LMFDB - Embedded newform 128.12.a.e.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [128,12,Mod(1,128)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("128.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	128.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-20,0,-1804] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$98.3479271116$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 4442x^{4} + 153566x^{3} - 1333532x^{2} - 4433532x + 49754286 $ x^6 - 2x^5 - 4442x^4 + 153566x^3 - 1333532x^2 - 4433532*x + 49754286
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{43}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-5.65189$ of defining polynomial
Character	$\chi$	$=$	128.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+551.275 q^{3} -3654.34 q^{5} -40487.8 q^{7} +126757. q^{9} +786434. q^{11} -7304.56 q^{13} -2.01455e6 q^{15} +2.76028e6 q^{17} +5.35085e6 q^{19} -2.23199e7 q^{21} -1.21321e7 q^{23} -3.54739e7 q^{25} -2.77786e7 q^{27} +4.12196e7 q^{29} -2.16997e8 q^{31} +4.33541e8 q^{33} +1.47956e8 q^{35} +7.29218e8 q^{37} -4.02682e6 q^{39} +7.55803e8 q^{41} +1.08585e9 q^{43} -4.63214e8 q^{45} -8.69907e8 q^{47} -3.38067e8 q^{49} +1.52167e9 q^{51} +4.14610e9 q^{53} -2.87390e9 q^{55} +2.94979e9 q^{57} +8.16388e9 q^{59} +2.17668e9 q^{61} -5.13212e9 q^{63} +2.66933e7 q^{65} -4.75936e9 q^{67} -6.68811e9 q^{69} +1.55138e10 q^{71} +1.64559e10 q^{73} -1.95559e10 q^{75} -3.18410e10 q^{77} +3.86007e10 q^{79} -3.77683e10 q^{81} -9.96841e9 q^{83} -1.00870e10 q^{85} +2.27234e10 q^{87} +5.42764e10 q^{89} +2.95745e8 q^{91} -1.19625e11 q^{93} -1.95538e10 q^{95} -1.28140e11 q^{97} +9.96862e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 20 q^{3} - 1804 q^{5} + 49368 q^{7} + 313814 q^{9} - 688460 q^{11} - 2290348 q^{13} + 4828264 q^{15} + 4127636 q^{17} + 9936364 q^{19} + 20325616 q^{21} + 9921320 q^{23} + 51633002 q^{25} - 132503384 q^{27}+ \cdots + 240482467988 q^{99}+O(q^{100}) $ 6 * q - 20 * q^3 - 1804 * q^5 + 49368 * q^7 + 313814 * q^9 - 688460 * q^11 - 2290348 * q^13 + 4828264 * q^15 + 4127636 * q^17 + 9936364 * q^19 + 20325616 * q^21 + 9921320 * q^23 + 51633002 * q^25 - 132503384 * q^27 + 25222532 * q^29 - 246133120 * q^31 - 337991288 * q^33 + 227061200 * q^35 - 660146172 * q^37 + 648086056 * q^39 + 486915836 * q^41 + 1147261348 * q^43 + 626323028 * q^45 + 603975088 * q^47 + 1846879062 * q^49 - 1556268056 * q^51 - 428043996 * q^53 - 1355415976 * q^55 - 783352648 * q^57 + 809510676 * q^59 - 2822052332 * q^61 + 14526601496 * q^63 + 1376905688 * q^65 + 9742675036 * q^67 - 8441748208 * q^69 + 16181067512 * q^71 + 35963668692 * q^73 + 51500880116 * q^75 - 72754679408 * q^77 - 6103946192 * q^79 + 81240620990 * q^81 + 67929536124 * q^83 - 76741413928 * q^85 + 132686631816 * q^87 + 22721416116 * q^89 + 44438236496 * q^91 - 37482073344 * q^93 + 282717822568 * q^95 + 218896625492 * q^97 + 240482467988 * q^99

