
$509$	−2.11935e31	−1.58106	−0.790529	+	0.612425i	$0.790194\pi$
$510$	0	0
$511$	−2.36283e31	−1.68497
$512$	0	0
$513$	5.71012e30	0.389310
$514$	0	0
$515$	−1.03557e30	−0.0675146
$516$	0	0
$517$	1.47753e31	0.921290
$518$	0	0
$519$	−2.33007e30	−0.138977
$520$	0	0
$521$	−1.22820e30	−0.0700869	−0.0350434	−	0.999386i	$-0.511157\pi$
$521$	−1.22820e30	−0.0700869	−0.0350434	+	0.999386i	$0.511157\pi$
$522$	0	0
$523$	−7.24558e30	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$523$	−7.24558e30	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$524$	0	0
$525$	−2.75937e30	−0.144204
$526$	0	0
$527$	−2.69175e28	−0.00134652
$528$	0	0
$529$	−1.15570e29	−0.00553483
$530$	0	0
$531$	2.58623e31	1.18599
$532$	0	0
$533$	−2.94399e31	−1.29292
$534$	0	0
$535$	−1.89641e30	−0.0797744
$536$	0	0
$537$	−4.82208e30	−0.194326
$538$	0	0
$539$	3.78397e29	0.0146109
$540$	0	0
$541$	9.30820e30	0.344427	0.172214	−	0.985060i	$-0.444908\pi$
$541$	9.30820e30	0.344427	0.172214	+	0.985060i	$0.444908\pi$
$542$	0	0
$543$	2.33296e30	0.0827389
$544$	0	0
$545$	−9.17839e30	−0.312038
$546$	0	0
$547$	2.67466e31	0.871796	0.435898	−	0.899996i	$-0.356431\pi$
$547$	2.67466e31	0.871796	0.435898	+	0.899996i	$0.356431\pi$
$548$	0	0
$549$	4.29177e31	1.34139
$550$	0	0
$551$	−2.59789e31	−0.778712
$552$	0	0
$553$	2.06340e31	0.593258
$554$	0	0
$555$	−5.34592e29	−0.0147453
$556$	0	0
$557$	2.40510e31	0.636501	0.318250	−	0.948007i	$-0.396905\pi$
$557$	2.40510e31	0.636501	0.318250	+	0.948007i	$0.396905\pi$
$558$	0	0
$559$	1.56916e31	0.398504
$560$	0	0
$561$	−5.87305e29	−0.0143150
$562$	0	0
$563$	−2.15437e31	−0.504049	−0.252025	−	0.967721i	$-0.581096\pi$
$563$	−2.15437e31	−0.504049	−0.252025	+	0.967721i	$0.581096\pi$
$564$	0	0
$565$	1.56076e30	0.0350574
$566$	0	0
$567$	4.23614e31	0.913618
$568$	0	0
$569$	1.70978e31	0.354120	0.177060	−	0.984200i	$-0.443341\pi$
$569$	1.70978e31	0.354120	0.177060	+	0.984200i	$0.443341\pi$
$570$	0	0
$571$	8.69138e31	1.72891	0.864457	−	0.502707i	$-0.167663\pi$
$571$	8.69138e31	1.72891	0.864457	+	0.502707i	$0.167663\pi$
$572$	0	0
$573$	4.18217e30	0.0799142
$574$	0	0
$575$	−4.96577e31	−0.911605
$576$	0	0
$577$	1.59167e31	0.280757	0.140378	−	0.990098i	$-0.455168\pi$
$577$	1.59167e31	0.280757	0.140378	+	0.990098i	$0.455168\pi$
$578$	0	0
$579$	−1.30637e31	−0.221443
$580$	0	0
$581$	−7.54602e31	−1.22940
$582$	0	0
$583$	3.86605e31	0.605451
$584$	0	0
$585$	2.33852e31	0.352086
$586$	0	0
$587$	−5.71080e31	−0.826721	−0.413361	−	0.910567i	$-0.635645\pi$
