LMFDB - Embedded newform 8.14.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,14,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, 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("8.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	8.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:	$8.57847431615$
Analytic rank:	$0$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-13.4732$ of defining polynomial
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2224.57 q^{3} +30700.8 q^{5} -341598. q^{7} +3.35438e6 q^{9} +O(q^{10})$	$q+2224.57 q^{3} +30700.8 q^{5} -341598. q^{7} +3.35438e6 q^{9} +9.18675e6 q^{11} -9.32183e6 q^{13} +6.82961e7 q^{15} -8.60213e6 q^{17} -2.27175e8 q^{19} -7.59908e8 q^{21} +5.58492e8 q^{23} -2.78163e8 q^{25} +3.91537e9 q^{27} -4.09280e9 q^{29} +2.31420e8 q^{31} +2.04366e10 q^{33} -1.04873e10 q^{35} -2.41367e10 q^{37} -2.07370e10 q^{39} -1.37946e10 q^{41} +1.90701e9 q^{43} +1.02982e11 q^{45} -2.27170e10 q^{47} +1.98003e10 q^{49} -1.91360e10 q^{51} +7.19059e10 q^{53} +2.82041e11 q^{55} -5.05367e11 q^{57} +3.71648e11 q^{59} -3.48930e11 q^{61} -1.14585e12 q^{63} -2.86188e11 q^{65} +8.75836e11 q^{67} +1.24240e12 q^{69} +1.10711e11 q^{71} +2.15665e12 q^{73} -6.18792e11 q^{75} -3.13818e12 q^{77} -3.20959e11 q^{79} +3.36204e12 q^{81} +1.63705e12 q^{83} -2.64092e11 q^{85} -9.10471e12 q^{87} -3.21830e12 q^{89} +3.18432e12 q^{91} +5.14810e11 q^{93} -6.97447e12 q^{95} -3.93052e12 q^{97} +3.08159e13 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2224.57	1.76180	0.880902	−	0.473299i	$-0.156937\pi$
$3$	2224.57	1.76180	0.880902	+	0.473299i	$0.156937\pi$
$4$	0	0
$5$	30700.8	0.878709	0.439354	−	0.898314i	$-0.355207\pi$
$5$	30700.8	0.878709	0.439354	+	0.898314i	$0.355207\pi$
$6$	0	0
$7$	−341598.	−1.09743	−0.548717	−	0.836008i	$-0.684883\pi$
$7$	−341598.	−1.09743	−0.548717	+	0.836008i	$0.684883\pi$
$8$	0	0
$9$	3.35438e6	2.10395
$10$	0	0
$11$	9.18675e6	1.56354	0.781771	−	0.623566i	$-0.214317\pi$
$11$	9.18675e6	1.56354	0.781771	+	0.623566i	$0.214317\pi$
$12$	0	0
$13$	−9.32183e6	−0.535635	−0.267818	−	0.963470i	$-0.586302\pi$
$13$	−9.32183e6	−0.535635	−0.267818	+	0.963470i	$0.586302\pi$
$14$	0	0
$15$	6.82961e7	1.54811
$16$	0	0
$17$	−8.60213e6	−0.0864347	−0.0432173	−	0.999066i	$-0.513761\pi$
$17$	−8.60213e6	−0.0864347	−0.0432173	+	0.999066i	$0.513761\pi$
$18$	0	0
$19$	−2.27175e8	−1.10780	−0.553902	−	0.832582i	$-0.686862\pi$
$19$	−2.27175e8	−1.10780	−0.553902	+	0.832582i	$0.686862\pi$
$20$	0	0
$21$	−7.59908e8	−1.93346
$22$	0	0
$23$	5.58492e8	0.786658	0.393329	−	0.919398i	$-0.371323\pi$
$23$	5.58492e8	0.786658	0.393329	+	0.919398i	$0.371323\pi$
$24$	0	0
$25$	−2.78163e8	−0.227871
$26$	0	0
$27$	3.91537e9	1.94495
$28$	0	0
$29$	−4.09280e9	−1.27771	−0.638857	−	0.769326i	$-0.720592\pi$
$29$	−4.09280e9	−1.27771	−0.638857	+	0.769326i	$0.720592\pi$
$30$	0	0
$31$	2.31420e8	0.0468328	0.0234164	−	0.999726i	$-0.492546\pi$
$31$	2.31420e8	0.0468328	0.0234164	+	0.999726i	$0.492546\pi$
$32$	0	0
$33$	2.04366e10	2.75465
$34$	0	0
$35$	−1.04873e10	−0.964324
$36$	0	0
$37$	−2.41367e10	−1.54656	−0.773278	−	0.634067i	$-0.781384\pi$
$37$	−2.41367e10	−1.54656	−0.773278	+	0.634067i	$0.781384\pi$
$38$	0	0
$39$	−2.07370e10	−0.943685
$40$	0	0
$41$	−1.37946e10	−0.453540	−0.226770	−	0.973948i	$-0.572817\pi$
$41$	−1.37946e10	−0.453540	−0.226770	+	0.973948i	$0.572817\pi$
$42$	0	0
$43$	1.90701e9	0.0460052	0.0230026	−	0.999735i	$-0.492677\pi$
$43$	1.90701e9	0.0460052	0.0230026	+	0.999735i	$0.492677\pi$
$44$	0	0
$45$	1.02982e11	1.84876
$46$	0	0
$47$	−2.27170e10	−0.307408	−0.153704	−	0.988117i	$-0.549120\pi$
$47$	−2.27170e10	−0.307408	−0.153704	+	0.988117i	$0.549120\pi$
$48$	0	0
$49$	1.98003e10	0.204360
$50$	0	0
$51$	−1.91360e10	−0.152281
$52$	0	0
$53$	7.19059e10	0.445627	0.222813	−	0.974861i	$-0.428476\pi$
$53$	7.19059e10	0.445627	0.222813	+	0.974861i	$0.428476\pi$
$54$	0	0
$55$	2.82041e11	1.37390
$56$	0	0
$57$	−5.05367e11	−1.95173
$58$	0	0
$59$	3.71648e11	1.14708	0.573541	−	0.819177i	$-0.305569\pi$
$59$	3.71648e11	1.14708	0.573541	+	0.819177i	$0.305569\pi$
$60$	0	0
$61$	−3.48930e11	−0.867151	−0.433576	−	0.901117i	$-0.642748\pi$
$61$	−3.48930e11	−0.867151	−0.433576	+	0.901117i	$0.642748\pi$
$62$	0	0
$63$	−1.14585e12	−2.30895
$64$	0	0
$65$	−2.86188e11	−0.470668
$66$	0	0
$67$	8.75836e11	1.18287	0.591434	−	0.806353i	$-0.298562\pi$
$67$	8.75836e11	1.18287	0.591434	+	0.806353i	$0.298562\pi$
$68$	0	0
$69$	1.24240e12	1.38594
$70$	0	0
$71$	1.10711e11	0.102568	0.0512838	−	0.998684i	$-0.483669\pi$
$71$	1.10711e11	0.102568	0.0512838	+	0.998684i	$0.483669\pi$
$72$	0	0
$73$	2.15665e12	1.66795	0.833973	−	0.551805i	$-0.186061\pi$
$73$	2.15665e12	1.66795	0.833973	+	0.551805i	$0.186061\pi$
$74$	0	0
$75$	−6.18792e11	−0.401464
$76$	0	0
$77$	−3.13818e12	−1.71588
$78$	0	0
$79$	−3.20959e11	−0.148550	−0.0742751	−	0.997238i	$-0.523664\pi$
