LMFDB - Embedded newform 16.14.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,14,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, 14, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 14 $
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:	$17.1569486323$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{781}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 195 $ x^2 - x - 195
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-13.4732$ 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)

$f(q)$	$=$	$q+1352.57 q^{3} -12224.8 q^{5} -452526. q^{7} +235118. q^{9} +O(q^{10})$	$q+1352.57 q^{3} -12224.8 q^{5} -452526. q^{7} +235118. q^{9} -7.28729e6 q^{11} +2.80664e7 q^{13} -1.65349e7 q^{15} -1.45191e8 q^{17} -1.08428e8 q^{19} -6.12072e8 q^{21} -1.59777e8 q^{23} -1.07126e9 q^{25} -1.83842e9 q^{27} +4.40214e9 q^{29} -5.53608e9 q^{31} -9.85656e9 q^{33} +5.53205e9 q^{35} +1.25151e10 q^{37} +3.79617e10 q^{39} +1.51058e10 q^{41} +3.15026e10 q^{43} -2.87427e9 q^{45} -3.50306e10 q^{47} +1.07891e11 q^{49} -1.96381e11 q^{51} -1.09912e11 q^{53} +8.90858e10 q^{55} -1.46656e11 q^{57} +1.18303e11 q^{59} -2.98314e11 q^{61} -1.06397e11 q^{63} -3.43107e11 q^{65} -7.44158e11 q^{67} -2.16109e11 q^{69} +1.15098e12 q^{71} +1.84863e12 q^{73} -1.44895e12 q^{75} +3.29769e12 q^{77} +2.20082e12 q^{79} -2.86144e12 q^{81} +1.92754e12 q^{83} +1.77494e12 q^{85} +5.95420e12 q^{87} -5.50546e12 q^{89} -1.27008e13 q^{91} -7.48793e12 q^{93} +1.32551e12 q^{95} +1.35322e13 q^{97} -1.71337e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 872 q^{3} + 18476 q^{5} - 110928 q^{7} + 3589498 q^{9}+O(q^{10}) $ 2 * q - 872 * q^3 + 18476 * q^5 - 110928 * q^7 + 3589498 * q^9	\( 2 q - 872 q^{3} + 18476 q^{5} - 110928 q^{7} + 3589498 q^{9} - 16474040 q^{11} + 18744572 q^{13} - 84830960 q^{15} - 153793628 q^{17} + 118747640 q^{19} - 1371980736 q^{21} - 718268912 q^{23} - 1349419874 q^{25} - 5753785616 q^{27} + 309341340 q^{29} - 5767504192 q^{31} + 10579993952 q^{33} + 16019391264 q^{35} - 11621553300 q^{37} + 58698760912 q^{39} + 1311168276 q^{41} + 29595620104 q^{43} + 100107952252 q^{45} - 12313617888 q^{47} + 127691156146 q^{49} - 177245377360 q^{51} - 38006007028 q^{53} - 192954931664 q^{55} - 652023429728 q^{57} - 253345911704 q^{59} - 647244384292 q^{61} + 1039453222896 q^{63} - 629294375512 q^{65} - 1619993806312 q^{67} + 1026293771456 q^{69} + 1040270142512 q^{71} + 4005283908692 q^{73} - 830155668760 q^{75} + 159512776896 q^{77} + 2521777572064 q^{79} + 500597403058 q^{81} + 290486230904 q^{83} + 1510847254552 q^{85} + 15058906809936 q^{87} - 8723755657740 q^{89} - 15885098476896 q^{91} - 6973121519360 q^{93} + 8299984202576 q^{95} + 9601712299972 q^{97} - 32529223326424 q^{99}+O(q^{100}) \) 2 * q - 872 * q^3 + 18476 * q^5 - 110928 * q^7 + 3589498 * q^9 - 16474040 * q^11 + 18744572 * q^13 - 84830960 * q^15 - 153793628 * q^17 + 118747640 * q^19 - 1371980736 * q^21 - 718268912 * q^23 - 1349419874 * q^25 - 5753785616 * q^27 + 309341340 * q^29 - 5767504192 * q^31 + 10579993952 * q^33 + 16019391264 * q^35 - 11621553300 * q^37 + 58698760912 * q^39 + 1311168276 * q^41 + 29595620104 * q^43 + 100107952252 * q^45 - 12313617888 * q^47 + 127691156146 * q^49 - 177245377360 * q^51 - 38006007028 * q^53 - 192954931664 * q^55 - 652023429728 * q^57 - 253345911704 * q^59 - 647244384292 * q^61 + 1039453222896 * q^63 - 629294375512 * q^65 - 1619993806312 * q^67 + 1026293771456 * q^69 + 1040270142512 * q^71 + 4005283908692 * q^73 - 830155668760 * q^75 + 159512776896 * q^77 + 2521777572064 * q^79 + 500597403058 * q^81 + 290486230904 * q^83 + 1510847254552 * q^85 + 15058906809936 * q^87 - 8723755657740 * q^89 - 15885098476896 * q^91 - 6973121519360 * q^93 + 8299984202576 * q^95 + 9601712299972 * q^97 - 32529223326424 * 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
$3$	1352.57	1.07120	0.535601	−	0.844471i	$-0.320085\pi$
$3$	1352.57	1.07120	0.535601	+	0.844471i	$0.320085\pi$
$4$	0	0
$5$	−12224.8	−0.349895	−0.174947	−	0.984578i	$-0.555976\pi$
$5$	−12224.8	−0.349895	−0.174947	+	0.984578i	$0.555976\pi$
$6$	0	0
$7$	−452526.	−1.45381	−0.726903	−	0.686740i	$-0.759041\pi$
$7$	−452526.	−1.45381	−0.726903	+	0.686740i	$0.759041\pi$
$8$	0	0
$9$	235118.	0.147472
$10$	0	0
$11$	−7.28729e6	−1.24026	−0.620131	−	0.784498i	$-0.712921\pi$
$11$	−7.28729e6	−1.24026	−0.620131	+	0.784498i	$0.712921\pi$
$12$	0	0
$13$	2.80664e7	1.61270	0.806352	−	0.591435i	$-0.201438\pi$
$13$	2.80664e7	1.61270	0.806352	+	0.591435i	$0.201438\pi$
$14$	0	0
$15$	−1.65349e7	−0.374808
$16$	0	0
$17$	−1.45191e8	−1.45889	−0.729446	−	0.684038i	$-0.760222\pi$
$17$	−1.45191e8	−1.45889	−0.729446	+	0.684038i	$0.760222\pi$
$18$	0	0
$19$	−1.08428e8	−0.528740	−0.264370	−	0.964421i	$-0.585164\pi$
$19$	−1.08428e8	−0.528740	−0.264370	+	0.964421i	$0.585164\pi$
$20$	0	0
$21$	−6.12072e8	−1.55732
$22$	0	0
$23$	−1.59777e8	−0.225052	−0.112526	−	0.993649i	$-0.535894\pi$
$23$	−1.59777e8	−0.225052	−0.112526	+	0.993649i	$0.535894\pi$
$24$	0	0
$25$	−1.07126e9	−0.877574
$26$	0	0
$27$	−1.83842e9	−0.913229
$28$	0	0
$29$	4.40214e9	1.37429	0.687143	−	0.726522i	$-0.258865\pi$
$29$	4.40214e9	1.37429	0.687143	+	0.726522i	$0.258865\pi$
$30$	0	0
$31$	−5.53608e9	−1.12034	−0.560172	−	0.828376i	$-0.689265\pi$
$31$	−5.53608e9	−1.12034	−0.560172	+	0.828376i	$0.689265\pi$
$32$	0	0
$33$	−9.85656e9	−1.32857
$34$	0	0
$35$	5.53205e9	0.508679