$587$	−5.71080e31	−0.826721	−0.413361	+	0.910567i	$0.635645\pi$
$588$	0	0
$589$	8.51202e29	0.0118497
$590$	0	0
$591$	−4.89991e30	−0.0656043
$592$	0	0
$593$	2.68653e31	0.345989	0.172995	−	0.984923i	$-0.444656\pi$
$593$	2.68653e31	0.345989	0.172995	+	0.984923i	$0.444656\pi$
$594$	0	0
$595$	−3.28223e30	−0.0406653
$596$	0	0
$597$	−2.18822e31	−0.260848
$598$	0	0
$599$	−1.35452e30	−0.0155373	−0.00776867	−	0.999970i	$-0.502473\pi$
$599$	−1.35452e30	−0.0155373	−0.00776867	+	0.999970i	$0.502473\pi$
$600$	0	0
$601$	−7.37959e31	−0.814659	−0.407329	−	0.913281i	$-0.633540\pi$
$601$	−7.37959e31	−0.814659	−0.407329	+	0.913281i	$0.633540\pi$
$602$	0	0
$603$	2.54507e31	0.270428
$604$	0	0
$605$	1.69527e31	0.173401
$606$	0	0
$607$	3.23323e31	0.318395	0.159198	−	0.987247i	$-0.449109\pi$
$607$	3.23323e31	0.318395	0.159198	+	0.987247i	$0.449109\pi$
$608$	0	0
$609$	1.04858e31	0.0994266
$610$	0	0
$611$	−1.94701e32	−1.77785
$612$	0	0
$613$	4.35574e31	0.383059	0.191530	−	0.981487i	$-0.438655\pi$
$613$	4.35574e31	0.383059	0.191530	+	0.981487i	$0.438655\pi$
$614$	0	0
$615$	5.78862e30	0.0490356
$616$	0	0
$617$	−2.29127e32	−1.86981	−0.934904	−	0.354902i	$-0.884514\pi$
$617$	−2.29127e32	−1.86981	−0.934904	+	0.354902i	$0.884514\pi$
$618$	0	0
$619$	−2.32366e32	−1.82697	−0.913484	−	0.406875i	$-0.866618\pi$
$619$	−2.32366e32	−1.82697	−0.913484	+	0.406875i	$0.866618\pi$
$620$	0	0
$621$	−4.14765e31	−0.314231
$622$	0	0
$623$	9.35970e30	0.0683360
$624$	0	0
$625$	1.06551e32	0.749787
$626$	0	0
$627$	1.85721e31	0.125975
$628$	0	0
$629$	6.77005e30	0.0442700
$630$	0	0
$631$	2.22944e32	1.40559	0.702793	−	0.711394i	$-0.251936\pi$
$631$	2.22944e32	1.40559	0.702793	+	0.711394i	$0.251936\pi$
$632$	0	0
$633$	3.90244e31	0.237243
$634$	0	0
$635$	4.37892e31	0.256726
$636$	0	0
$637$	−4.98629e30	−0.0281951
$638$	0	0
$639$	2.10161e32	1.14628
$640$	0	0
$641$	−2.75021e32	−1.44710	−0.723550	−	0.690272i	$-0.757491\pi$
$641$	−2.75021e32	−1.44710	−0.723550	+	0.690272i	$0.757491\pi$
$642$	0	0
$643$	−1.47614e32	−0.749377	−0.374689	−	0.927151i	$-0.622250\pi$
$643$	−1.47614e32	−0.749377	−0.374689	+	0.927151i	$0.622250\pi$
$644$	0	0
$645$	−3.08537e30	−0.0151137
$646$	0	0
$647$	−1.82358e32	−0.862036	−0.431018	−	0.902343i	$-0.641846\pi$
$647$	−1.82358e32	−0.862036	−0.431018	+	0.902343i	$0.641846\pi$
$648$	0	0
$649$	1.70432e32	0.777567
$650$	0	0
$651$	−3.43569e29	−0.00151298
$652$	0	0
$653$	−4.35899e31	−0.185304	−0.0926519	−	0.995699i	$-0.529534\pi$
$653$	−4.35899e31	−0.185304	−0.0926519	+	0.995699i	$0.529534\pi$
$654$	0	0