$79$	−3.20959e11	−0.148550	−0.0742751	+	0.997238i	$0.523664\pi$
$80$	0	0
$81$	3.36204e12	1.32266
$82$	0	0
$83$	1.63705e12	0.549611	0.274805	−	0.961500i	$-0.411387\pi$
$83$	1.63705e12	0.549611	0.274805	+	0.961500i	$0.411387\pi$
$84$	0	0
$85$	−2.64092e11	−0.0759509
$86$	0	0
$87$	−9.10471e12	−2.25108
$88$	0	0
$89$	−3.21830e12	−0.686422	−0.343211	−	0.939258i	$-0.611515\pi$
$89$	−3.21830e12	−0.686422	−0.343211	+	0.939258i	$0.611515\pi$
$90$	0	0
$91$	3.18432e12	0.587824
$92$	0	0
$93$	5.14810e11	0.0825102
$94$	0	0
$95$	−6.97447e12	−0.973437
$96$	0	0
$97$	−3.93052e12	−0.479109	−0.239554	−	0.970883i	$-0.577001\pi$
$97$	−3.93052e12	−0.479109	−0.239554	+	0.970883i	$0.577001\pi$
$98$	0	0
$99$	3.08159e13	3.28962
$100$	0	0
$101$	2.16615e12	0.203049	0.101524	−	0.994833i	$-0.467628\pi$
$101$	2.16615e12	0.203049	0.101524	+	0.994833i	$0.467628\pi$
$102$	0	0
$103$	2.09777e13	1.73107	0.865537	−	0.500846i	$-0.166978\pi$
$103$	2.09777e13	1.73107	0.865537	+	0.500846i	$0.166978\pi$
$104$	0	0
$105$	−2.33298e13	−1.69895
$106$	0	0
$107$	−1.62952e13	−1.04970	−0.524850	−	0.851194i	$-0.675879\pi$
$107$	−1.62952e13	−1.04970	−0.524850	+	0.851194i	$0.675879\pi$
$108$	0	0
$109$	3.34141e13	1.90835	0.954174	−	0.299252i	$-0.0967371\pi$
$109$	3.34141e13	1.90835	0.954174	+	0.299252i	$0.0967371\pi$
$110$	0	0
$111$	−5.36936e13	−2.72473
$112$	0	0
$113$	3.92919e12	0.177539	0.0887693	−	0.996052i	$-0.471707\pi$
$113$	3.92919e12	0.177539	0.0887693	+	0.996052i	$0.471707\pi$
$114$	0	0
$115$	1.71462e13	0.691243
$116$	0	0
$117$	−3.12690e13	−1.12695
$118$	0	0
$119$	2.93847e12	0.0948563
$120$	0	0
$121$	4.98737e13	1.44466
$122$	0	0
$123$	−3.06871e13	−0.799048
$124$	0	0
$125$	−4.60164e13	−1.07894
$126$	0	0
$127$	−3.18567e13	−0.673716	−0.336858	−	0.941555i	$-0.609364\pi$
$127$	−3.18567e13	−0.673716	−0.336858	+	0.941555i	$0.609364\pi$
$128$	0	0
$129$	4.24227e12	0.0810522
$130$	0	0
$131$	5.05827e13	0.874458	0.437229	−	0.899350i	$-0.355960\pi$
$131$	5.05827e13	0.874458	0.437229	+	0.899350i	$0.355960\pi$
$132$	0	0
$133$	7.76027e13	1.21574
$134$	0	0
$135$	1.20205e14	1.70904
$136$	0	0
$137$	8.90097e13	1.15015	0.575074	−	0.818101i	$-0.304973\pi$
$137$	8.90097e13	1.15015	0.575074	+	0.818101i	$0.304973\pi$
$138$	0	0
$139$	−7.27644e13	−0.855703	−0.427851	−	0.903849i	$-0.640729\pi$
$139$	−7.27644e13	−0.855703	−0.427851	+	0.903849i	$0.640729\pi$
$140$	0	0
$141$	−5.05355e13	−0.541593
$142$	0	0
$143$	−8.56373e13	−0.837488
$144$	0	0
$145$	−1.25652e14	−1.12274
$146$	0	0
$147$	4.40471e13	0.360043
$148$	0	0
$149$	−2.52533e14	−1.89063	−0.945317	−	0.326153i	$-0.894248\pi$
$149$	−2.52533e14	−1.89063	−0.945317	+	0.326153i	$0.894248\pi$
$150$	0	0
$151$	1.28515e14	0.882276	0.441138	−	0.897439i	$-0.354575\pi$
$151$	1.28515e14	0.882276	0.441138	+	0.897439i	$0.354575\pi$
$152$	0	0
$153$	−2.88548e13	−0.181854
$154$	0	0
$155$	7.10478e12	0.0411524
$156$	0	0
$157$	9.35290e13	0.498423	0.249212	−	0.968449i	$-0.419829\pi$
$157$	9.35290e13	0.498423	0.249212	+	0.968449i	$0.419829\pi$
$158$	0	0
$159$	1.59960e14	0.785107
$160$	0	0
$161$	−1.90780e14	−0.863305
$162$	0	0
$163$	−4.06073e14	−1.69584	−0.847919	−	0.530126i	$-0.822145\pi$
$163$	−4.06073e14	−1.69584	−0.847919	+	0.530126i	$0.822145\pi$
$164$	0	0
$165$	6.27419e14	2.42054
$166$	0	0
$167$	4.27425e14	1.52477	0.762383	−	0.647126i	$-0.224029\pi$
$167$	4.27425e14	1.52477	0.762383	+	0.647126i	$0.224029\pi$
$168$	0	0
$169$	−2.15979e14	−0.713095
$170$	0	0
$171$	−7.62033e14	−2.33077
$172$	0	0
$173$	4.96614e14	1.40838	0.704189	−	0.710013i	$-0.251311\pi$
$173$	4.96614e14	1.40838	0.704189	+	0.710013i	$0.251311\pi$
$174$	0	0
$175$	9.50199e13	0.250073
$176$	0	0
$177$	8.26757e14	2.02093
$178$	0	0
$179$	7.04957e14	1.60184	0.800918	−	0.598774i	$-0.204345\pi$
$179$	7.04957e14	1.60184	0.800918	+	0.598774i	$0.204345\pi$
$180$	0	0
$181$	−5.97913e14	−1.26394	−0.631971	−	0.774992i	$-0.717754\pi$
$181$	−5.97913e14	−1.26394	−0.631971	+	0.774992i	$0.717754\pi$
$182$	0	0
$183$	−7.76220e14	−1.52775
$184$	0	0
$185$	−7.41015e14	−1.35897
$186$	0	0
$187$	−7.90256e13	−0.135144
$188$	0	0
$189$	−1.33748e15	−2.13445
$190$	0	0
$191$	4.68786e14	0.698647	0.349323	−	0.937002i	$-0.386412\pi$
$191$	4.68786e14	0.698647	0.349323	+	0.937002i	$0.386412\pi$
$192$	0	0
$193$	4.66164e14	0.649257	0.324629	−	0.945842i	$-0.394761\pi$
$193$	4.66164e14	0.649257	0.324629	+	0.945842i	$0.394761\pi$
$194$	0	0
$195$	−6.36644e14	−0.829224
$196$	0	0
$197$	1.99579e14	0.243268	0.121634	−	0.992575i	$-0.461187\pi$
$197$	1.99579e14	0.243268	0.121634	+	0.992575i	$0.461187\pi$
$198$	0	0
$199$	−3.01889e14	−0.344590	−0.172295	−	0.985045i	$-0.555118\pi$
$199$	−3.01889e14	−0.344590	−0.172295	+	0.985045i	$0.555118\pi$
$200$	0	0
$201$	1.94836e15	2.08398
$202$	0	0
$203$	1.39809e15	1.40221
$204$	0	0
$205$	−4.23507e14	−0.398529
$206$	0	0
$207$	1.87339e15	1.65509