$36$	0	0
$37$	1.25151e10	0.801905	0.400953	−	0.916099i	$-0.368679\pi$
$37$	1.25151e10	0.801905	0.400953	+	0.916099i	$0.368679\pi$
$38$	0	0
$39$	3.79617e10	1.72753
$40$	0	0
$41$	1.51058e10	0.496648	0.248324	−	0.968677i	$-0.420120\pi$
$41$	1.51058e10	0.496648	0.248324	+	0.968677i	$0.420120\pi$
$42$	0	0
$43$	3.15026e10	0.759979	0.379990	−	0.924991i	$-0.375928\pi$
$43$	3.15026e10	0.759979	0.379990	+	0.924991i	$0.375928\pi$
$44$	0	0
$45$	−2.87427e9	−0.0515996
$46$	0	0
$47$	−3.50306e10	−0.474037	−0.237018	−	0.971505i	$-0.576170\pi$
$47$	−3.50306e10	−0.474037	−0.237018	+	0.971505i	$0.576170\pi$
$48$	0	0
$49$	1.07891e11	1.11355
$50$	0	0
$51$	−1.96381e11	−1.56277
$52$	0	0
$53$	−1.09912e11	−0.681164	−0.340582	−	0.940215i	$-0.610624\pi$
$53$	−1.09912e11	−0.681164	−0.340582	+	0.940215i	$0.610624\pi$
$54$	0	0
$55$	8.90858e10	0.433961
$56$	0	0
$57$	−1.46656e11	−0.566387
$58$	0	0
$59$	1.18303e11	0.365137	0.182569	−	0.983193i	$-0.441559\pi$
$59$	1.18303e11	0.365137	0.182569	+	0.983193i	$0.441559\pi$
$60$	0	0
$61$	−2.98314e11	−0.741361	−0.370680	−	0.928760i	$-0.620875\pi$
$61$	−2.98314e11	−0.741361	−0.370680	+	0.928760i	$0.620875\pi$
$62$	0	0
$63$	−1.06397e11	−0.214395
$64$	0	0
$65$	−3.43107e11	−0.564277
$66$	0	0
$67$	−7.44158e11	−1.00503	−0.502515	−	0.864569i	$-0.667592\pi$
$67$	−7.44158e11	−1.00503	−0.502515	+	0.864569i	$0.667592\pi$
$68$	0	0
$69$	−2.16109e11	−0.241076
$70$	0	0
$71$	1.15098e12	1.06632	0.533162	−	0.846013i	$-0.321004\pi$
$71$	1.15098e12	1.06632	0.533162	+	0.846013i	$0.321004\pi$
$72$	0	0
$73$	1.84863e12	1.42972	0.714861	−	0.699267i	$-0.246490\pi$
$73$	1.84863e12	1.42972	0.714861	+	0.699267i	$0.246490\pi$
$74$	0	0
$75$	−1.44895e12	−0.940058
$76$	0	0
$77$	3.29769e12	1.80310
$78$	0	0
$79$	2.20082e12	1.01861	0.509305	−	0.860586i	$-0.329902\pi$
$79$	2.20082e12	1.01861	0.509305	+	0.860586i	$0.329902\pi$
$80$	0	0
$81$	−2.86144e12	−1.12572
$82$	0	0
$83$	1.92754e12	0.647136	0.323568	−	0.946205i	$-0.395117\pi$
$83$	1.92754e12	0.647136	0.323568	+	0.946205i	$0.395117\pi$
$84$	0	0
$85$	1.77494e12	0.510459
$86$	0	0
$87$	5.95420e12	1.47214
$88$	0	0
$89$	−5.50546e12	−1.17424	−0.587122	−	0.809498i	$-0.699739\pi$
$89$	−5.50546e12	−1.17424	−0.587122	+	0.809498i	$0.699739\pi$
$90$	0	0
$91$	−1.27008e13	−2.34456
$92$	0	0
$93$	−7.48793e12	−1.20011
$94$	0	0
$95$	1.32551e12	0.185003
$96$	0	0
$97$	1.35322e13	1.64950	0.824752	−	0.565495i	$-0.191315\pi$
$97$	1.35322e13	1.64950	0.824752	+	0.565495i	$0.191315\pi$
$98$	0	0
$99$	−1.71337e12	−0.182904
$100$	0	0
$101$	−8.36824e12	−0.784414	−0.392207	−	0.919877i	$-0.628288\pi$
$101$	−8.36824e12	−0.784414	−0.392207	+	0.919877i	$0.628288\pi$
$102$	0	0
$103$	−2.33937e12	−0.193044	−0.0965222	−	0.995331i	$-0.530772\pi$
$103$	−2.33937e12	−0.193044	−0.0965222	+	0.995331i	$0.530772\pi$
$104$	0	0
$105$	7.48247e12	0.544897
$106$	0	0
$107$	−2.43985e13	−1.57170	−0.785849	−	0.618418i	$-0.787774\pi$
$107$	−2.43985e13	−1.57170	−0.785849	+	0.618418i	$0.787774\pi$
$108$	0	0
$109$	1.59545e11	0.00911196	0.00455598	−	0.999990i	$-0.498550\pi$
$109$	1.59545e11	0.00911196	0.00455598	+	0.999990i	$0.498550\pi$
$110$	0	0
$111$	1.69275e13	0.859002
$112$	0	0
$113$	−2.90424e13	−1.31227	−0.656133	−	0.754645i	$-0.727809\pi$
$113$	−2.90424e13	−1.31227	−0.656133	+	0.754645i	$0.727809\pi$
$114$	0	0
$115$	1.95325e12	0.0787446
$116$	0	0
$117$	6.59890e12	0.237828
$118$	0	0
$119$	6.57029e13	2.12095
$120$	0	0
$121$	1.85819e13	0.538251
$122$	0	0
$123$	2.04316e13	0.532010
$124$	0	0
$125$	2.80188e13	0.656953
$126$	0	0
$127$	−5.45015e13	−1.15261	−0.576307	−	0.817233i	$-0.695507\pi$
$127$	−5.45015e13	−1.15261	−0.576307	+	0.817233i	$0.695507\pi$
$128$	0	0
$129$	4.26094e13	0.814091
$130$	0	0
$131$	1.87712e12	0.0324510	0.0162255	−	0.999868i	$-0.494835\pi$
$131$	1.87712e12	0.0324510	0.0162255	+	0.999868i	$0.494835\pi$
$132$	0	0
$133$	4.90664e13	0.768686
$134$	0	0
$135$	2.24743e13	0.319534
$136$	0	0
$137$	−8.61421e13	−1.11309	−0.556547	−	0.830816i	$-0.687874\pi$
$137$	−8.61421e13	−1.11309	−0.556547	+	0.830816i	$0.687874\pi$
$138$	0	0
$139$	−6.45595e13	−0.759214	−0.379607	−	0.925148i	$-0.623941\pi$
$139$	−6.45595e13	−0.759214	−0.379607	+	0.925148i	$0.623941\pi$
$140$	0	0
$141$	−4.73813e13	−0.507789
$142$	0	0
$143$	−2.04528e14	−2.00018
$144$	0	0
$145$	−5.38154e13	−0.480855
$146$	0	0
$147$	1.45930e14	1.19284
$148$	0	0
$149$	−7.18629e13	−0.538015	−0.269007	−	0.963138i	$-0.586696\pi$
$149$	−7.18629e13	−0.538015	−0.269007	+	0.963138i	$0.586696\pi$
$150$	0	0
$151$	1.05962e14	0.727445	0.363723	−	0.931507i	$-0.381506\pi$
$151$	1.05962e14	0.727445	0.363723	+	0.931507i	$0.381506\pi$
$152$	0	0
$153$	−3.41371e13	−0.215145
$154$	0	0
$155$	6.76776e13	0.392003
$156$	0	0
$157$	−9.36766e13	−0.499210	−0.249605	−	0.968348i	$-0.580301\pi$
$157$	−9.36766e13	−0.499210	−0.249605	+	0.968348i	$0.580301\pi$
$158$	0	0
$159$	−1.48663e14	−0.729663
$160$	0	0
$161$	7.23033e13	0.327182
$162$	0	0
$163$	1.96048e14	0.818734	0.409367	−	0.912370i	$-0.365750\pi$
$163$	1.96048e14	0.818734	0.409367	+	0.912370i	$0.365750\pi$
$164$	0	0
$165$	1.20495e14	0.464860
$166$	0	0
$167$	−4.90223e14	−1.74878	−0.874392	−	0.485220i	$-0.838739\pi$
$167$	−4.90223e14	−1.74878	−0.874392	+	0.485220i	$0.838739\pi$
$168$	0	0
$169$	4.84848e14	1.60082
$170$	0	0
$171$	−2.54933e13	−0.0779743
$172$	0	0
$173$	−7.90181e13	−0.224092	−0.112046	−	0.993703i	$-0.535740\pi$
$173$	−7.90181e13	−0.224092	−0.112046	+	0.993703i	$0.535740\pi$