$655$	8.42287e31	0.345689
$656$	0	0
$657$	4.19193e32	1.66116
$658$	0	0
$659$	3.58880e31	0.137331	0.0686653	−	0.997640i	$-0.478126\pi$
$659$	3.58880e31	0.137331	0.0686653	+	0.997640i	$0.478126\pi$
$660$	0	0
$661$	2.78803e32	1.03034	0.515171	−	0.857088i	$-0.327729\pi$
$661$	2.78803e32	1.03034	0.515171	+	0.857088i	$0.327729\pi$
$662$	0	0
$663$	7.73917e30	0.0276241
$664$	0	0
$665$	1.03793e32	0.357864
$666$	0	0
$667$	1.88703e32	0.628537
$668$	0	0
$669$	2.78612e31	0.0896601
$670$	0	0
$671$	2.82827e32	0.879452
$672$	0	0
$673$	1.90131e31	0.0571320	0.0285660	−	0.999592i	$-0.490906\pi$
$673$	1.90131e31	0.0571320	0.0285660	+	0.999592i	$0.490906\pi$
$674$	0	0
$675$	9.91881e31	0.288049
$676$	0	0
$677$	−3.87840e32	−1.08864	−0.544318	−	0.838879i	$-0.683212\pi$
$677$	−3.87840e32	−1.08864	−0.544318	+	0.838879i	$0.683212\pi$
$678$	0	0
$679$	4.26927e32	1.15838
$680$	0	0
$681$	8.91968e31	0.233969
$682$	0	0
$683$	−6.51681e32	−1.65271	−0.826356	−	0.563148i	$-0.809590\pi$
$683$	−6.51681e32	−1.65271	−0.826356	+	0.563148i	$0.809590\pi$
$684$	0	0
$685$	3.89794e31	0.0955857
$686$	0	0
$687$	5.45123e31	0.129268
$688$	0	0
$689$	−5.09446e32	−1.16836
$690$	0	0
$691$	6.18205e31	0.137131	0.0685654	−	0.997647i	$-0.478158\pi$
$691$	6.18205e31	0.137131	0.0685654	+	0.997647i	$0.478158\pi$
$692$	0	0
$693$	2.86853e32	0.615499
$694$	0	0
$695$	−1.19085e32	−0.247191
$696$	0	0
$697$	−7.33069e31	−0.147220
$698$	0	0
$699$	−7.99177e31	−0.155294
$700$	0	0
$701$	5.26483e32	0.989982	0.494991	−	0.868898i	$-0.335171\pi$
$701$	5.26483e32	0.989982	0.494991	+	0.868898i	$0.335171\pi$
$702$	0	0
$703$	−2.14087e32	−0.389586
$704$	0	0
$705$	3.82831e31	0.0674268
$706$	0	0
$707$	2.50415e32	0.426910
$708$	0	0
$709$	−7.09958e32	−1.17166	−0.585829	−	0.810435i	$-0.699231\pi$
$709$	−7.09958e32	−1.17166	−0.585829	+	0.810435i	$0.699231\pi$
$710$	0	0
$711$	−3.66070e32	−0.584876
$712$	0	0
$713$	−6.18287e30	−0.00956446
$714$	0	0
$715$	1.54108e32	0.230837
$716$	0	0
$717$	−1.26598e32	−0.183636
$718$	0	0
$719$	−4.58875e32	−0.644633	−0.322316	−	0.946632i	$-0.604461\pi$
$719$	−4.58875e32	−0.644633	−0.322316	+	0.946632i	$0.604461\pi$
$720$	0	0
$721$	1.67390e32	0.227758
$722$	0	0
$723$	9.14583e31	0.120540
$724$	0	0
$725$	−4.51268e32	−0.576166
$726$	0	0
$727$	−1.25284e33	−1.54970	−0.774851	−	0.632144i	$-0.782175\pi$
$727$	−1.25284e33	−1.54970	−0.774851	+	0.632144i	$0.782175\pi$
$728$	0	0
$729$	−7.09581e32	−0.850424
$730$	0	0
$731$	3.90730e31	0.0453761
$732$	0	0
$733$	−1.67809e30	−0.00188852	−0.000944258	−	1.00000i	$-0.500301\pi$