$208$	0	0
$209$	−2.08700e15	−1.73210
$210$	0	0
$211$	−1.80118e15	−1.40515	−0.702573	−	0.711611i	$-0.747966\pi$
$211$	−1.80118e15	−1.40515	−0.702573	+	0.711611i	$0.747966\pi$
$212$	0	0
$213$	2.46283e14	0.180704
$214$	0	0
$215$	5.85467e13	0.0404252
$216$	0	0
$217$	−7.90526e13	−0.0513959
$218$	0	0
$219$	4.79762e15	2.93859
$220$	0	0
$221$	8.01876e13	0.0462975
$222$	0	0
$223$	−2.36916e15	−1.29007	−0.645035	−	0.764153i	$-0.723157\pi$
$223$	−2.36916e15	−1.29007	−0.645035	+	0.764153i	$0.723157\pi$
$224$	0	0
$225$	−9.33064e14	−0.479430
$226$	0	0
$227$	−1.88352e15	−0.913697	−0.456849	−	0.889544i	$-0.651022\pi$
$227$	−1.88352e15	−0.913697	−0.456849	+	0.889544i	$0.651022\pi$
$228$	0	0
$229$	4.08258e15	1.87070	0.935349	−	0.353725i	$-0.115085\pi$
$229$	4.08258e15	1.87070	0.935349	+	0.353725i	$0.115085\pi$
$230$	0	0
$231$	−6.98109e15	−3.02305
$232$	0	0
$233$	−6.16749e14	−0.252520	−0.126260	−	0.991997i	$-0.540297\pi$
$233$	−6.16749e14	−0.252520	−0.126260	+	0.991997i	$0.540297\pi$
$234$	0	0
$235$	−6.97431e14	−0.270122
$236$	0	0
$237$	−7.13995e14	−0.261716
$238$	0	0
$239$	5.72458e14	0.198681	0.0993407	−	0.995053i	$-0.468327\pi$
$239$	5.72458e14	0.198681	0.0993407	+	0.995053i	$0.468327\pi$
$240$	0	0
$241$	−6.98649e14	−0.229693	−0.114847	−	0.993383i	$-0.536638\pi$
$241$	−6.98649e14	−0.229693	−0.114847	+	0.993383i	$0.536638\pi$
$242$	0	0
$243$	1.23672e15	0.385327
$244$	0	0
$245$	6.07884e14	0.179573
$246$	0	0
$247$	2.11769e15	0.593379
$248$	0	0
$249$	3.64174e15	0.968307
$250$	0	0
$251$	4.42788e15	1.11768	0.558839	−	0.829276i	$-0.311247\pi$
$251$	4.42788e15	1.11768	0.558839	+	0.829276i	$0.311247\pi$
$252$	0	0
$253$	5.13072e15	1.22997
$254$	0	0
$255$	−5.87491e14	−0.133811
$256$	0	0
$257$	−3.45343e15	−0.747628	−0.373814	−	0.927504i	$-0.621950\pi$
$257$	−3.45343e15	−0.747628	−0.373814	+	0.927504i	$0.621950\pi$
$258$	0	0
$259$	8.24504e15	1.69724
$260$	0	0
$261$	−1.37288e16	−2.68825
$262$	0	0
$263$	2.68676e14	0.0500629	0.0250315	−	0.999687i	$-0.492031\pi$
$263$	2.68676e14	0.0500629	0.0250315	+	0.999687i	$0.492031\pi$
$264$	0	0
$265$	2.20757e15	0.391576
$266$	0	0
$267$	−7.15932e15	−1.20934
$268$	0	0
$269$	−4.84800e15	−0.780140	−0.390070	−	0.920785i	$-0.627549\pi$
$269$	−4.84800e15	−0.780140	−0.390070	+	0.920785i	$0.627549\pi$
$270$	0	0
$271$	−8.45435e14	−0.129652	−0.0648261	−	0.997897i	$-0.520649\pi$
$271$	−8.45435e14	−0.129652	−0.0648261	+	0.997897i	$0.520649\pi$
$272$	0	0
$273$	7.08373e15	1.03563
$274$	0	0
$275$	−2.55541e15	−0.356286
$276$	0	0
$277$	9.23216e15	1.22796	0.613981	−	0.789321i	$-0.289567\pi$
$277$	9.23216e15	1.22796	0.613981	+	0.789321i	$0.289567\pi$
$278$	0	0
$279$	7.76271e14	0.0985340
$280$	0	0
$281$	−6.24571e15	−0.756817	−0.378408	−	0.925639i	$-0.623529\pi$
$281$	−6.24571e15	−0.756817	−0.378408	+	0.925639i	$0.623529\pi$
$282$	0	0
$283$	−2.27398e15	−0.263133	−0.131567	−	0.991307i	$-0.542001\pi$
$283$	−2.27398e15	−0.263133	−0.131567	+	0.991307i	$0.542001\pi$
$284$	0	0
$285$	−1.55152e16	−1.71501
$286$	0	0
$287$	4.71222e15	0.497730
$288$	0	0
$289$	−9.83058e15	−0.992529
$290$	0	0
$291$	−8.74371e15	−0.844095
$292$	0	0
$293$	6.05443e15	0.559028	0.279514	−	0.960142i	$-0.409827\pi$
$293$	6.05443e15	0.559028	0.279514	+	0.960142i	$0.409827\pi$
$294$	0	0
$295$	1.14099e16	1.00795
$296$	0	0
$297$	3.59695e16	3.04101
$298$	0	0
$299$	−5.20616e15	−0.421362
$300$	0	0
$301$	−6.51430e14	−0.0504877
$302$	0	0
$303$	4.81875e15	0.357732
$304$	0	0
$305$	−1.07124e16	−0.761973
$306$	0	0
$307$	1.40774e16	0.959671	0.479835	−	0.877359i	$-0.340696\pi$
$307$	1.40774e16	0.959671	0.479835	+	0.877359i	$0.340696\pi$
$308$	0	0
$309$	4.66663e16	3.04981
$310$	0	0
$311$	−9.12928e15	−0.572129	−0.286064	−	0.958210i	$-0.592347\pi$
$311$	−9.12928e15	−0.572129	−0.286064	+	0.958210i	$0.592347\pi$
$312$	0	0
$313$	−2.46408e16	−1.48121	−0.740606	−	0.671940i	$-0.765461\pi$
$313$	−2.46408e16	−1.48121	−0.740606	+	0.671940i	$0.765461\pi$
$314$	0	0
$315$	−3.51785e16	−2.02889
$316$	0	0
$317$	1.17144e16	0.648388	0.324194	−	0.945991i	$-0.394907\pi$
$317$	1.17144e16	0.648388	0.324194	+	0.945991i	$0.394907\pi$
$318$	0	0
$319$	−3.75995e16	−1.99776
$320$	0	0
$321$	−3.62498e16	−1.84937
$322$	0	0
$323$	1.95419e15	0.0957527
$324$	0	0
$325$	2.59299e15	0.122056
$326$	0	0
$327$	7.43320e16	3.36214
$328$	0	0
$329$	7.76009e15	0.337360
$330$	0	0
$331$	−1.59924e16	−0.668392	−0.334196	−	0.942504i	$-0.608465\pi$
$331$	−1.59924e16	−0.668392	−0.334196	+	0.942504i	$0.608465\pi$
$332$	0	0
$333$	−8.09635e16	−3.25388
$334$	0	0
$335$	2.68889e16	1.03940
$336$	0	0
$337$	5.65737e15	0.210388	0.105194	−	0.994452i	$-0.466454\pi$
$337$	5.65737e15	0.210388	0.105194	+	0.994452i	$0.466454\pi$
$338$	0	0
$339$	8.74074e15	0.312788
$340$	0	0
$341$	2.12600e15	0.0732250
$342$	0	0
$343$	2.63334e16	0.873162
$344$	0	0
$345$	3.81428e16	1.21783
$346$	0	0