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
$3$	551.275	1.30979	0.654895	−	0.755720i	$-0.272713\pi$
$3$	551.275	1.30979	0.654895	+	0.755720i	$0.272713\pi$
$4$	0	0
$5$	−3654.34	−0.522967	−0.261483	−	0.965208i	$-0.584212\pi$
$5$	−3654.34	−0.522967	−0.261483	+	0.965208i	$0.584212\pi$
$6$	0	0
$7$	−40487.8	−0.910510	−0.455255	−	0.890361i	$-0.650452\pi$
$7$	−40487.8	−0.910510	−0.455255	+	0.890361i	$0.650452\pi$
$8$	0	0
$9$	126757.	0.715548
$10$	0	0
$11$	786434.	1.47232	0.736160	−	0.676807i	$-0.236637\pi$
$11$	786434.	1.47232	0.736160	+	0.676807i	$0.236637\pi$
$12$	0	0
$13$	−7304.56	−0.00545639	−0.00272820	−	0.999996i	$-0.500868\pi$
$13$	−7304.56	−0.00545639	−0.00272820	+	0.999996i	$0.500868\pi$
$14$	0	0
$15$	−2.01455e6	−0.684976
$16$	0	0
$17$	2.76028e6	0.471502	0.235751	−	0.971813i	$-0.424245\pi$
$17$	2.76028e6	0.471502	0.235751	+	0.971813i	$0.424245\pi$
$18$	0	0
$19$	5.35085e6	0.495767	0.247884	−	0.968790i	$-0.420265\pi$
$19$	5.35085e6	0.495767	0.247884	+	0.968790i	$0.420265\pi$
$20$	0	0
$21$	−2.23199e7	−1.19258
$22$	0	0
$23$	−1.21321e7	−0.393035	−0.196518	−	0.980500i	$-0.562963\pi$
$23$	−1.21321e7	−0.393035	−0.196518	+	0.980500i	$0.562963\pi$
$24$	0	0
$25$	−3.54739e7	−0.726506
$26$	0	0
$27$	−2.77786e7	−0.372572
$28$	0	0
$29$	4.12196e7	0.373177	0.186589	−	0.982438i	$-0.440257\pi$
$29$	4.12196e7	0.373177	0.186589	+	0.982438i	$0.440257\pi$
$30$	0	0
$31$	−2.16997e8	−1.36133	−0.680667	−	0.732593i	$-0.738310\pi$
$31$	−2.16997e8	−1.36133	−0.680667	+	0.732593i	$0.738310\pi$
$32$	0	0
$33$	4.33541e8	1.92843
$34$	0	0
$35$	1.47956e8	0.476166
$36$	0	0
$37$	7.29218e8	1.72881	0.864406	−	0.502794i	$-0.167694\pi$
$37$	7.29218e8	1.72881	0.864406	+	0.502794i	$0.167694\pi$
$38$	0	0
$39$	−4.02682e6	−0.00714673
$40$	0	0
$41$	7.55803e8	1.01882	0.509410	−	0.860524i	$-0.329864\pi$
$41$	7.55803e8	1.01882	0.509410	+	0.860524i	$0.329864\pi$
$42$	0	0
$43$	1.08585e9	1.12641	0.563203	−	0.826318i	$-0.309569\pi$
$43$	1.08585e9	1.12641	0.563203	+	0.826318i	$0.309569\pi$
$44$	0	0
$45$	−4.63214e8	−0.374208
$46$	0	0
$47$	−8.69907e8	−0.553267	−0.276633	−	0.960976i	$-0.589219\pi$
$47$	−8.69907e8	−0.553267	−0.276633	+	0.960976i	$0.589219\pi$
$48$	0	0
$49$	−3.38067e8	−0.170972
$50$	0	0
$51$	1.52167e9	0.617568
$52$	0	0
$53$	4.14610e9	1.36183	0.680914	−	0.732364i	$-0.261583\pi$