$733$	−1.67809e30	−0.00188852	−0.000944258		1.00000i	$0.500301\pi$
$734$	0	0
$735$	9.80431e29	0.00106933
$736$	0	0
$737$	1.67720e32	0.177300
$738$	0	0
$739$	−2.96573e32	−0.303893	−0.151946	−	0.988389i	$-0.548554\pi$
$739$	−2.96573e32	−0.303893	−0.151946	+	0.988389i	$0.548554\pi$
$740$	0	0
$741$	−2.44733e32	−0.243099
$742$	0	0
$743$	4.48312e32	0.431727	0.215863	−	0.976424i	$-0.430743\pi$
$743$	4.48312e32	0.431727	0.215863	+	0.976424i	$0.430743\pi$
$744$	0	0
$745$	−4.89633e32	−0.457166
$746$	0	0
$747$	1.33875e33	1.21203
$748$	0	0
$749$	3.06536e32	0.269116
$750$	0	0
$751$	3.50465e32	0.298391	0.149195	−	0.988808i	$-0.452332\pi$
$751$	3.50465e32	0.298391	0.149195	+	0.988808i	$0.452332\pi$
$752$	0	0
$753$	2.01329e32	0.166251
$754$	0	0
$755$	−4.00654e32	−0.320907
$756$	0	0
$757$	4.37490e32	0.339911	0.169956	−	0.985452i	$-0.445638\pi$
$757$	4.37490e32	0.339911	0.169956	+	0.985452i	$0.445638\pi$
$758$	0	0
$759$	−1.34902e32	−0.101681
$760$	0	0
$761$	−1.53089e31	−0.0111949	−0.00559744	−	0.999984i	$-0.501782\pi$
$761$	−1.53089e31	−0.0111949	−0.00559744	+	0.999984i	$0.501782\pi$
$762$	0	0
$763$	1.48360e33	1.05265
$764$	0	0
$765$	5.82304e31	0.0400907
$766$	0	0
$767$	−2.24586e33	−1.50050
$768$	0	0
$769$	2.67481e33	1.73436	0.867180	−	0.497995i	$-0.165930\pi$
$769$	2.67481e33	1.73436	0.867180	+	0.497995i	$0.165930\pi$
$770$	0	0
$771$	−2.49929e32	−0.157287
$772$	0	0
$773$	8.38808e31	0.0512387	0.0256194	−	0.999672i	$-0.491844\pi$
$773$	8.38808e31	0.0512387	0.0256194	+	0.999672i	$0.491844\pi$
$774$	0	0
$775$	1.47859e31	0.00876753
$776$	0	0
$777$	8.64114e31	0.0497427
$778$	0	0
$779$	2.31816e33	1.29557
$780$	0	0
$781$	1.38496e33	0.751534
$782$	0	0
$783$	−3.76921e32	−0.198605
$784$	0	0
$785$	−6.14797e30	−0.00314580
$786$	0	0
$787$	3.07440e33	1.52775	0.763874	−	0.645366i	$-0.223295\pi$
$787$	3.07440e33	1.52775	0.763874	+	0.645366i	$0.223295\pi$
$788$	0	0
$789$	−4.87412e32	−0.235240
$790$	0	0
$791$	−2.52281e32	−0.118265
$792$	0	0
$793$	−3.72693e33	−1.69711
$794$	0	0
$795$	1.00170e32	0.0443114
$796$	0	0
$797$	2.31415e33	0.994535	0.497268	−	0.867597i	$-0.334337\pi$
$797$	2.31415e33	0.994535	0.497268	+	0.867597i	$0.334337\pi$
$798$	0	0
$799$	−4.84816e32	−0.202437
$800$	0	0
$801$	−1.66052e32	−0.0673704
$802$	0	0
$803$	2.76247e33	1.08910
$804$	0	0
$805$	−7.53918e32	−0.288849
$806$	0	0
$807$	3.14611e31	0.0117146
$808$	0	0
$809$	3.80662e33	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$809$	3.80662e33	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$810$	0	0