$347$	−1.91864e16	−0.590000	−0.295000	−	0.955497i	$-0.595320\pi$
$347$	−1.91864e16	−0.590000	−0.295000	+	0.955497i	$0.595320\pi$
$348$	0	0
$349$	−1.62333e16	−0.480885	−0.240442	−	0.970663i	$-0.577292\pi$
$349$	−1.62333e16	−0.480885	−0.240442	+	0.970663i	$0.577292\pi$
$350$	0	0
$351$	−3.64984e16	−1.04178
$352$	0	0
$353$	4.71540e16	1.29713	0.648564	−	0.761160i	$-0.275370\pi$
$353$	4.71540e16	1.29713	0.648564	+	0.761160i	$0.275370\pi$
$354$	0	0
$355$	3.39891e15	0.0901270
$356$	0	0
$357$	6.53683e15	0.167118
$358$	0	0
$359$	−3.06168e16	−0.754825	−0.377413	−	0.926045i	$-0.623186\pi$
$359$	−3.06168e16	−0.754825	−0.377413	+	0.926045i	$0.623186\pi$
$360$	0	0
$361$	9.55572e15	0.227231
$362$	0	0
$363$	1.10947e17	2.54521
$364$	0	0
$365$	6.62110e16	1.46564
$366$	0	0
$367$	−9.05053e15	−0.193350	−0.0966751	−	0.995316i	$-0.530821\pi$
$367$	−9.05053e15	−0.193350	−0.0966751	+	0.995316i	$0.530821\pi$
$368$	0	0
$369$	−4.62725e16	−0.954226
$370$	0	0
$371$	−2.45629e16	−0.489046
$372$	0	0
$373$	−5.96602e16	−1.14704	−0.573519	−	0.819192i	$-0.694422\pi$
$373$	−5.96602e16	−1.14704	−0.573519	+	0.819192i	$0.694422\pi$
$374$	0	0
$375$	−1.02367e17	−1.90088
$376$	0	0
$377$	3.81524e16	0.684389
$378$	0	0
$379$	−3.59746e16	−0.623507	−0.311753	−	0.950163i	$-0.600916\pi$
$379$	−3.59746e16	−0.623507	−0.311753	+	0.950163i	$0.600916\pi$
$380$	0	0
$381$	−7.08675e16	−1.18696
$382$	0	0
$383$	−3.65533e16	−0.591745	−0.295872	−	0.955227i	$-0.595610\pi$
$383$	−3.65533e16	−0.591745	−0.295872	+	0.955227i	$0.595610\pi$
$384$	0	0
$385$	−9.63446e16	−1.50776
$386$	0	0
$387$	6.39683e15	0.0967929
$388$	0	0
$389$	1.30128e17	1.90414	0.952072	−	0.305873i	$-0.0989483\pi$
$389$	1.30128e17	1.90414	0.952072	+	0.305873i	$0.0989483\pi$
$390$	0	0
$391$	−4.80422e15	−0.0679945
$392$	0	0
$393$	1.12525e17	1.54062
$394$	0	0
$395$	−9.85371e15	−0.130532
$396$	0	0
$397$	4.04022e16	0.517924	0.258962	−	0.965887i	$-0.416619\pi$
$397$	4.04022e16	0.517924	0.258962	+	0.965887i	$0.416619\pi$
$398$	0	0
$399$	1.72633e17	2.14190
$400$	0	0
$401$	−1.10742e17	−1.33007	−0.665037	−	0.746811i	$-0.731584\pi$
$401$	−1.10742e17	−1.33007	−0.665037	+	0.746811i	$0.731584\pi$
$402$	0	0
$403$	−2.15726e15	−0.0250853
$404$	0	0
$405$	1.03217e17	1.16224
$406$	0	0
$407$	−2.21737e17	−2.41811
$408$	0	0
$409$	2.65227e16	0.280166	0.140083	−	0.990140i	$-0.455263\pi$
$409$	2.65227e16	0.280166	0.140083	+	0.990140i	$0.455263\pi$
$410$	0	0
$411$	1.98008e17	2.02634
$412$	0	0
$413$	−1.26954e17	−1.25885
$414$	0	0
$415$	5.02589e16	0.482948
$416$	0	0
$417$	−1.61869e17	−1.50758
$418$	0	0
$419$	1.41472e17	1.27726	0.638631	−	0.769513i	$-0.279501\pi$
$419$	1.41472e17	1.27726	0.638631	+	0.769513i	$0.279501\pi$
$420$	0	0
$421$	1.28484e17	1.12465	0.562323	−	0.826918i	$-0.309908\pi$
$421$	1.28484e17	1.12465	0.562323	+	0.826918i	$0.309908\pi$
$422$	0	0
$423$	−7.62015e16	−0.646772
$424$	0	0
$425$	2.39279e15	0.0196960
$426$	0	0
$427$	1.19194e17	0.951641
$428$	0	0
$429$	−1.90506e17	−1.47549
$430$	0	0
$431$	1.43879e17	1.08117	0.540587	−	0.841288i	$-0.318202\pi$
$431$	1.43879e17	1.08117	0.540587	+	0.841288i	$0.318202\pi$
$432$	0	0
$433$	−2.64520e15	−0.0192880	−0.00964402	−	0.999953i	$-0.503070\pi$
$433$	−2.64520e15	−0.0192880	−0.00964402	+	0.999953i	$0.503070\pi$
$434$	0	0
$435$	−2.79522e17	−1.97804
$436$	0	0
$437$	−1.26876e17	−0.871463
$438$	0	0
$439$	1.34376e17	0.895989	0.447994	−	0.894036i	$-0.352138\pi$
$439$	1.34376e17	0.895989	0.447994	+	0.894036i	$0.352138\pi$
$440$	0	0
$441$	6.64176e16	0.429965
$442$	0	0
$443$	−8.81369e16	−0.554030	−0.277015	−	0.960866i	$-0.589345\pi$
$443$	−8.81369e16	−0.554030	−0.277015	+	0.960866i	$0.589345\pi$
$444$	0	0
$445$	−9.88044e16	−0.603165
$446$	0	0
$447$	−5.61777e17	−3.33093
$448$	0	0
$449$	1.36300e17	0.785044	0.392522	−	0.919743i	$-0.371603\pi$
$449$	1.36300e17	0.785044	0.392522	+	0.919743i	$0.371603\pi$
$450$	0	0
$451$	−1.26728e17	−0.709128
$452$	0	0
$453$	2.85891e17	1.55440
$454$	0	0
$455$	9.77612e16	0.516526
$456$	0	0
$457$	3.64371e17	1.87106	0.935532	−	0.353243i	$-0.114921\pi$
$457$	3.64371e17	1.87106	0.935532	+	0.353243i	$0.114921\pi$
$458$	0	0
$459$	−3.36805e16	−0.168111
$460$	0	0
$461$	−4.15694e16	−0.201706	−0.100853	−	0.994901i	$-0.532157\pi$
$461$	−4.15694e16	−0.201706	−0.100853	+	0.994901i	$0.532157\pi$
$462$	0	0
$463$	4.15266e17	1.95907	0.979534	−	0.201279i	$-0.0645098\pi$
$463$	4.15266e17	1.95907	0.979534	+	0.201279i	$0.0645098\pi$
$464$	0	0
$465$	1.58051e16	0.0725024
$466$	0	0
$467$	1.46646e17	0.654202	0.327101	−	0.944989i	$-0.393928\pi$
$467$	1.46646e17	0.654202	0.327101	+	0.944989i	$0.393928\pi$
$468$	0	0
$469$	−2.99184e17	−1.29812
$470$	0	0
$471$	2.08062e17	0.878124
$472$	0	0
$473$	1.75192e16	0.0719311
$474$	0	0
$475$	6.31918e16	0.252437
$476$	0	0
$477$	2.41200e17	0.937578
$478$	0	0
$479$	−1.66742e17	−0.630758	−0.315379	−	0.948966i	$-0.602132\pi$