$174$	0	0
$175$	4.84772e14	1.27582
$176$	0	0
$177$	1.60012e14	0.391135
$178$	0	0
$179$	3.61625e14	0.821700	0.410850	−	0.911703i	$-0.365232\pi$
$179$	3.61625e14	0.821700	0.410850	+	0.911703i	$0.365232\pi$
$180$	0	0
$181$	−1.55078e14	−0.327822	−0.163911	−	0.986475i	$-0.552411\pi$
$181$	−1.55078e14	−0.327822	−0.163911	+	0.986475i	$0.552411\pi$
$182$	0	0
$183$	−4.03490e14	−0.794147
$184$	0	0
$185$	−1.52995e14	−0.280582
$186$	0	0
$187$	1.05805e15	1.80941
$188$	0	0
$189$	8.31932e14	1.32766
$190$	0	0
$191$	−1.82528e14	−0.272027	−0.136013	−	0.990707i	$-0.543429\pi$
$191$	−1.82528e14	−0.272027	−0.136013	+	0.990707i	$0.543429\pi$
$192$	0	0
$193$	−1.28326e15	−1.78728	−0.893639	−	0.448786i	$-0.851857\pi$
$193$	−1.28326e15	−1.78728	−0.893639	+	0.448786i	$0.851857\pi$
$194$	0	0
$195$	−4.64075e14	−0.604454
$196$	0	0
$197$	−3.04303e14	−0.370916	−0.185458	−	0.982652i	$-0.559377\pi$
$197$	−3.04303e14	−0.370916	−0.185458	+	0.982652i	$0.559377\pi$
$198$	0	0
$199$	−6.78433e13	−0.0774394	−0.0387197	−	0.999250i	$-0.512328\pi$
$199$	−6.78433e13	−0.0774394	−0.0387197	+	0.999250i	$0.512328\pi$
$200$	0	0
$201$	−1.00652e15	−1.07659
$202$	0	0
$203$	−1.99208e15	−1.99794
$204$	0	0
$205$	−1.84666e14	−0.173775
$206$	0	0
$207$	−3.75664e13	−0.0331889
$208$	0	0
$209$	7.90145e14	0.655777
$210$	0	0
$211$	−8.63768e14	−0.673847	−0.336924	−	0.941532i	$-0.609386\pi$
$211$	−8.63768e14	−0.673847	−0.336924	+	0.941532i	$0.609386\pi$
$212$	0	0
$213$	1.55678e15	1.14225
$214$	0	0
$215$	−3.85114e14	−0.265913
$216$	0	0
$217$	2.50522e15	1.62876
$218$	0	0
$219$	2.50040e15	1.53152
$220$	0	0
$221$	−4.07500e15	−2.35276
$222$	0	0
$223$	2.22939e15	1.21396	0.606981	−	0.794717i	$-0.292381\pi$
$223$	2.22939e15	1.21396	0.606981	+	0.794717i	$0.292381\pi$
$224$	0	0
$225$	−2.51871e14	−0.129417
$226$	0	0
$227$	−3.44059e15	−1.66903	−0.834516	−	0.550983i	$-0.814253\pi$
$227$	−3.44059e15	−1.66903	−0.834516	+	0.550983i	$0.814253\pi$
$228$	0	0
$229$	7.09971e14	0.325320	0.162660	−	0.986682i	$-0.447993\pi$
$229$	7.09971e14	0.325320	0.162660	+	0.986682i	$0.447993\pi$
$230$	0	0
$231$	4.46035e15	1.93148
$232$	0	0
$233$	−8.27446e14	−0.338786	−0.169393	−	0.985549i	$-0.554181\pi$
$233$	−8.27446e14	−0.338786	−0.169393	+	0.985549i	$0.554181\pi$
$234$	0	0
$235$	4.28243e14	0.165863
$236$	0	0
$237$	2.97676e15	1.09114
$238$	0	0
$239$	1.13283e15	0.393167	0.196584	−	0.980487i	$-0.437015\pi$
$239$	1.13283e15	0.393167	0.196584	+	0.980487i	$0.437015\pi$
$240$	0	0
$241$	2.81149e13	0.00924329	0.00462164	−	0.999989i	$-0.498529\pi$
$241$	2.81149e13	0.00924329	0.00462164	+	0.999989i	$0.498529\pi$
$242$	0	0
$243$	−9.39259e14	−0.292647
$244$	0	0
$245$	−1.31895e15	−0.389626
$246$	0	0
$247$	−3.04318e15	−0.852702
$248$	0	0
$249$	2.60713e15	0.693213
$250$	0	0
$251$	5.45538e14	0.137704	0.0688519	−	0.997627i	$-0.478066\pi$
$251$	5.45538e14	0.137704	0.0688519	+	0.997627i	$0.478066\pi$
$252$	0	0
$253$	1.16434e15	0.279124
$254$	0	0
$255$	2.40073e15	0.546804
$256$	0	0
$257$	4.41616e14	0.0956048	0.0478024	−	0.998857i	$-0.484778\pi$
$257$	4.41616e14	0.0956048	0.0478024	+	0.998857i	$0.484778\pi$
$258$	0	0
$259$	−5.66341e15	−1.16581
$260$	0	0
$261$	1.03502e15	0.202668
$262$	0	0
$263$	−1.77399e15	−0.330551	−0.165276	−	0.986247i	$-0.552851\pi$
$263$	−1.77399e15	−0.330551	−0.165276	+	0.986247i	$0.552851\pi$
$264$	0	0
$265$	1.34365e15	0.238336
$266$	0	0
$267$	−7.44651e15	−1.25785
$268$	0	0
$269$	−1.01131e16	−1.62741	−0.813703	−	0.581281i	$-0.802552\pi$
$269$	−1.01131e16	−1.62741	−0.813703	+	0.581281i	$0.802552\pi$
$270$	0	0
$271$	3.06778e15	0.470461	0.235231	−	0.971940i	$-0.424415\pi$
$271$	3.06778e15	0.470461	0.235231	+	0.971940i	$0.424415\pi$
$272$	0	0
$273$	−1.71787e16	−2.51150
$274$	0	0
$275$	7.80656e15	1.08842
$276$	0	0
$277$	7.55487e15	1.00487	0.502433	−	0.864616i	$-0.332438\pi$
$277$	7.55487e15	1.00487	0.502433	+	0.864616i	$0.332438\pi$
$278$	0	0
$279$	−1.30163e15	−0.165219
$280$	0	0
$281$	−6.02256e15	−0.729777	−0.364888	−	0.931051i	$-0.618893\pi$
$281$	−6.02256e15	−0.729777	−0.364888	+	0.931051i	$0.618893\pi$
$282$	0	0
$283$	7.97392e14	0.0922698	0.0461349	−	0.998935i	$-0.485310\pi$
$283$	7.97392e14	0.0922698	0.0461349	+	0.998935i	$0.485310\pi$
$284$	0	0
$285$	1.79284e15	0.198176
$286$	0	0
$287$	−6.83577e15	−0.722030
$288$	0	0
$289$	1.11760e16	1.12837
$290$	0	0
$291$	1.83033e16	1.76695
$292$	0	0
$293$	1.71127e16	1.58008	0.790039	−	0.613057i	$-0.210060\pi$
$293$	1.71127e16	1.58008	0.790039	+	0.613057i	$0.210060\pi$
$294$	0	0
$295$	−1.44623e15	−0.127760
$296$	0	0
$297$	1.33971e16	1.13264
$298$	0	0
$299$	−4.48437e15	−0.362943
$300$	0	0
$301$	−1.42558e16	−1.10486
$302$	0	0
$303$	−1.13186e16	−0.840265
$304$	0	0
$305$	3.64683e15	0.259398
$306$	0	0
$307$	1.08209e16	0.737676	0.368838	−	0.929494i	$-0.379756\pi$
$307$	1.08209e16	0.737676	0.368838	+	0.929494i	$0.379756\pi$
$308$	0	0
$309$	−3.16416e15	−0.206789
$310$	0	0
$311$	1.10167e16	0.690411	0.345206	−	0.938527i	$-0.387809\pi$
$311$	1.10167e16	0.690411	0.345206	+	0.938527i	$0.387809\pi$
$312$	0	0
$313$	2.64690e16	1.59111	0.795555	−	0.605882i	$-0.207180\pi$
$313$	2.64690e16	1.59111	0.795555	+	0.605882i	$0.207180\pi$
$314$	0	0
$315$	1.30068e15	0.0750158
$316$	0	0
$317$	−3.88517e15	−0.215043	−0.107521	−	0.994203i	$-0.534291\pi$
$317$	−3.88517e15	−0.215043	−0.107521	+	0.994203i	$0.534291\pi$
$318$	0	0
$319$	−3.20797e16	−1.70447
$320$	0	0
$321$	−3.30007e16	−1.68361