$811$	−4.24454e33	−1.49311	−0.746554	−	0.665325i	$-0.768293\pi$
$811$	−4.24454e33	−1.49311	−0.746554	+	0.665325i	$0.768293\pi$
$812$	0	0
$813$	−5.83886e32	−0.199658
$814$	0	0
$815$	−7.28284e32	−0.242096
$816$	0	0
$817$	−1.23559e33	−0.399321
$818$	0	0
$819$	−3.77998e33	−1.18775
$820$	0	0
$821$	−2.99736e33	−0.915784	−0.457892	−	0.889008i	$-0.651395\pi$
$821$	−2.99736e33	−0.915784	−0.457892	+	0.889008i	$0.651395\pi$
$822$	0	0
$823$	5.19319e32	0.154289	0.0771447	−	0.997020i	$-0.475420\pi$
$823$	5.19319e32	0.154289	0.0771447	+	0.997020i	$0.475420\pi$
$824$	0	0
$825$	3.22609e32	0.0932086
$826$	0	0
$827$	−4.81781e32	−0.135374	−0.0676872	−	0.997707i	$-0.521562\pi$
$827$	−4.81781e32	−0.135374	−0.0676872	+	0.997707i	$0.521562\pi$
$828$	0	0
$829$	−2.15627e33	−0.589287	−0.294644	−	0.955607i	$-0.595201\pi$
$829$	−2.15627e33	−0.589287	−0.294644	+	0.955607i	$0.595201\pi$
$830$	0	0
$831$	−2.59274e32	−0.0689202
$832$	0	0
$833$	−1.24161e31	−0.00321048
$834$	0	0
$835$	−3.77467e32	−0.0949477
$836$	0	0
$837$	1.23499e31	0.00302218
$838$	0	0
$839$	2.38371e32	0.0567532	0.0283766	−	0.999597i	$-0.490966\pi$
$839$	2.38371e32	0.0567532	0.0283766	+	0.999597i	$0.490966\pi$
$840$	0	0
$841$	−2.60187e33	−0.602743
$842$	0	0
$843$	4.73702e32	0.106780
$844$	0	0
$845$	−6.94915e32	−0.152433
$846$	0	0
$847$	−2.74023e33	−0.584962
$848$	0	0
$849$	1.28501e33	0.266974
$850$	0	0
$851$	1.55506e33	0.314454
$852$	0	0
$853$	−1.94872e32	−0.0383563	−0.0191781	−	0.999816i	$-0.506105\pi$
$853$	−1.94872e32	−0.0383563	−0.0191781	+	0.999816i	$0.506105\pi$
$854$	0	0
$855$	−1.84140e33	−0.352807
$856$	0	0
$857$	−1.87495e33	−0.349711	−0.174855	−	0.984594i	$-0.555946\pi$
$857$	−1.87495e33	−0.349711	−0.174855	+	0.984594i	$0.555946\pi$
$858$	0	0
$859$	−6.46779e33	−1.17445	−0.587225	−	0.809424i	$-0.699780\pi$
$859$	−6.46779e33	−1.17445	−0.587225	+	0.809424i	$0.699780\pi$
$860$	0	0
$861$	−9.35673e32	−0.165420
$862$	0	0
$863$	1.04050e34	1.79110	0.895550	−	0.444961i	$-0.146782\pi$
$863$	1.04050e34	1.79110	0.895550	+	0.444961i	$0.146782\pi$
$864$	0	0
$865$	1.52244e33	0.255184
$866$	0	0
$867$	−9.58438e32	−0.156439
$868$	0	0
$869$	−2.41239e33	−0.383461
$870$	0	0
$871$	−2.21012e33	−0.342142
$872$	0	0
$873$	−7.57417e33	−1.14201
$874$	0	0
$875$	3.77522e33	0.554434
$876$	0	0
$877$	8.52438e33	1.21946	0.609730	−	0.792609i	$-0.291278\pi$
$877$	8.52438e33	1.21946	0.609730	+	0.792609i	$0.291278\pi$
$878$	0	0
$879$	1.56761e33	0.218457
$880$	0	0
$881$	1.20706e34	1.63872	0.819362	−	0.573276i	$-0.194328\pi$