$479$	−1.66742e17	−0.630758	−0.315379	+	0.948966i	$0.602132\pi$
$480$	0	0
$481$	2.24998e17	0.828391
$482$	0	0
$483$	−4.24403e17	−1.52097
$484$	0	0
$485$	−1.20670e17	−0.420997
$486$	0	0
$487$	−1.02648e17	−0.348667	−0.174334	−	0.984687i	$-0.555777\pi$
$487$	−1.02648e17	−0.348667	−0.174334	+	0.984687i	$0.555777\pi$
$488$	0	0
$489$	−9.03336e17	−2.98773
$490$	0	0
$491$	−3.03049e17	−0.976073	−0.488037	−	0.872823i	$-0.662287\pi$
$491$	−3.03049e17	−0.976073	−0.488037	+	0.872823i	$0.662287\pi$
$492$	0	0
$493$	3.52068e16	0.110439
$494$	0	0
$495$	9.46072e17	2.89062
$496$	0	0
$497$	−3.78185e16	−0.112561
$498$	0	0
$499$	−1.25532e17	−0.364001	−0.182001	−	0.983298i	$-0.558257\pi$
$499$	−1.25532e17	−0.364001	−0.182001	+	0.983298i	$0.558257\pi$
$500$	0	0
$501$	9.50837e17	2.68634
$502$	0	0
$503$	−2.18681e16	−0.0602032	−0.0301016	−	0.999547i	$-0.509583\pi$
$503$	−2.18681e16	−0.0602032	−0.0301016	+	0.999547i	$0.509583\pi$
$504$	0	0
$505$	6.65027e16	0.178421
$506$	0	0
$507$	−4.80459e17	−1.25633
$508$	0	0
$509$	−5.98743e17	−1.52607	−0.763035	−	0.646357i	$-0.776292\pi$
$509$	−5.98743e17	−1.52607	−0.763035	+	0.646357i	$0.776292\pi$
$510$	0	0
$511$	−7.36709e17	−1.83046
$512$	0	0
$513$	−8.89476e17	−2.15462
$514$	0	0
$515$	6.44032e17	1.52111
$516$	0	0
$517$	−2.08696e17	−0.480645
$518$	0	0
$519$	1.10475e18	2.48128
$520$	0	0
$521$	−8.32756e15	−0.0182420	−0.00912100	−	0.999958i	$-0.502903\pi$
$521$	−8.32756e15	−0.0182420	−0.00912100	+	0.999958i	$0.502903\pi$
$522$	0	0
$523$	−1.41495e17	−0.302328	−0.151164	−	0.988509i	$-0.548302\pi$
$523$	−1.41495e17	−0.302328	−0.151164	+	0.988509i	$0.548302\pi$
$524$	0	0
$525$	2.11378e17	0.440580
$526$	0	0
$527$	−1.99070e15	−0.00404797
$528$	0	0
$529$	−1.92123e17	−0.381169
$530$	0	0
$531$	1.24665e18	2.41341
$532$	0	0
$533$	1.28591e17	0.242932
$534$	0	0
$535$	−5.00276e17	−0.922381
$536$	0	0
$537$	1.56823e18	2.82212
$538$	0	0
$539$	1.81900e17	0.319526
$540$	0	0
$541$	4.45803e17	0.764471	0.382236	−	0.924065i	$-0.375154\pi$
$541$	4.45803e17	0.764471	0.382236	+	0.924065i	$0.375154\pi$
$542$	0	0
$543$	−1.33010e18	−2.22682
$544$	0	0
$545$	1.02584e18	1.67688
$546$	0	0
$547$	−9.13831e17	−1.45864	−0.729320	−	0.684172i	$-0.760164\pi$
$547$	−9.13831e17	−1.45864	−0.729320	+	0.684172i	$0.760164\pi$
$548$	0	0
$549$	−1.17045e18	−1.82445
$550$	0	0
$551$	9.29784e17	1.41546
$552$	0	0
$553$	1.09639e17	0.163024
$554$	0	0
$555$	−1.64844e18	−2.39424
$556$	0	0
$557$	−6.27602e17	−0.890482	−0.445241	−	0.895411i	$-0.646882\pi$
$557$	−6.27602e17	−0.890482	−0.445241	+	0.895411i	$0.646882\pi$
$558$	0	0
$559$	−1.77768e16	−0.0246420
$560$	0	0
$561$	−1.75798e17	−0.238098
$562$	0	0
$563$	6.33220e17	0.838011	0.419006	−	0.907984i	$-0.362379\pi$
$563$	6.33220e17	0.838011	0.419006	+	0.907984i	$0.362379\pi$
$564$	0	0
$565$	1.20629e17	0.156005
$566$	0	0
$567$	−1.14847e18	−1.45154
$568$	0	0
$569$	−5.18457e17	−0.640446	−0.320223	−	0.947342i	$-0.603758\pi$
$569$	−5.18457e17	−0.640446	−0.320223	+	0.947342i	$0.603758\pi$
$570$	0	0
$571$	1.98029e17	0.239108	0.119554	−	0.992828i	$-0.461854\pi$
$571$	1.98029e17	0.239108	0.119554	+	0.992828i	$0.461854\pi$
$572$	0	0
$573$	1.04285e18	1.23088
$574$	0	0
$575$	−1.55352e17	−0.179257
$576$	0	0
$577$	−4.09548e16	−0.0462021	−0.0231011	−	0.999733i	$-0.507354\pi$
$577$	−4.09548e16	−0.0462021	−0.0231011	+	0.999733i	$0.507354\pi$
$578$	0	0
$579$	1.03701e18	1.14386
$580$	0	0
$581$	−5.59215e17	−0.603161
$582$	0	0
$583$	6.60581e17	0.696756
$584$	0	0
$585$	−9.59983e17	−0.990262
$586$	0	0
$587$	1.00006e18	1.00897	0.504485	−	0.863420i	$-0.331682\pi$
$587$	1.00006e18	1.00897	0.504485	+	0.863420i	$0.331682\pi$
$588$	0	0
$589$	−5.25730e16	−0.0518816
$590$	0	0
$591$	4.43977e17	0.428590
$592$	0	0
$593$	1.62146e17	0.153126	0.0765631	−	0.997065i	$-0.475605\pi$
$593$	1.62146e17	0.153126	0.0765631	+	0.997065i	$0.475605\pi$
$594$	0	0
$595$	9.02135e16	0.0833510
$596$	0	0
$597$	−6.71573e17	−0.607100
$598$	0	0
$599$	−1.85462e18	−1.64052	−0.820260	−	0.571992i	$-0.806171\pi$
$599$	−1.85462e18	−1.64052	−0.820260	+	0.571992i	$0.806171\pi$
$600$	0	0
$601$	−1.05677e18	−0.914737	−0.457369	−	0.889277i	$-0.651208\pi$
$601$	−1.05677e18	−0.914737	−0.457369	+	0.889277i	$0.651208\pi$
$602$	0	0
$603$	2.93789e18	2.48870
$604$	0	0
$605$	1.53116e18	1.26944
$606$	0	0
$607$	−7.72501e17	−0.626863	−0.313431	−	0.949611i	$-0.601479\pi$
$607$	−7.72501e17	−0.626863	−0.313431	+	0.949611i	$0.601479\pi$
$608$	0	0
$609$	3.11015e18	2.47041
$610$	0	0
$611$	2.11764e17	0.164659
$612$	0	0
$613$	−1.15792e18	−0.881428	−0.440714	−	0.897648i	$-0.645275\pi$
$613$	−1.15792e18	−0.881428	−0.440714	+	0.897648i	$0.645275\pi$
$614$	0	0
$615$	−9.42120e17	−0.702131
$616$	0	0
$617$	−1.42912e18	−1.04283	−0.521416	−	0.853303i	$-0.674596\pi$
$617$	−1.42912e18	−1.04283	−0.521416	+	0.853303i	$0.674596\pi$
$618$	0	0
$619$	1.90926e18	1.36420	0.682098	−	0.731261i	$-0.261068\pi$