$322$	0	0
$323$	1.57428e16	0.771375
$324$	0	0
$325$	−3.00663e16	−1.41527
$326$	0	0
$327$	2.15796e14	0.00976074
$328$	0	0
$329$	1.58523e16	0.689157
$330$	0	0
$331$	3.69659e16	1.54497	0.772483	−	0.635035i	$-0.219014\pi$
$331$	3.69659e16	1.54497	0.772483	+	0.635035i	$0.219014\pi$
$332$	0	0
$333$	2.94252e15	0.118258
$334$	0	0
$335$	9.09720e15	0.351655
$336$	0	0
$337$	−3.50293e16	−1.30268	−0.651340	−	0.758786i	$-0.725793\pi$
$337$	−3.50293e16	−1.30268	−0.651340	+	0.758786i	$0.725793\pi$
$338$	0	0
$339$	−3.92818e16	−1.40570
$340$	0	0
$341$	4.03431e16	1.38952
$342$	0	0
$343$	−4.97864e15	−0.165082
$344$	0	0
$345$	2.64190e15	0.0843513
$346$	0	0
$347$	−3.29030e16	−1.01180	−0.505899	−	0.862593i	$-0.668839\pi$
$347$	−3.29030e16	−1.01180	−0.505899	+	0.862593i	$0.668839\pi$
$348$	0	0
$349$	−4.00420e16	−1.18618	−0.593090	−	0.805136i	$-0.702092\pi$
$349$	−4.00420e16	−1.18618	−0.593090	+	0.805136i	$0.702092\pi$
$350$	0	0
$351$	−5.15978e16	−1.47277
$352$	0	0
$353$	−1.37884e16	−0.379296	−0.189648	−	0.981852i	$-0.560735\pi$
$353$	−1.37884e16	−0.379296	−0.189648	+	0.981852i	$0.560735\pi$
$354$	0	0
$355$	−1.40705e16	−0.373101
$356$	0	0
$357$	8.88677e16	2.27196
$358$	0	0
$359$	4.30837e14	0.0106218	0.00531092	−	0.999986i	$-0.498309\pi$
$359$	4.30837e14	0.0106218	0.00531092	+	0.999986i	$0.498309\pi$
$360$	0	0
$361$	−3.02964e16	−0.720434
$362$	0	0
$363$	2.51333e16	0.576575
$364$	0	0
$365$	−2.25992e16	−0.500252
$366$	0	0
$367$	6.94463e16	1.48361	0.741804	−	0.670616i	$-0.233970\pi$
$367$	6.94463e16	1.48361	0.741804	+	0.670616i	$0.233970\pi$
$368$	0	0
$369$	3.55164e15	0.0732416
$370$	0	0
$371$	4.97380e16	0.990280
$372$	0	0
$373$	2.24900e16	0.432396	0.216198	−	0.976350i	$-0.430634\pi$
$373$	2.24900e16	0.432396	0.216198	+	0.976350i	$0.430634\pi$
$374$	0	0
$375$	3.78973e16	0.703729
$376$	0	0
$377$	1.23552e17	2.21632
$378$	0	0
$379$	−4.69095e16	−0.813029	−0.406514	−	0.913644i	$-0.633256\pi$
$379$	−4.69095e16	−0.813029	−0.406514	+	0.913644i	$0.633256\pi$
$380$	0	0
$381$	−7.37169e16	−1.23468
$382$	0	0
$383$	6.37867e16	1.03261	0.516307	−	0.856403i	$-0.327306\pi$
$383$	6.37867e16	1.03261	0.516307	+	0.856403i	$0.327306\pi$
$384$	0	0
$385$	−4.03137e16	−0.630895
$386$	0	0
$387$	7.40682e15	0.112075
$388$	0	0
$389$	−1.51321e16	−0.221425	−0.110712	−	0.993852i	$-0.535313\pi$
$389$	−1.51321e16	−0.221425	−0.110712	+	0.993852i	$0.535313\pi$
$390$	0	0
$391$	2.31983e16	0.328327
$392$	0	0
$393$	2.53893e15	0.0347616
$394$	0	0
$395$	−2.69046e16	−0.356406
$396$	0	0
$397$	1.79854e16	0.230558	0.115279	−	0.993333i	$-0.463224\pi$
$397$	1.79854e16	0.230558	0.115279	+	0.993333i	$0.463224\pi$
$398$	0	0
$399$	6.63657e16	0.823417
$400$	0	0
$401$	−8.16229e15	−0.0980332	−0.0490166	−	0.998798i	$-0.515609\pi$
$401$	−8.16229e15	−0.0980332	−0.0490166	+	0.998798i	$0.515609\pi$
$402$	0	0
$403$	−1.55378e17	−1.80679
$404$	0	0
$405$	3.49806e16	0.393885
$406$	0	0
$407$	−9.12012e16	−0.994573
$408$	0	0
$409$	1.55109e16	0.163845	0.0819227	−	0.996639i	$-0.473894\pi$
$409$	1.55109e16	0.163845	0.0819227	+	0.996639i	$0.473894\pi$
$410$	0	0
$411$	−1.16513e17	−1.19235
$412$	0	0
$413$	−5.35350e16	−0.530839
$414$	0	0
$415$	−2.35638e16	−0.226430
$416$	0	0
$417$	−8.73212e16	−0.813271
$418$	0	0
$419$	−1.34978e16	−0.121863	−0.0609314	−	0.998142i	$-0.519407\pi$
$419$	−1.34978e16	−0.121863	−0.0609314	+	0.998142i	$0.519407\pi$
$420$	0	0
$421$	8.44210e16	0.738953	0.369477	−	0.929240i	$-0.379537\pi$
$421$	8.44210e16	0.738953	0.369477	+	0.929240i	$0.379537\pi$
$422$	0	0
$423$	−8.23632e15	−0.0699070
$424$	0	0
$425$	1.55537e17	1.28029
$426$	0	0
$427$	1.34995e17	1.07779
$428$	0	0
$429$	−2.76638e17	−2.14259
$430$	0	0
$431$	2.51699e17	1.89139	0.945693	−	0.325062i	$-0.105385\pi$
$431$	2.51699e17	1.89139	0.945693	+	0.325062i	$0.105385\pi$
$432$	0	0
$433$	−1.04387e17	−0.761162	−0.380581	−	0.924748i	$-0.624276\pi$
$433$	−1.04387e17	−0.761162	−0.380581	+	0.924748i	$0.624276\pi$
$434$	0	0
$435$	−7.27890e16	−0.515093
$436$	0	0
$437$	1.73243e16	0.118994
$438$	0	0
$439$	−1.50949e17	−1.00650	−0.503248	−	0.864142i	$-0.667862\pi$
$439$	−1.50949e17	−1.00650	−0.503248	+	0.864142i	$0.667862\pi$
$440$	0	0
$441$	2.53670e16	0.164217
$442$	0	0
$443$	1.21966e17	0.766681	0.383341	−	0.923607i	$-0.374774\pi$
$443$	1.21966e17	0.766681	0.383341	+	0.923607i	$0.374774\pi$
$444$	0	0
$445$	6.73032e16	0.410862
$446$	0	0
$447$	−9.71995e16	−0.576322
$448$	0	0
$449$	−2.40890e17	−1.38745	−0.693724	−	0.720241i	$-0.744031\pi$
$449$	−2.40890e17	−1.38745	−0.693724	+	0.720241i	$0.744031\pi$
$450$	0	0
$451$	−1.10080e17	−0.615974
$452$	0	0
$453$	1.43321e17	0.779240
$454$	0	0
$455$	1.55265e17	0.820349
$456$	0	0
$457$	1.12095e17	0.575615	0.287807	−	0.957688i	$-0.407074\pi$
$457$	1.12095e17	0.575615	0.287807	+	0.957688i	$0.407074\pi$
$458$	0	0
$459$	2.66923e17	1.33230
$460$	0	0
$461$	2.60788e17	1.26541	0.632706	−	0.774392i	$-0.281944\pi$
$461$	2.60788e17	1.26541	0.632706	+	0.774392i	$0.281944\pi$
$462$	0	0
$463$	−1.50149e17	−0.708348	−0.354174	−	0.935180i	$-0.615238\pi$
$463$	−1.50149e17	−0.708348	−0.354174	+	0.935180i	$0.615238\pi$
$464$	0	0
$465$	9.15386e16	0.419914
$466$	0	0
$467$	1.57516e17	0.702691	0.351346	−	0.936246i	$-0.385724\pi$
$467$	1.57516e17	0.702691	0.351346	+	0.936246i	$0.385724\pi$
$468$	0	0
$469$	3.36751e17	1.46112
$470$	0	0
$471$	−1.26704e17	−0.534754
$472$	0	0
$473$	−2.29569e17	−0.942574
$474$	0	0
$475$	1.16154e17	0.464009
$476$	0	0