$881$	1.20706e34	1.63872	0.819362	+	0.573276i	$0.194328\pi$
$882$	0	0
$883$	−4.60310e33	−0.608839	−0.304420	−	0.952538i	$-0.598462\pi$
$883$	−4.60310e33	−0.608839	−0.304420	+	0.952538i	$0.598462\pi$
$884$	0	0
$885$	4.41593e32	0.0569081
$886$	0	0
$887$	7.25037e33	0.910413	0.455207	−	0.890386i	$-0.349565\pi$
$887$	7.25037e33	0.910413	0.455207	+	0.890386i	$0.349565\pi$
$888$	0	0
$889$	−7.07809e33	−0.866055
$890$	0	0
$891$	−4.95263e33	−0.590530
$892$	0	0
$893$	1.53312e34	1.78149
$894$	0	0
$895$	3.15069e33	0.356813
$896$	0	0
$897$	1.77766e33	0.196217
$898$	0	0
$899$	−5.61873e31	−0.00604507
$900$	0	0
$901$	−1.26855e33	−0.133037
$902$	0	0
$903$	4.98720e32	0.0509856
$904$	0	0
$905$	−1.52433e33	−0.151922
$906$	0	0
$907$	3.03876e33	0.295266	0.147633	−	0.989042i	$-0.452835\pi$
$907$	3.03876e33	0.295266	0.147633	+	0.989042i	$0.452835\pi$
$908$	0	0
$909$	−4.44264e33	−0.420878
$910$	0	0
$911$	1.60065e34	1.47855	0.739273	−	0.673406i	$-0.235169\pi$
$911$	1.60065e34	1.47855	0.739273	+	0.673406i	$0.235169\pi$
$912$	0	0
$913$	8.82233e33	0.794638
$914$	0	0
$915$	7.32809e32	0.0643648
$916$	0	0
$917$	−1.36147e34	−1.16617
$918$	0	0
$919$	1.54454e34	1.29024	0.645120	−	0.764081i	$-0.276807\pi$
$919$	1.54454e34	1.29024	0.645120	+	0.764081i	$0.276807\pi$
$920$	0	0
$921$	5.65088e32	0.0460395
$922$	0	0
$923$	−1.82501e34	−1.45026
$924$	0	0
$925$	−3.71881e33	−0.288253
$926$	0	0
$927$	−2.96969e33	−0.224540
$928$	0	0
$929$	1.19786e34	0.883535	0.441768	−	0.897129i	$-0.354352\pi$
$929$	1.19786e34	0.883535	0.441768	+	0.897129i	$0.354352\pi$
$930$	0	0
$931$	3.92631e32	0.0282529
$932$	0	0
$933$	−1.77653e33	−0.124719
$934$	0	0
$935$	3.83738e32	0.0262846
$936$	0	0
$937$	−2.83646e33	−0.189570	−0.0947852	−	0.995498i	$-0.530216\pi$
$937$	−2.83646e33	−0.189570	−0.0947852	+	0.995498i	$0.530216\pi$
$938$	0	0
$939$	1.25407e33	0.0817839
$940$	0	0
$941$	2.88589e34	1.83653	0.918267	−	0.395961i	$-0.129588\pi$
$941$	2.88589e34	1.83653	0.918267	+	0.395961i	$0.129588\pi$
$942$	0	0
$943$	−1.68384e34	−1.04572
$944$	0	0
$945$	1.50590e33	0.0912707
$946$	0	0
$947$	−2.29990e34	−1.36045	−0.680226	−	0.733002i	$-0.738118\pi$
$947$	−2.29990e34	−1.36045	−0.680226	+	0.733002i	$0.738118\pi$
$948$	0	0
$949$	−3.64023e34	−2.10168
$950$	0	0
$951$	−1.42695e32	−0.00804144
$952$	0	0
$953$	−2.39769e34	−1.31894	−0.659470	−	0.751731i	$-0.729219\pi$
$953$	−2.39769e34	−1.31894	−0.659470	+	0.751731i	$0.729219\pi$
$954$	0	0
$955$	−2.73258e33	−0.146735
$956$	0	0
$957$	−1.22593e33	−0.0642658
$958$	0	0