$619$	1.90926e18	1.36420	0.682098	+	0.731261i	$0.261068\pi$
$620$	0	0
$621$	2.18670e18	1.53001
$622$	0	0
$623$	1.09936e18	0.753303
$624$	0	0
$625$	−1.07319e18	−0.720204
$626$	0	0
$627$	−4.64268e18	−3.05162
$628$	0	0
$629$	2.07627e17	0.133676
$630$	0	0
$631$	1.10754e18	0.698501	0.349251	−	0.937029i	$-0.386436\pi$
$631$	1.10754e18	0.698501	0.349251	+	0.937029i	$0.386436\pi$
$632$	0	0
$633$	−4.00685e18	−2.47559
$634$	0	0
$635$	−9.78028e17	−0.592000
$636$	0	0
$637$	−1.84575e17	−0.109463
$638$	0	0
$639$	3.71365e17	0.215797
$640$	0	0
$641$	−3.03185e18	−1.72636	−0.863178	−	0.504900i	$-0.831529\pi$
$641$	−3.03185e18	−1.72636	−0.863178	+	0.504900i	$0.831529\pi$
$642$	0	0
$643$	−1.83914e17	−0.102622	−0.0513112	−	0.998683i	$-0.516340\pi$
$643$	−1.83914e17	−0.102622	−0.0513112	+	0.998683i	$0.516340\pi$
$644$	0	0
$645$	1.30241e17	0.0712213
$646$	0	0
$647$	−1.82632e18	−0.978810	−0.489405	−	0.872057i	$-0.662786\pi$
$647$	−1.82632e18	−0.978810	−0.489405	+	0.872057i	$0.662786\pi$
$648$	0	0
$649$	3.41424e18	1.79351
$650$	0	0
$651$	−1.75858e17	−0.0905494
$652$	0	0
$653$	8.09505e16	0.0408587	0.0204293	−	0.999791i	$-0.493497\pi$
$653$	8.09505e16	0.0408587	0.0204293	+	0.999791i	$0.493497\pi$
$654$	0	0
$655$	1.55293e18	0.768394
$656$	0	0
$657$	7.23424e18	3.50928
$658$	0	0
$659$	6.86521e17	0.326512	0.163256	−	0.986584i	$-0.447800\pi$
$659$	6.86521e17	0.326512	0.163256	+	0.986584i	$0.447800\pi$
$660$	0	0
$661$	7.88664e17	0.367775	0.183888	−	0.982947i	$-0.441132\pi$
$661$	7.88664e17	0.367775	0.183888	+	0.982947i	$0.441132\pi$
$662$	0	0
$663$	1.78383e17	0.0815671
$664$	0	0
$665$	2.38247e18	1.06828
$666$	0	0
$667$	−2.28580e18	−1.00512
$668$	0	0
$669$	−5.27036e18	−2.27285
$670$	0	0
$671$	−3.20554e18	−1.35583
$672$	0	0
$673$	2.18239e18	0.905388	0.452694	−	0.891666i	$-0.350463\pi$
$673$	2.18239e18	0.905388	0.452694	+	0.891666i	$0.350463\pi$
$674$	0	0
$675$	−1.08911e18	−0.443197
$676$	0	0
$677$	3.05717e18	1.22038	0.610188	−	0.792256i	$-0.291094\pi$
$677$	3.05717e18	1.22038	0.610188	+	0.792256i	$0.291094\pi$
$678$	0	0
$679$	1.34266e18	0.525790
$680$	0	0
$681$	−4.19002e18	−1.60976
$682$	0	0
$683$	1.48644e18	0.560290	0.280145	−	0.959958i	$-0.409617\pi$
$683$	1.48644e18	0.560290	0.280145	+	0.959958i	$0.409617\pi$
$684$	0	0
$685$	2.73267e18	1.01064
$686$	0	0
$687$	9.08197e18	3.29580
$688$	0	0
$689$	−6.70294e17	−0.238693
$690$	0	0
$691$	−2.56014e16	−0.00894655	−0.00447328	−	0.999990i	$-0.501424\pi$
$691$	−2.56014e16	−0.00894655	−0.00447328	+	0.999990i	$0.501424\pi$
$692$	0	0
$693$	−1.05266e19	−3.61014
$694$	0	0
$695$	−2.23393e18	−0.751914
$696$	0	0
$697$	1.18663e17	0.0392016
$698$	0	0
$699$	−1.37200e18	−0.444890
$700$	0	0
$701$	2.23789e18	0.712313	0.356156	−	0.934426i	$-0.384087\pi$
$701$	2.23789e18	0.712313	0.356156	+	0.934426i	$0.384087\pi$
$702$	0	0
$703$	5.48326e18	1.71328
$704$	0	0
$705$	−1.55148e18	−0.475902
$706$	0	0
$707$	−7.39954e17	−0.222832
$708$	0	0
$709$	−1.45363e18	−0.429786	−0.214893	−	0.976638i	$-0.568940\pi$
$709$	−1.45363e18	−0.429786	−0.214893	+	0.976638i	$0.568940\pi$
$710$	0	0
$711$	−1.07662e18	−0.312543
$712$	0	0
$713$	1.29246e17	0.0368414
$714$	0	0
$715$	−2.62914e18	−0.735908
$716$	0	0
$717$	1.27347e18	0.350038
$718$	0	0
$719$	−6.32656e18	−1.70777	−0.853885	−	0.520461i	$-0.825760\pi$
$719$	−6.32656e18	−1.70777	−0.853885	+	0.520461i	$0.825760\pi$
$720$	0	0
$721$	−7.16593e18	−1.89974
$722$	0	0
$723$	−1.55419e18	−0.404675
$724$	0	0
$725$	1.13847e18	0.291154
$726$	0	0
$727$	3.41363e18	0.857517	0.428758	−	0.903419i	$-0.358951\pi$
$727$	3.41363e18	0.857517	0.428758	+	0.903419i	$0.358951\pi$
$728$	0	0
$729$	−2.60901e18	−0.643793
$730$	0	0
$731$	−1.64043e16	−0.00397645
$732$	0	0
$733$	7.93022e18	1.88847	0.944234	−	0.329276i	$-0.106805\pi$
$733$	7.93022e18	1.88847	0.944234	+	0.329276i	$0.106805\pi$
$734$	0	0
$735$	1.35228e18	0.316373
$736$	0	0
$737$	8.04608e18	1.84946
$738$	0	0
$739$	5.86152e17	0.132380	0.0661898	−	0.997807i	$-0.478916\pi$
$739$	5.86152e17	0.132380	0.0661898	+	0.997807i	$0.478916\pi$
$740$	0	0
$741$	4.71095e18	1.04542
$742$	0	0
$743$	−5.31469e17	−0.115891	−0.0579456	−	0.998320i	$-0.518455\pi$
$743$	−5.31469e17	−0.115891	−0.0579456	+	0.998320i	$0.518455\pi$
$744$	0	0
$745$	−7.75297e18	−1.66132
$746$	0	0
$747$	5.49130e18	1.15636
$748$	0	0
$749$	5.56642e18	1.15198
$750$	0	0
$751$	−4.47766e18	−0.910734	−0.455367	−	0.890304i	$-0.650492\pi$
$751$	−4.47766e18	−0.910734	−0.455367	+	0.890304i	$0.650492\pi$
$752$	0	0
$753$	9.85013e18	1.96913
$754$	0	0
$755$	3.94552e18	0.775264
$756$	0	0
$757$	−4.54650e18	−0.878120	−0.439060	−	0.898458i	$-0.644689\pi$
$757$	−4.54650e18	−0.878120	−0.439060	+	0.898458i	$0.644689\pi$
$758$	0	0
$759$	1.14136e19	2.16697
$760$	0	0
$761$	−7.91093e18	−1.47648	−0.738239	−	0.674539i	$-0.764342\pi$
$761$	−7.91093e18	−1.47648	−0.738239	+	0.674539i	$0.764342\pi$
$762$	0	0