$477$	−2.58422e16	−0.100452
$478$	0	0
$479$	−3.75827e17	−1.42170	−0.710848	−	0.703345i	$-0.751689\pi$
$479$	−3.75827e17	−1.42170	−0.710848	+	0.703345i	$0.751689\pi$
$480$	0	0
$481$	3.51254e17	1.29324
$482$	0	0
$483$	9.77951e16	0.350478
$484$	0	0
$485$	−1.65429e17	−0.577153
$486$	0	0
$487$	4.53837e17	1.54156	0.770782	−	0.637099i	$-0.219866\pi$
$487$	4.53837e17	1.54156	0.770782	+	0.637099i	$0.219866\pi$
$488$	0	0
$489$	2.65168e17	0.877029
$490$	0	0
$491$	−5.98005e17	−1.92608	−0.963042	−	0.269351i	$-0.913191\pi$
$491$	−5.98005e17	−1.92608	−0.963042	+	0.269351i	$0.913191\pi$
$492$	0	0
$493$	−6.39153e17	−2.00493
$494$	0	0
$495$	2.09456e16	0.0639970
$496$	0	0
$497$	−5.20849e17	−1.55023
$498$	0	0
$499$	4.80727e17	1.39394	0.696971	−	0.717099i	$-0.254530\pi$
$499$	4.80727e17	1.39394	0.696971	+	0.717099i	$0.254530\pi$
$500$	0	0
$501$	−6.63059e17	−1.87330
$502$	0	0
$503$	−5.47213e17	−1.50649	−0.753243	−	0.657743i	$-0.771511\pi$
$503$	−5.47213e17	−1.50649	−0.753243	+	0.657743i	$0.771511\pi$
$504$	0	0
$505$	1.02300e17	0.274462
$506$	0	0
$507$	6.55790e17	1.71480
$508$	0	0
$509$	−8.68173e16	−0.221279	−0.110640	−	0.993861i	$-0.535290\pi$
$509$	−8.68173e16	−0.221279	−0.110640	+	0.993861i	$0.535290\pi$
$510$	0	0
$511$	−8.36553e17	−2.07854
$512$	0	0
$513$	1.99336e17	0.482861
$514$	0	0
$515$	2.85984e16	0.0675452
$516$	0	0
$517$	2.55278e17	0.587930
$518$	0	0
$519$	−1.06877e17	−0.240048
$520$	0	0
$521$	1.21497e17	0.266147	0.133074	−	0.991106i	$-0.457515\pi$
$521$	1.21497e17	0.266147	0.133074	+	0.991106i	$0.457515\pi$
$522$	0	0
$523$	−4.79553e17	−1.02465	−0.512325	−	0.858792i	$-0.671216\pi$
$523$	−4.79553e17	−1.02465	−0.512325	+	0.858792i	$0.671216\pi$
$524$	0	0
$525$	6.55687e17	1.36666
$526$	0	0
$527$	8.03792e17	1.63446
$528$	0	0
$529$	−4.78508e17	−0.949351
$530$	0	0
$531$	2.78150e16	0.0538474
$532$	0	0
$533$	4.23966e17	0.800947
$534$	0	0
$535$	2.98268e17	0.549929
$536$	0	0
$537$	4.89122e17	0.880206
$538$	0	0
$539$	−7.86232e17	−1.38110
$540$	0	0
$541$	−5.38054e16	−0.0922664	−0.0461332	−	0.998935i	$-0.514690\pi$
$541$	−5.38054e16	−0.0922664	−0.0461332	+	0.998935i	$0.514690\pi$
$542$	0	0
$543$	−2.09753e17	−0.351164
$544$	0	0
$545$	−1.95041e15	−0.00318823
$546$	0	0
$547$	9.62528e17	1.53637	0.768185	−	0.640228i	$-0.221160\pi$
$547$	9.62528e17	1.53637	0.768185	+	0.640228i	$0.221160\pi$
$548$	0	0
$549$	−7.01389e16	−0.109330
$550$	0	0
$551$	−4.77315e17	−0.726640
$552$	0	0
$553$	−9.95928e17	−1.48086
$554$	0	0
$555$	−2.06936e17	−0.300560
$556$	0	0
$557$	3.40799e17	0.483548	0.241774	−	0.970333i	$-0.422271\pi$
$557$	3.40799e17	0.483548	0.241774	+	0.970333i	$0.422271\pi$
$558$	0	0
$559$	8.84165e17	1.22562
$560$	0	0
$561$	1.43109e18	1.93824
$562$	0	0
$563$	2.05601e15	0.00272095	0.00136048	−	0.999999i	$-0.499567\pi$
$563$	2.05601e15	0.00272095	0.00136048	+	0.999999i	$0.499567\pi$
$564$	0	0
$565$	3.55038e17	0.459155
$566$	0	0
$567$	1.29488e18	1.63658
$568$	0	0
$569$	−1.51190e18	−1.86764	−0.933821	−	0.357741i	$-0.883547\pi$
$569$	−1.51190e18	−1.86764	−0.933821	+	0.357741i	$0.883547\pi$
$570$	0	0
$571$	8.11054e17	0.979298	0.489649	−	0.871920i	$-0.337125\pi$
$571$	8.11054e17	0.979298	0.489649	+	0.871920i	$0.337125\pi$
$572$	0	0
$573$	−2.46881e17	−0.291395
$574$	0	0
$575$	1.71162e17	0.197500
$576$	0	0
$577$	1.32718e17	0.149723	0.0748614	−	0.997194i	$-0.476149\pi$
$577$	1.32718e17	0.149723	0.0748614	+	0.997194i	$0.476149\pi$
$578$	0	0
$579$	−1.73570e18	−1.91453
$580$	0	0
$581$	−8.72262e17	−0.940811
$582$	0	0
$583$	8.00960e17	0.844822
$584$	0	0
$585$	−8.06704e16	−0.0832149
$586$	0	0
$587$	−1.10453e18	−1.11437	−0.557187	−	0.830387i	$-0.688119\pi$
$587$	−1.10453e18	−1.11437	−0.557187	+	0.830387i	$0.688119\pi$
$588$	0	0
$589$	6.00266e17	0.592371
$590$	0	0
$591$	−4.11591e17	−0.397326
$592$	0	0
$593$	1.44365e18	1.36335	0.681675	−	0.731656i	$-0.261252\pi$
$593$	1.44365e18	1.36335	0.681675	+	0.731656i	$0.261252\pi$
$594$	0	0
$595$	−8.03207e17	−0.742108
$596$	0	0
$597$	−9.17627e16	−0.0829532
$598$	0	0
$599$	−1.84559e17	−0.163253	−0.0816266	−	0.996663i	$-0.526011\pi$
$599$	−1.84559e17	−0.163253	−0.0816266	+	0.996663i	$0.526011\pi$
$600$	0	0
$601$	5.19182e17	0.449403	0.224701	−	0.974428i	$-0.427859\pi$
$601$	5.19182e17	0.449403	0.224701	+	0.974428i	$0.427859\pi$
$602$	0	0
$603$	−1.74965e17	−0.148213
$604$	0	0
$605$	−2.27160e17	−0.188331
$606$	0	0
$607$	−1.80021e18	−1.46082	−0.730410	−	0.683009i	$-0.760671\pi$
$607$	−1.80021e18	−1.46082	−0.730410	+	0.683009i	$0.760671\pi$
$608$	0	0
$609$	−2.69443e18	−2.14020
$610$	0	0
$611$	−9.83184e17	−0.764481
$612$	0	0
$613$	−1.41838e18	−1.07969	−0.539847	−	0.841763i	$-0.681518\pi$
$613$	−1.41838e18	−1.07969	−0.539847	+	0.841763i	$0.681518\pi$
$614$	0	0
$615$	−2.49773e17	−0.186148
$616$	0	0
$617$	8.74021e17	0.637776	0.318888	−	0.947792i	$-0.396691\pi$
$617$	8.74021e17	0.637776	0.318888	+	0.947792i	$0.396691\pi$
$618$	0	0
$619$	8.56739e16	0.0612152	0.0306076	−	0.999531i	$-0.490256\pi$
$619$	8.56739e16	0.0612152	0.0306076	+	0.999531i	$0.490256\pi$
$620$	0	0
$621$	2.93737e17	0.205524
$622$	0	0
$623$	2.49136e18	1.70712
$624$	0	0
$625$	9.65162e17	0.647709
$626$	0	0
$627$	1.06873e18	0.702469
$628$	0	0
$629$	−1.81709e18	−1.16989
$630$	0	0
$631$	2.34850e17	0.148115	0.0740575	−	0.997254i	$-0.476405\pi$
$631$	2.34850e17	0.148115	0.0740575	+	0.997254i	$0.476405\pi$
$632$	0	0
$633$	−1.16831e18	−0.721826
$634$	0	0
$635$	6.66270e17	0.403294
$636$	0	0