$959$	−6.30062e33	−0.322455
$960$	0	0
$961$	−2.00115e34	−0.999908
$962$	0	0
$963$	−5.43829e33	−0.265314
$964$	0	0
$965$	8.53563e33	0.406604
$966$	0	0
$967$	6.66196e33	0.309883	0.154942	−	0.987924i	$-0.450481\pi$
$967$	6.66196e33	0.309883	0.154942	+	0.987924i	$0.450481\pi$
$968$	0	0
$969$	−6.09398e32	−0.0276808
$970$	0	0
$971$	3.35172e34	1.48678	0.743390	−	0.668858i	$-0.233217\pi$
$971$	3.35172e34	1.48678	0.743390	+	0.668858i	$0.233217\pi$
$972$	0	0
$973$	1.92490e34	0.833892
$974$	0	0
$975$	−4.25115e33	−0.179868
$976$	0	0
$977$	1.17263e34	0.484590	0.242295	−	0.970203i	$-0.422100\pi$
$977$	1.17263e34	0.484590	0.242295	+	0.970203i	$0.422100\pi$
$978$	0	0
$979$	−1.09428e33	−0.0441699
$980$	0	0
$981$	−2.63207e34	−1.03778
$982$	0	0
$983$	−1.51439e34	−0.583274	−0.291637	−	0.956529i	$-0.594200\pi$
$983$	−1.51439e34	−0.583274	−0.291637	+	0.956529i	$0.594200\pi$
$984$	0	0
$985$	3.20154e33	0.120460
$986$	0	0
$987$	−6.18808e33	−0.227462
$988$	0	0
$989$	8.97497e33	0.322311
$990$	0	0
$991$	−2.60260e34	−0.913187	−0.456593	−	0.889675i	$-0.650931\pi$
$991$	−2.60260e34	−0.913187	−0.456593	+	0.889675i	$0.650931\pi$
$992$	0	0
$993$	−1.96122e33	−0.0672374
$994$	0	0
$995$	1.42976e34	0.478957
$996$	0	0
$997$	−1.96863e34	−0.644422	−0.322211	−	0.946668i	$-0.604426\pi$
$997$	−1.96863e34	−0.644422	−0.322211	+	0.946668i	$0.604426\pi$
$998$	0	0
$999$	−3.10613e33	−0.0993613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.24.a.b.1.2		2
4.3	odd	2		1.24.a.a.1.2	✓	2
8.3	odd	2		64.24.a.d.1.2		2
8.5	even	2		64.24.a.g.1.1		2
12.11	even	2		9.24.a.b.1.1		2
20.3	even	4		25.24.b.a.24.1		4
20.7	even	4		25.24.b.a.24.4		4
20.19	odd	2		25.24.a.a.1.1		2
28.27	even	2		49.24.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.24.a.a.1.2	✓	2	4.3	odd	2
9.24.a.b.1.1		2	12.11	even	2
16.24.a.b.1.2		2	1.1	even	1	trivial
25.24.a.a.1.1		2	20.19	odd	2
25.24.b.a.24.1		4	20.3	even	4
25.24.b.a.24.4		4	20.7	even	4
49.24.a.b.1.2		2	28.27	even	2
64.24.a.d.1.2		2	8.3	odd	2
64.24.a.g.1.1		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+48964.9 q^{3} -3.19930e7 q^{5} +5.17135e9 q^{7} -9.17456e10 q^{9} +O(q^{10})\)	\(q+48964.9 q^{3} -3.19930e7 q^{5} +5.17135e9 q^{7} -9.17456e10 q^{9} -6.04602e11 q^{11} +7.96710e12 q^{13} -1.56653e12 q^{15} +1.98385e13 q^{17} -6.27346e14 q^{19} +2.53215e14 q^{21} +4.55685e15 q^{23} -1.08974e16 q^{25} -9.10202e15 q^{27} +4.14107e16 q^{29} -1.35683e15 q^{31} -2.96043e16 q^{33} -1.65447e17 q^{35} +3.41258e17 q^{37} +3.90108e17 q^{39} -3.69518e18 