$763$	−1.14142e19	−2.09429
$764$	0	0
$765$	−8.85866e17	−0.159797
$766$	0	0
$767$	−3.46444e18	−0.614418
$768$	0	0
$769$	−6.56634e18	−1.14499	−0.572495	−	0.819908i	$-0.694024\pi$
$769$	−6.56634e18	−1.14499	−0.572495	+	0.819908i	$0.694024\pi$
$770$	0	0
$771$	−7.68239e18	−1.31717
$772$	0	0
$773$	−6.09277e18	−1.02718	−0.513592	−	0.858035i	$-0.671685\pi$
$773$	−6.09277e18	−1.02718	−0.513592	+	0.858035i	$0.671685\pi$
$774$	0	0
$775$	−6.43725e16	−0.0106718
$776$	0	0
$777$	1.83416e19	2.99021
$778$	0	0
$779$	3.13381e18	0.502433
$780$	0	0
$781$	1.01707e18	0.160369
$782$	0	0
$783$	−1.60248e19	−2.48509
$784$	0	0
$785$	2.87142e18	0.437969
$786$	0	0
$787$	5.62269e18	0.843545	0.421773	−	0.906702i	$-0.361408\pi$
$787$	5.62269e18	0.843545	0.421773	+	0.906702i	$0.361408\pi$
$788$	0	0
$789$	5.97688e17	0.0882010
$790$	0	0
$791$	−1.34220e18	−0.194837
$792$	0	0
$793$	3.25267e18	0.464477
$794$	0	0
$795$	4.91089e18	0.689880
$796$	0	0
$797$	−1.28588e19	−1.77713	−0.888567	−	0.458746i	$-0.848299\pi$
$797$	−1.28588e19	−1.77713	−0.888567	+	0.458746i	$0.848299\pi$
$798$	0	0
$799$	1.95415e17	0.0265707
$800$	0	0
$801$	−1.07954e19	−1.44420
$802$	0	0
$803$	1.98126e19	2.60790
$804$	0	0
$805$	−5.85709e18	−0.758593
$806$	0	0
$807$	−1.07847e19	−1.37445
$808$	0	0
$809$	5.07548e18	0.636520	0.318260	−	0.948003i	$-0.396902\pi$
$809$	5.07548e18	0.636520	0.318260	+	0.948003i	$0.396902\pi$
$810$	0	0
$811$	−7.26888e18	−0.897082	−0.448541	−	0.893762i	$-0.648056\pi$
$811$	−7.26888e18	−0.897082	−0.448541	+	0.893762i	$0.648056\pi$
$812$	0	0
$813$	−1.88073e18	−0.228422
$814$	0	0
$815$	−1.24668e19	−1.49015
$816$	0	0
$817$	−4.33225e17	−0.0509648
$818$	0	0
$819$	1.06814e19	1.23675
$820$	0	0
$821$	1.50633e19	1.71668	0.858339	−	0.513083i	$-0.171497\pi$
$821$	1.50633e19	1.71668	0.858339	+	0.513083i	$0.171497\pi$
$822$	0	0
$823$	1.23976e19	1.39072	0.695359	−	0.718662i	$-0.255245\pi$
$823$	1.23976e19	1.39072	0.695359	+	0.718662i	$0.255245\pi$
$824$	0	0
$825$	−5.68469e18	−0.627706
$826$	0	0
$827$	1.01774e19	1.10625	0.553124	−	0.833099i	$-0.313436\pi$
$827$	1.01774e19	1.10625	0.553124	+	0.833099i	$0.313436\pi$
$828$	0	0
$829$	−9.57706e18	−1.02477	−0.512386	−	0.858755i	$-0.671238\pi$
$829$	−9.57706e18	−1.02477	−0.512386	+	0.858755i	$0.671238\pi$
$830$	0	0
$831$	2.05376e19	2.16343
$832$	0	0
$833$	−1.70324e17	−0.0176638
$834$	0	0
$835$	1.31223e19	1.33982
$836$	0	0
$837$	9.06094e17	0.0910873
$838$	0	0
$839$	1.29416e19	1.28095	0.640477	−	0.767977i	$-0.278737\pi$
$839$	1.29416e19	1.28095	0.640477	+	0.767977i	$0.278737\pi$
$840$	0	0
$841$	6.49038e18	0.632552
$842$	0	0
$843$	−1.38940e19	−1.33336
$844$	0	0
$845$	−6.63072e18	−0.626602
$846$	0	0
$847$	−1.70367e19	−1.58542
$848$	0	0
$849$	−5.05863e18	−0.463589
$850$	0	0
$851$	−1.34801e19	−1.21661
$852$	0	0
$853$	1.74188e19	1.54828	0.774138	−	0.633016i	$-0.218183\pi$
$853$	1.74188e19	1.54828	0.774138	+	0.633016i	$0.218183\pi$
$854$	0	0
$855$	−2.33950e19	−2.04807
$856$	0	0
$857$	−4.71644e18	−0.406667	−0.203334	−	0.979110i	$-0.565178\pi$
$857$	−4.71644e18	−0.406667	−0.203334	+	0.979110i	$0.565178\pi$
$858$	0	0
$859$	−1.14265e19	−0.970414	−0.485207	−	0.874399i	$-0.661256\pi$
$859$	−1.14265e19	−0.970414	−0.485207	+	0.874399i	$0.661256\pi$
$860$	0	0
$861$	1.04827e19	0.876902
$862$	0	0
$863$	4.59537e18	0.378661	0.189330	−	0.981913i	$-0.439368\pi$
$863$	4.59537e18	0.378661	0.189330	+	0.981913i	$0.439368\pi$
$864$	0	0
$865$	1.52464e19	1.23755
$866$	0	0
$867$	−2.18688e19	−1.74864
$868$	0	0
$869$	−2.94857e18	−0.232265
$870$	0	0
$871$	−8.16439e18	−0.633586
$872$	0	0
$873$	−1.31845e19	−1.00802
$874$	0	0
$875$	1.57191e19	1.18407
$876$	0	0
$877$	−1.03578e19	−0.768726	−0.384363	−	0.923182i	$-0.625579\pi$
$877$	−1.03578e19	−0.768726	−0.384363	+	0.923182i	$0.625579\pi$
$878$	0	0
$879$	1.34685e19	0.984898
$880$	0	0
$881$	−1.80889e19	−1.30337	−0.651686	−	0.758489i	$-0.725938\pi$
$881$	−1.80889e19	−1.30337	−0.651686	+	0.758489i	$0.725938\pi$
$882$	0	0
$883$	−8.36653e18	−0.594019	−0.297010	−	0.954874i	$-0.595989\pi$
$883$	−8.36653e18	−0.594019	−0.297010	+	0.954874i	$0.595989\pi$
$884$	0	0
$885$	2.53821e19	1.77581
$886$	0	0
$887$	−1.65265e19	−1.13941	−0.569703	−	0.821851i	$-0.692942\pi$
$887$	−1.65265e19	−1.13941	−0.569703	+	0.821851i	$0.692942\pi$
$888$	0	0
$889$	1.08822e19	0.739359
$890$	0	0
$891$	3.08862e19	2.06804
$892$	0	0
$893$	5.16075e18	0.340548
$894$	0	0
$895$	2.16428e19	1.40755
$896$	0	0
$897$	−1.15815e19	−0.742357
$898$	0	0
$899$	−9.47156e17	−0.0598389
$900$	0	0
$901$	−6.18544e17	−0.0385176
$902$	0	0
$903$	−1.44915e18	−0.0889494
$904$	0	0
$905$	−1.83564e19	−1.11064
$906$	0	0
$907$	1.48578e18	0.0886147	0.0443074	−	0.999018i	$-0.485892\pi$
$907$	1.48578e18	0.0886147	0.0443074	+	0.999018i	$0.485892\pi$
$908$	0	0
$909$	7.26610e18	0.427205
$910$	0	0
$911$	−1.98362e19	−1.14971	−0.574855	−	0.818256i	$-0.694941\pi$