$637$	3.02811e18	1.79583
$638$	0	0
$639$	2.70616e17	0.157253
$640$	0	0
$641$	2.86283e18	1.63011	0.815057	−	0.579380i	$-0.196705\pi$
$641$	2.86283e18	1.63011	0.815057	+	0.579380i	$0.196705\pi$
$642$	0	0
$643$	−4.45824e17	−0.248766	−0.124383	−	0.992234i	$-0.539695\pi$
$643$	−4.45824e17	−0.248766	−0.124383	+	0.992234i	$0.539695\pi$
$644$	0	0
$645$	−5.20893e17	−0.284846
$646$	0	0
$647$	2.70439e18	1.44941	0.724705	−	0.689059i	$-0.241976\pi$
$647$	2.70439e18	1.44941	0.724705	+	0.689059i	$0.241976\pi$
$648$	0	0
$649$	−8.62105e17	−0.452866
$650$	0	0
$651$	3.38848e18	1.74473
$652$	0	0
$653$	−2.96211e17	−0.149508	−0.0747541	−	0.997202i	$-0.523817\pi$
$653$	−2.96211e17	−0.149508	−0.0747541	+	0.997202i	$0.523817\pi$
$654$	0	0
$655$	−2.29475e16	−0.0113544
$656$	0	0
$657$	4.34645e17	0.210843
$658$	0	0
$659$	−1.93623e18	−0.920875	−0.460438	−	0.887692i	$-0.652308\pi$
$659$	−1.93623e18	−0.920875	−0.460438	+	0.887692i	$0.652308\pi$
$660$	0	0
$661$	−2.14190e18	−0.998825	−0.499412	−	0.866364i	$-0.666451\pi$
$661$	−2.14190e18	−0.998825	−0.499412	+	0.866364i	$0.666451\pi$
$662$	0	0
$663$	−5.51172e18	−2.52028
$664$	0	0
$665$	−5.99828e17	−0.268959
$666$	0	0
$667$	−7.03361e17	−0.309286
$668$	0	0
$669$	3.01540e18	1.30040
$670$	0	0
$671$	2.17390e18	0.919482
$672$	0	0
$673$	3.09007e18	1.28195	0.640974	−	0.767563i	$-0.278531\pi$
$673$	3.09007e18	1.28195	0.640974	+	0.767563i	$0.278531\pi$
$674$	0	0
$675$	1.96942e18	0.801426
$676$	0	0
$677$	4.35504e18	1.73847	0.869233	−	0.494403i	$-0.164613\pi$
$677$	4.35504e18	1.73847	0.869233	+	0.494403i	$0.164613\pi$
$678$	0	0
$679$	−6.12369e18	−2.39806
$680$	0	0
$681$	−4.65363e18	−1.78787
$682$	0	0
$683$	1.73059e18	0.652319	0.326159	−	0.945315i	$-0.394245\pi$
$683$	1.73059e18	0.652319	0.326159	+	0.945315i	$0.394245\pi$
$684$	0	0
$685$	1.05307e18	0.389466
$686$	0	0
$687$	9.60284e17	0.348483
$688$	0	0
$689$	−3.08483e18	−1.09852
$690$	0	0
$691$	3.55737e17	0.124315	0.0621573	−	0.998066i	$-0.480202\pi$
$691$	3.55737e17	0.124315	0.0621573	+	0.998066i	$0.480202\pi$
$692$	0	0
$693$	7.75345e17	0.265906
$694$	0	0
$695$	7.89229e17	0.265645
$696$	0	0
$697$	−2.19324e18	−0.724556
$698$	0	0
$699$	−1.11918e18	−0.362908
$700$	0	0
$701$	−2.63200e18	−0.837759	−0.418880	−	0.908042i	$-0.637577\pi$
$701$	−2.63200e18	−0.837759	−0.418880	+	0.908042i	$0.637577\pi$
$702$	0	0
$703$	−1.35699e18	−0.424000
$704$	0	0
$705$	5.79228e17	0.177673
$706$	0	0
$707$	3.78685e18	1.14039
$708$	0	0
$709$	−6.27661e18	−1.85577	−0.927886	−	0.372863i	$-0.878376\pi$
$709$	−6.27661e18	−1.85577	−0.927886	+	0.372863i	$0.878376\pi$
$710$	0	0
$711$	5.17451e17	0.150216
$712$	0	0
$713$	8.84539e17	0.252136
$714$	0	0
$715$	2.50032e18	0.699852
$716$	0	0
$717$	1.53223e18	0.421161
$718$	0	0
$719$	−5.58024e18	−1.50631	−0.753156	−	0.657842i	$-0.771469\pi$
$719$	−5.58024e18	−1.50631	−0.753156	+	0.657842i	$0.771469\pi$
$720$	0	0
$721$	1.05863e18	0.280649
$722$	0	0
$723$	3.80274e16	0.00990142
$724$	0	0
$725$	−4.71582e18	−1.20604
$726$	0	0
$727$	−2.15542e18	−0.541449	−0.270725	−	0.962657i	$-0.587263\pi$
$727$	−2.15542e18	−0.541449	−0.270725	+	0.962657i	$0.587263\pi$
$728$	0	0
$729$	3.29165e18	0.812240
$730$	0	0
$731$	−4.57391e18	−1.10873
$732$	0	0
$733$	−3.43698e18	−0.818467	−0.409234	−	0.912430i	$-0.634204\pi$
$733$	−3.43698e18	−0.818467	−0.409234	+	0.912430i	$0.634204\pi$
$734$	0	0
$735$	−1.78396e18	−0.417367
$736$	0	0
$737$	5.42289e18	1.24650
$738$	0	0
$739$	2.34366e18	0.529305	0.264653	−	0.964344i	$-0.414743\pi$
$739$	2.34366e18	0.529305	0.264653	+	0.964344i	$0.414743\pi$
$740$	0	0
$741$	−4.11611e18	−0.913416
$742$	0	0
$743$	6.88793e18	1.50197	0.750985	−	0.660319i	$-0.229579\pi$
$743$	6.88793e18	1.50197	0.750985	+	0.660319i	$0.229579\pi$
$744$	0	0
$745$	8.78511e17	0.188248
$746$	0	0
$747$	4.53199e17	0.0954343
$748$	0	0
$749$	1.10410e19	2.28494
$750$	0	0
$751$	1.85684e17	0.0377671	0.0188836	−	0.999822i	$-0.493989\pi$
$751$	1.85684e17	0.0377671	0.0188836	+	0.999822i	$0.493989\pi$
$752$	0	0
$753$	7.37878e17	0.147508
$754$	0	0
$755$	−1.29537e18	−0.254529
$756$	0	0
$757$	8.78485e18	1.69672	0.848362	−	0.529417i	$-0.177589\pi$
$757$	8.78485e18	1.69672	0.848362	+	0.529417i	$0.177589\pi$
$758$	0	0
$759$	1.57485e18	0.298998
$760$	0	0
$761$	1.88495e17	0.0351803	0.0175901	−	0.999845i	$-0.494401\pi$
$761$	1.88495e17	0.0351803	0.0175901	+	0.999845i	$0.494401\pi$
$762$	0	0
$763$	−7.21984e16	−0.0132470
$764$	0	0
$765$	4.17320e17	0.0752782
$766$	0	0
$767$	3.32033e18	0.588859
$768$	0	0
$769$	5.16573e17	0.0900763	0.0450381	−	0.998985i	$-0.485659\pi$
$769$	5.16573e17	0.0900763	0.0450381	+	0.998985i	$0.485659\pi$
$770$	0	0
$771$	5.97316e17	0.102412
$772$	0	0
$773$	2.41255e18	0.406733	0.203366	−	0.979103i	$-0.434812\pi$
$773$	2.41255e18	0.406733	0.203366	+	0.979103i	$0.434812\pi$
$774$	0	0
$775$	5.93057e18	0.983185
$776$	0	0
$777$	−7.66015e18	−1.24882
$778$	0	0
$779$	−1.63789e18	−0.262598
$780$	0	0
$781$	−8.38753e18	−1.32252
$782$	0	0
$783$	−8.09297e18	−1.25504
$784$	0	0
$785$	1.14518e18	0.174671
$786$	0	0
$787$	1.24281e18	0.186453	0.0932265	−	0.995645i	$-0.470282\pi$
$787$	1.24281e18	0.186453	0.0932265	+	0.995645i	$0.470282\pi$
$788$	0	0
$789$	−2.39944e18	−0.354087
$790$	0	0
$791$	1.31424e19	1.90778
$792$	0	0
$793$	−8.37260e18	−1.19560
$794$	0	0
$795$	1.81738e18	0.255305
$796$	0	0
$797$	−5.76552e17	−0.0796818	−0.0398409	−	0.999206i	$-0.512685\pi$
$797$	−5.76552e17	−0.0796818	−0.0398409	+	0.999206i	$0.512685\pi$