q^{41} +1.96955e18 q^{43} +2.93522e18 q^{45} -2.44381e19 q^{47} -6.25860e17 q^{49} +9.71391e17 q^{51} -6.39437e19 q^{53} +1.93431e19 q^{55} -3.07179e19 q^{57} -2.81892e20 q^{59} -4.67790e20 q^{61} -4.74449e20 q^{63} -2.54892e20 q^{65} -2.77406e20 q^{67} +2.23126e20 q^{69} -2.29069e21 q^{71} -4.56908e21 q^{73} -5.33588e20 q^{75} -3.12661e21 q^{77} +3.99005e21 q^{79} +8.19155e21 q^{81} -1.45920e22 q^{83} -6.34694e20 q^{85} +2.02767e21 q^{87} +1.80991e21 q^{89} +4.12007e22 q^{91} -6.64369e19 q^{93} +2.00707e22 q^{95} +8.25561e22 q^{97} +5.54696e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 339480 q^{3} + 73069020 q^{5} + 1359184400 q^{7} - 34999394166 q^{9}+O(q^{10}) \) 2 * q - 339480 * q^3 + 73069020 * q^5 + 1359184400 * q^7 - 34999394166 * q^9	\( 2 q - 339480 q^{3} + 73069020 q^{5} + 1359184400 q^{7} - 34999394166 q^{9} - 856801968264 q^{11} + 4376109322060 q^{13} - 42377338985040 q^{15} + 254028147597540 q^{17} - 4260600979960 q^{19} + 17\!\cdots\!44 q^{21}+ \cdots + 41\!\cdots\!12 q^{99}+O(q^{100}) \) 2 * q - 339480 * q^3 + 73069020 * q^5 + 1359184400 * q^7 - 34999394166 * q^9 - 856801968264 * q^11 + 4376109322060 * q^13 - 42377338985040 * q^15 + 254028147597540 * q^17 - 4260600979960 * q^19 + 1734031637722944 * q^21 + 8144713079008560 * q^23 - 11780274628800850 * q^25 + 5424634982716560 * q^27 + 20818433601623340 * q^29 - 137714017177000384 * q^31 + 68361366766001760 * q^33 - 565961271250425120 * q^35 - 897721264408967780 * q^37 + 1785011473665029232 * q^39 - 2294435477168314956 * q^41 + 1750760768619855800 * q^43 + 8897092690294206540 * q^45 - 15759744217656780960 * q^47 - 13461981704376200814 * q^49 - 89998362845078292144 * q^51 - 140287253401646796420 * q^53 - 7153550955060182640 * q^55 - 272752401448627175520 * q^57 - 280872989971340771880 * q^59 - 180452892516502223636 * q^61 - 690775113933935014320 * q^63 - 632168834809440380760 * q^65 - 1754233163431557625240 * q^67 - 1170560672172404223552 * q^69 - 3055033510194143328624 * q^71 - 8063408253877606149260 * q^73 - 190631089350520885800 * q^75 - 2165184764357449665600 * q^77 - 6244916814559639980640 * q^79 - 2793528580929833975598 * q^81 - 6875994082418498976120 * q^83 + 23969743087870314902520 * q^85 + 10026640653837674384880 * q^87 + 6395093086173070004820 * q^89 + 54890178162704560146016 * q^91 + 52900811441357852079360 * q^93 + 85533361066700858502000 * q^95 - 31147288846254030500540 * q^97 + 41158245132312135981912 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more