$911$	−1.98362e19	−1.14971	−0.574855	+	0.818256i	$0.694941\pi$
$912$	0	0
$913$	1.50392e19	0.859340
$914$	0	0
$915$	−2.38306e19	−1.34245
$916$	0	0
$917$	−1.72790e19	−0.959659
$918$	0	0
$919$	1.49392e19	0.818044	0.409022	−	0.912524i	$-0.365870\pi$
$919$	1.49392e19	0.818044	0.409022	+	0.912524i	$0.365870\pi$
$920$	0	0
$921$	3.13161e19	1.69075
$922$	0	0
$923$	−1.03202e18	−0.0549388
$924$	0	0
$925$	6.71392e18	0.352416
$926$	0	0
$927$	7.03671e19	3.64210
$928$	0	0
$929$	1.18869e19	0.606692	0.303346	−	0.952881i	$-0.401896\pi$
$929$	1.18869e19	0.606692	0.303346	+	0.952881i	$0.401896\pi$
$930$	0	0
$931$	−4.49814e18	−0.226391
$932$	0	0
$933$	−2.03087e19	−1.00798
$934$	0	0
$935$	−2.42615e18	−0.118752
$936$	0	0
$937$	2.64517e19	1.27687	0.638434	−	0.769677i	$-0.279583\pi$
$937$	2.64517e19	1.27687	0.638434	+	0.769677i	$0.279583\pi$
$938$	0	0
$939$	−5.48152e19	−2.60960
$940$	0	0
$941$	−2.39169e19	−1.12298	−0.561489	−	0.827484i	$-0.689771\pi$
$941$	−2.39169e19	−1.12298	−0.561489	+	0.827484i	$0.689771\pi$
$942$	0	0
$943$	−7.70420e18	−0.356781
$944$	0	0
$945$	−4.10618e19	−1.87556
$946$	0	0
$947$	3.42855e19	1.54467	0.772336	−	0.635214i	$-0.219088\pi$
$947$	3.42855e19	1.54467	0.772336	+	0.635214i	$0.219088\pi$
$948$	0	0
$949$	−2.01040e19	−0.893411
$950$	0	0
$951$	2.60595e19	1.14233
$952$	0	0
$953$	2.90816e19	1.25752	0.628760	−	0.777599i	$-0.283563\pi$
$953$	2.90816e19	1.25752	0.628760	+	0.777599i	$0.283563\pi$
$954$	0	0
$955$	1.43921e19	0.613907
$956$	0	0
$957$	−8.36427e19	−3.51966
$958$	0	0
$959$	−3.04056e19	−1.26221
$960$	0	0
$961$	−2.43640e19	−0.997807
$962$	0	0
$963$	−5.46604e19	−2.20852
$964$	0	0
$965$	1.43116e19	0.570508
$966$	0	0
$967$	−9.38894e17	−0.0369270	−0.0184635	−	0.999830i	$-0.505877\pi$
$967$	−9.38894e17	−0.0369270	−0.0184635	+	0.999830i	$0.505877\pi$
$968$	0	0
$969$	4.34723e18	0.168697
$970$	0	0
$971$	4.07232e19	1.55925	0.779626	−	0.626245i	$-0.215409\pi$
$971$	4.07232e19	1.55925	0.779626	+	0.626245i	$0.215409\pi$
$972$	0	0
$973$	2.48562e19	0.939077
$974$	0	0
$975$	5.76828e18	0.215038
$976$	0	0
$977$	−1.18079e19	−0.434370	−0.217185	−	0.976131i	$-0.569687\pi$
$977$	−1.18079e19	−0.434370	−0.217185	+	0.976131i	$0.569687\pi$
$978$	0	0
$979$	−2.95657e19	−1.07325
$980$	0	0
$981$	1.12084e20	4.01507
$982$	0	0
$983$	3.30186e19	1.16724	0.583621	−	0.812026i	$-0.301635\pi$
$983$	3.30186e19	1.16724	0.583621	+	0.812026i	$0.301635\pi$
$984$	0	0
$985$	6.12724e18	0.213761
$986$	0	0
$987$	1.72628e19	0.594362
$988$	0	0
$989$	1.06505e18	0.0361904
$990$	0	0
$991$	1.46516e19	0.491367	0.245684	−	0.969350i	$-0.420988\pi$
$991$	1.46516e19	0.491367	0.245684	+	0.969350i	$0.420988\pi$
$992$	0	0
$993$	−3.55761e19	−1.17758
$994$	0	0
$995$	−9.26825e18	−0.302794
$996$	0	0
$997$	2.12523e19	0.685312	0.342656	−	0.939461i	$-0.388674\pi$
$997$	2.12523e19	0.685312	0.342656	+	0.939461i	$0.388674\pi$
$998$	0	0
$999$	−9.45039e19	−3.00797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.14.a.b.1.2	✓	2
3.2	odd	2		72.14.a.c.1.1		2
4.3	odd	2		16.14.a.e.1.1		2
5.2	odd	4		200.14.c.b.49.1		4
5.3	odd	4		200.14.c.b.49.4		4
5.4	even	2		200.14.a.b.1.1		2
8.3	odd	2		64.14.a.l.1.2		2
8.5	even	2		64.14.a.j.1.1		2
12.11	even	2		144.14.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.14.a.b.1.2	✓	2	1.1	even	1	trivial
16.14.a.e.1.1		2	4.3	odd	2
64.14.a.j.1.1		2	8.5	even	2
64.14.a.l.1.2		2	8.3	odd	2
72.14.a.c.1.1		2	3.2	odd	2
144.14.a.n.1.1		2	12.11	even	2
200.14.a.b.1.1		2	5.4	even	2
200.14.c.b.49.1		4	5.2	odd	4
200.14.c.b.49.4		4	5.3	odd	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:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q+2224.57 q^{3} +30700.8 q^{5} -341598. q^{7} +3.35438e6 q^{9} +O(q^{10})\)	\(q+2224.57 q^{3} +30700.8 q^{5} -341598. q^{7} +3.35438e6 q^{9} +9.18675e6 q^{11} -9.32183e6 q^{13} +6.82961e7 q^{15} -8.60213e6 q^{17} -2.27175e8 q^{19} -7.59908e8 q^{21} +5.58492e8 q^{23} -2.78163e8 q^{25} +3.91537e9 q^{27} -4.09280e9 q^{29} +2.31420e8 q^{31} +2.04366e10 q^{33} -1.04873e10 q^{35} -2.41367e10 q^{37} -2.07370e10 q^{39} -1.37946e10 q^{41} +1.90701e9 q^{43} +1.02982e11 q^{45} -2.27170e10 q^{47} +1.98003e10 q^{49} -1.91360e10 q^{51} +7.19059e10 q^{53} +2.82041e11 q^{55} -5.05367e11 q^{57} +3.71648e11 q^{59} -3.48930e11 q^{61} -1.14585e12 q^{63} -2.86188e11 q^{65} +8.75836e11 q^{67} +1.24240e12 q^{69} +1.10711e11 q^{71} +2.15665e12 q^{73} -6.18792e11 q^{75} -3.13818e12 q^{77} -3.20959e11 q^{79} +3.36204e12 q^{81} +1.63705e12 q^{83} -2.64092e11 q^{85} -9.10471e12 q^{87} -3.21830e12 q^{89} +3.18432e12 q^{91} +5.14810e11 q^{93} -6.97447e12 q^{95} -3.93052e12 q^{97} +3.08159e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 872 q^{3} + 18476 q^{5} + 110928 q^{7} + 3589498 q^{9}+O(q^{10}) \) 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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