$798$	0	0
$799$	5.08615e18	0.691568
$800$	0	0
$801$	−1.29443e18	−0.173168
$802$	0	0
$803$	−1.34715e19	−1.77323
$804$	0	0
$805$	−8.83895e17	−0.114479
$806$	0	0
$807$	−1.36787e19	−1.74328
$808$	0	0
$809$	4.34559e18	0.544983	0.272492	−	0.962158i	$-0.412152\pi$
$809$	4.34559e18	0.544983	0.272492	+	0.962158i	$0.412152\pi$
$810$	0	0
$811$	−1.69949e18	−0.209741	−0.104871	−	0.994486i	$-0.533443\pi$
$811$	−1.69949e18	−0.209741	−0.104871	+	0.994486i	$0.533443\pi$
$812$	0	0
$813$	4.14938e18	0.503958
$814$	0	0
$815$	−2.39665e18	−0.286471
$816$	0	0
$817$	−3.41576e18	−0.401832
$818$	0	0
$819$	−2.98618e18	−0.345756
$820$	0	0
$821$	−3.06171e17	−0.0348926	−0.0174463	−	0.999848i	$-0.505554\pi$
$821$	−3.06171e17	−0.0348926	−0.0174463	+	0.999848i	$0.505554\pi$
$822$	0	0
$823$	4.19904e18	0.471033	0.235516	−	0.971870i	$-0.424322\pi$
$823$	4.19904e18	0.471033	0.235516	+	0.971870i	$0.424322\pi$
$824$	0	0
$825$	1.05589e19	1.16592
$826$	0	0
$827$	−1.76597e19	−1.91954	−0.959772	−	0.280780i	$-0.909407\pi$
$827$	−1.76597e19	−1.91954	−0.959772	+	0.280780i	$0.909407\pi$
$828$	0	0
$829$	−1.95676e18	−0.209379	−0.104689	−	0.994505i	$-0.533385\pi$
$829$	−1.95676e18	−0.209379	−0.104689	+	0.994505i	$0.533385\pi$
$830$	0	0
$831$	1.02185e19	1.07641
$832$	0	0
$833$	−1.56648e19	−1.62455
$834$	0	0
$835$	5.99288e18	0.611890
$836$	0	0
$837$	1.01776e19	1.02313
$838$	0	0
$839$	8.20221e18	0.811855	0.405927	−	0.913905i	$-0.366949\pi$
$839$	8.20221e18	0.811855	0.405927	+	0.913905i	$0.366949\pi$
$840$	0	0
$841$	9.11822e18	0.888660
$842$	0	0
$843$	−8.14592e18	−0.781737
$844$	0	0
$845$	−5.92717e18	−0.560117
$846$	0	0
$847$	−8.40879e18	−0.782513
$848$	0	0
$849$	1.07853e18	0.0988396
$850$	0	0
$851$	−1.99963e18	−0.180471
$852$	0	0
$853$	6.81541e18	0.605792	0.302896	−	0.953024i	$-0.402047\pi$
$853$	6.81541e18	0.605792	0.302896	+	0.953024i	$0.402047\pi$
$854$	0	0
$855$	3.11651e17	0.0272828
$856$	0	0
$857$	−4.95407e18	−0.427156	−0.213578	−	0.976926i	$-0.568512\pi$
$857$	−4.95407e18	−0.427156	−0.213578	+	0.976926i	$0.568512\pi$
$858$	0	0
$859$	1.25946e19	1.06962	0.534809	−	0.844973i	$-0.320384\pi$
$859$	1.25946e19	1.06962	0.534809	+	0.844973i	$0.320384\pi$
$860$	0	0
$861$	−9.24585e18	−0.773440
$862$	0	0
$863$	9.02789e18	0.743903	0.371951	−	0.928252i	$-0.378689\pi$
$863$	9.02789e18	0.743903	0.371951	+	0.928252i	$0.378689\pi$
$864$	0	0
$865$	9.65982e17	0.0784087
$866$	0	0
$867$	1.51163e19	1.20871
$868$	0	0
$869$	−1.60380e19	−1.26334
$870$	0	0
$871$	−2.08858e19	−1.62082
$872$	0	0
$873$	3.18167e18	0.243255
$874$	0	0
$875$	−1.26792e19	−0.955082
$876$	0	0
$877$	4.25224e18	0.315588	0.157794	−	0.987472i	$-0.449562\pi$
$877$	4.25224e18	0.315588	0.157794	+	0.987472i	$0.449562\pi$
$878$	0	0
$879$	2.31461e19	1.69258
$880$	0	0
$881$	1.10187e19	0.793937	0.396968	−	0.917832i	$-0.370062\pi$
$881$	1.10187e19	0.793937	0.396968	+	0.917832i	$0.370062\pi$
$882$	0	0
$883$	1.99194e19	1.41427	0.707135	−	0.707079i	$-0.249987\pi$
$883$	1.99194e19	1.41427	0.707135	+	0.707079i	$0.249987\pi$
$884$	0	0
$885$	−1.95612e18	−0.136856
$886$	0	0
$887$	2.18455e19	1.50612	0.753058	−	0.657954i	$-0.228578\pi$
$887$	2.18455e19	1.50612	0.753058	+	0.657954i	$0.228578\pi$
$888$	0	0
$889$	2.46633e19	1.67568
$890$	0	0
$891$	2.08521e19	1.39619
$892$	0	0
$893$	3.79830e18	0.250642
$894$	0	0
$895$	−4.42080e18	−0.287509
$896$	0	0
$897$	−6.06541e18	−0.388785
$898$	0	0
$899$	−2.43706e19	−1.53967
$900$	0	0
$901$	1.59583e19	0.993744
$902$	0	0
$903$	−1.92819e19	−1.18353
$904$	0	0
$905$	1.89580e18	0.114703
$906$	0	0
$907$	−1.13902e19	−0.679336	−0.339668	−	0.940545i	$-0.610315\pi$
$907$	−1.13902e19	−0.679336	−0.339668	+	0.940545i	$0.610315\pi$
$908$	0	0
$909$	−1.96752e18	−0.115679
$910$	0	0
$911$	−3.34308e19	−1.93766	−0.968829	−	0.247732i	$-0.920315\pi$
$911$	−3.34308e19	−1.93766	−0.968829	+	0.247732i	$0.920315\pi$
$912$	0	0
$913$	−1.40465e19	−0.802619
$914$	0	0
$915$	4.93259e18	0.277868
$916$	0	0
$917$	−8.49446e17	−0.0471775
$918$	0	0
$919$	−2.89154e19	−1.58336	−0.791678	−	0.610939i	$-0.790792\pi$
$919$	−2.89154e19	−1.58336	−0.791678	+	0.610939i	$0.790792\pi$
$920$	0	0
$921$	1.46361e19	0.790199
$922$	0	0
$923$	3.23039e19	1.71966
$924$	0	0
$925$	−1.34069e19	−0.703731
$926$	0	0
$927$	−5.50028e17	−0.0284686
$928$	0	0
$929$	−2.80750e19	−1.43291	−0.716453	−	0.697635i	$-0.754236\pi$
$929$	−2.80750e19	−1.43291	−0.716453	+	0.697635i	$0.754236\pi$
$930$	0	0
$931$	−1.16984e19	−0.588780
$932$	0	0
$933$	1.49008e19	0.739569
$934$	0	0
$935$	−1.29345e19	−0.633103
$936$	0	0
$937$	−2.72300e19	−1.31444	−0.657220	−	0.753699i	$-0.728268\pi$
$937$	−2.72300e19	−1.31444	−0.657220	+	0.753699i	$0.728268\pi$
$938$	0	0
$939$	3.58012e19	1.70440
$940$	0	0
$941$	−2.67460e19	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$941$	−2.67460e19	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$942$	0	0
$943$	−2.41356e18	−0.111772
$944$	0	0
$945$	−1.01702e19	−0.464540
$946$	0	0
$947$	−3.01628e19	−1.35893	−0.679464	−	0.733709i	$-0.737788\pi$
$947$	−3.01628e19	−1.35893	−0.679464	+	0.733709i	$0.737788\pi$
$948$	0	0
$949$	5.18844e19	2.30572
$950$	0	0
$951$	−5.25496e18	−0.230354
$952$	0	0
$953$	−1.66980e19	−0.722037	−0.361019	−	0.932559i	$-0.617571\pi$
$953$	−1.66980e19	−0.722037	−0.361019	+	0.932559i	$0.617571\pi$
$954$	0	0
$955$	2.23137e18	0.0951807
$956$	0	0
$957$	−4.33900e19	−1.82584
$958$	0	0
$959$	3.89815e19	1.61822
$960$	0	0
$961$	6.23068e18	0.255172
$962$	0	0
$963$	−5.73653e18	−0.231781
$964$	0	0
$965$	1.56876e19	0.625359
$966$	0	0
$967$	2.79699e19	1.10007	0.550033	−	0.835143i	$-0.314615\pi$
$967$	2.79699e19	1.10007	0.550033	+	0.835143i	$0.314615\pi$
$968$	0	0
$969$	2.12932e19	0.826298
$970$	0	0
$971$	5.42017e18	0.207534	0.103767	−	0.994602i	$-0.466910\pi$
$971$	5.42017e18	0.207534	0.103767	+	0.994602i	$0.466910\pi$
$972$	0	0
$973$	2.92149e19	1.10375
$974$	0	0
$975$	−4.06668e19	−1.51604
$976$	0	0
$977$	−2.85933e19	−1.05184	−0.525921	−	0.850534i	$-0.676279\pi$
$977$	−2.85933e19	−1.05184	−0.525921	+	0.850534i	$0.676279\pi$
$978$	0	0
$979$	4.01199e19	1.45637
$980$	0	0
$981$	3.75119e16	0.00134376
$982$	0	0
$983$	−7.34661e18	−0.259710	−0.129855	−	0.991533i	$-0.541451\pi$
$983$	−7.34661e18	−0.259710	−0.129855	+	0.991533i	$0.541451\pi$
$984$	0	0
$985$	3.72005e18	0.129782
$986$	0	0
$987$	2.14413e19	0.738226
$988$	0	0
$989$	−5.03340e18	−0.171035
$990$	0	0
$991$	−1.78440e19	−0.598431	−0.299215	−	0.954186i	$-0.596725\pi$
$991$	−1.78440e19	−0.598431	−0.299215	+	0.954186i	$0.596725\pi$
$992$	0	0
$993$	4.99989e19	1.65497
$994$	0	0
$995$	8.29372e17	0.0270956
$996$	0	0
$997$	−5.53089e19	−1.78351	−0.891757	−	0.452514i	$-0.850527\pi$
$997$	−5.53089e19	−1.78351	−0.891757	+	0.452514i	$0.850527\pi$
$998$	0	0
$999$	−2.30080e19	−0.732323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.14.a.e.1.2		2
3.2	odd	2		144.14.a.n.1.2		2
4.3	odd	2		8.14.a.b.1.1	✓	2
8.3	odd	2		64.14.a.j.1.2		2
8.5	even	2		64.14.a.l.1.1		2
12.11	even	2		72.14.a.c.1.2		2
20.3	even	4		200.14.c.b.49.2		4
20.7	even	4		200.14.c.b.49.3		4
20.19	odd	2		200.14.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.14.a.b.1.1	✓	2	4.3	odd	2
16.14.a.e.1.2		2	1.1	even	1	trivial
64.14.a.j.1.2		2	8.3	odd	2
64.14.a.l.1.1		2	8.5	even	2
72.14.a.c.1.2		2	12.11	even	2
144.14.a.n.1.2		2	3.2	odd	2
200.14.a.b.1.2		2	20.19	odd	2
200.14.c.b.49.2		4	20.3	even	4
200.14.c.b.49.3		4	20.7	even	4

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:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+1352.57 q^{3} -12224.8 q^{5} -452526. q^{7} +235118. q^{9} +O(q^{10})\)	\(q+1352.57 q^{3} -12224.8 q^{5} -452526. q^{7} +235118. q^{9} -7.28729e6 q^{11} +2.80664e7 q^{13} -1.65349e7 q^{15} -1.45191e8 q^{17} -1.08428e8 q^{19} -6.12072e8 q^{21} -1.59777e8 q^{23} -1.07126e9 q^{25} -1.83842e9 q^{27} +4.40214e9 q^{29} -5.53608e9 q^{31} -9.85656e9 q^{33} +5.53205e9 q^{35} +1.25151e10 q^{37} +3.79617e10 q^{39} +1.51058e10 q^{41} +3.15026e10 q^{43} -2.87427e9 q^{45} -3.50306e10 q^{47} +1.07891e11 q^{49} -1.96381e11 q^{51} -1.09912e11 q^{53} +8.90858e10 q^{55} -1.46656e11 q^{57} +1.18303e11 q^{59} -2.98314e11 q^{61} -1.06397e11 q^{63} -3.43107e11 q^{65} -7.44158e11 q^{67} -2.16109e11 q^{69} +1.15098e12 q^{71} +1.84863e12 q^{73} -1.44895e12 q^{75} +3.29769e12 q^{77} +2.20082e12 q^{79} -2.86144e12 q^{81} +1.92754e12 q^{83} +1.77494e12 q^{85} +5.95420e12 q^{87} -5.50546e12 q^{89} -1.27008e13 q^{91} -7.48793e12 q^{93} +1.32551e12 q^{95} +1.35322e13 q^{97} -1.71337e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 872 q^{3} + 18476 q^{5} - 110928 q^{7} + 3589498 q^{9}+O(q^{10}) \) 2 * q - 872 * q^3 + 18476 * q^5 - 110928 * q^7 + 3589498 * q^9	\( 2 q - 872 q^{3} + 18476 q^{5} - 110928 q^{7} + 3589498 q^{9} - 16474040 q^{11} + 18744572 q^{13} - 84830960 q^{15} - 153793628 q^{17} + 118747640 q^{19} - 1371980736 q^{21} - 718268912 q^{23} - 1349419874 q^{25} - 5753785616 q^{27} + 309341340 q^{29} - 5767504192 q^{31} + 10579993952 q^{33} + 16019391264 q^{35} - 11621553300 q^{37} + 58698760912 q^{39} + 1311168276 q^{41} + 29595620104 q^{43} + 100107952252 q^{45} - 12313617888 q^{47} + 127691156146 q^{49} - 177245377360 q^{51} - 38006007028 q^{53} - 192954931664 q^{55} - 652023429728 q^{57} - 253345911704 q^{59} - 647244384292 q^{61} + 1039453222896 q^{63} - 629294375512 q^{65} - 1619993806312 q^{67} + 1026293771456 q^{69} + 1040270142512 q^{71} + 4005283908692 q^{73} - 830155668760 q^{75} + 159512776896 q^{77} + 2521777572064 q^{79} + 500597403058 q^{81} + 290486230904 q^{83} + 1510847254552 q^{85} + 15058906809936 q^{87} - 8723755657740 q^{89} - 15885098476896 q^{91} - 6973121519360 q^{93} + 8299984202576 q^{95} + 9601712299972 q^{97} - 32529223326424 q^{99}+O(q^{100}) \) 2 * q - 872 * q^3 + 18476 * q^5 - 110928 * q^7 + 3589498 * q^9 - 16474040 * q^11 + 18744572 * q^13 - 84830960 * q^15 - 153793628 * q^17 + 118747640 * q^19 - 1371980736 * q^21 - 718268912 * q^23 - 1349419874 * q^25 - 5753785616 * q^27 + 309341340 * q^29 - 5767504192 * q^31 + 10579993952 * q^33 + 16019391264 * q^35 - 11621553300 * q^37 + 58698760912 * q^39 + 1311168276 * q^41 + 29595620104 * q^43 + 100107952252 * q^45 - 12313617888 * q^47 + 127691156146 * q^49 - 177245377360 * q^51 - 38006007028 * q^53 - 192954931664 * q^55 - 652023429728 * q^57 - 253345911704 * q^59 - 647244384292 * q^61 + 1039453222896 * q^63 - 629294375512 * q^65 - 1619993806312 * q^67 + 1026293771456 * q^69 + 1040270142512 * q^71 + 4005283908692 * q^73 - 830155668760 * q^75 + 159512776896 * q^77 + 2521777572064 * q^79 + 500597403058 * q^81 + 290486230904 * q^83 + 1510847254552 * q^85 + 15058906809936 * q^87 - 8723755657740 * q^89 - 15885098476896 * q^91 - 6973121519360 * q^93 + 8299984202576 * q^95 + 9601712299972 * q^97 - 32529223326424 * q^99

Introduction

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