$53$	4.14610e9	1.36183	0.680914	+	0.732364i	$0.261583\pi$
$54$	0	0
$55$	−2.87390e9	−0.769974
$56$	0	0
$57$	2.94979e9	0.649351
$58$	0	0
$59$	8.16388e9	1.48666	0.743328	−	0.668927i	$-0.233246\pi$
$59$	8.16388e9	1.48666	0.743328	+	0.668927i	$0.233246\pi$
$60$	0	0
$61$	2.17668e9	0.329975	0.164987	−	0.986296i	$-0.447242\pi$
$61$	2.17668e9	0.329975	0.164987	+	0.986296i	$0.447242\pi$
$62$	0	0
$63$	−5.13212e9	−0.651514
$64$	0	0
$65$	2.66933e7	0.00285351
$66$	0	0
$67$	−4.75936e9	−0.430663	−0.215331	−	0.976541i	$-0.569083\pi$
$67$	−4.75936e9	−0.430663	−0.215331	+	0.976541i	$0.569083\pi$
$68$	0	0
$69$	−6.68811e9	−0.514794
$70$	0	0
$71$	1.55138e10	1.02046	0.510232	−	0.860037i	$-0.329560\pi$
$71$	1.55138e10	1.02046	0.510232	+	0.860037i	$0.329560\pi$
$72$	0	0
$73$	1.64559e10	0.929067	0.464533	−	0.885556i	$-0.346222\pi$
$73$	1.64559e10	0.929067	0.464533	+	0.885556i	$0.346222\pi$
$74$	0	0
$75$	−1.95559e10	−0.951570
$76$	0	0
$77$	−3.18410e10	−1.34056
$78$	0	0
$79$	3.86007e10	1.41139	0.705693	−	0.708518i	$-0.250636\pi$
$79$	3.86007e10	1.41139	0.705693	+	0.708518i	$0.250636\pi$
$80$	0	0
$81$	−3.77683e10	−1.20354
$82$	0	0
$83$	−9.96841e9	−0.277777	−0.138888	−	0.990308i	$-0.544353\pi$
$83$	−9.96841e9	−0.277777	−0.138888	+	0.990308i	$0.544353\pi$
$84$	0	0
$85$	−1.00870e10	−0.246580
$86$	0	0
$87$	2.27234e10	0.488784
$88$	0	0
$89$	5.42764e10	1.03030	0.515152	−	0.857099i	$-0.327735\pi$
$89$	5.42764e10	1.03030	0.515152	+	0.857099i	$0.327735\pi$
$90$	0	0
$91$	2.95745e8	0.00496810
$92$	0	0
$93$	−1.19625e11	−1.78306
$94$	0	0
$95$	−1.95538e10	−0.259270
$96$	0	0
$97$	−1.28140e11	−1.51510	−0.757549	−	0.652778i	$-0.773603\pi$
$97$	−1.28140e11	−1.51510	−0.757549	+	0.652778i	$0.773603\pi$
$98$	0	0
$99$	9.96862e10	1.05352
$100$	0	0
$101$	1.11960e11	1.05997	0.529986	−	0.848006i	$-0.322197\pi$
$101$	1.11960e11	1.05997	0.529986	+	0.848006i	$0.322197\pi$
$102$	0	0
$103$	7.18102e10	0.610354	0.305177	−	0.952296i	$-0.401284\pi$
$103$	7.18102e10	0.610354	0.305177	+	0.952296i	$0.401284\pi$
$104$	0	0
$105$	8.15645e10	0.623677
$106$	0	0
$107$	8.71862e10	0.600948	0.300474	−	0.953790i	$-0.402855\pi$
$107$	8.71862e10	0.600948	0.300474	+	0.953790i	$0.402855\pi$
$108$	0	0
$109$	−2.95112e11	−1.83714	−0.918568	−	0.395263i	$-0.870653\pi$
$109$	−2.95112e11	−1.83714	−0.918568	+	0.395263i	$0.870653\pi$
$110$	0	0
$111$	4.02000e11	2.26438
$112$	0	0
$113$	2.87657e11	1.46874	0.734368	−	0.678752i	$-0.237479\pi$
$113$	2.87657e11	1.46874	0.734368	+	0.678752i	$0.237479\pi$
$114$	0	0
$115$	4.43347e10	0.205544
$116$	0	0
$117$	−9.25906e8	−0.00390431
$118$	0	0
$119$	−1.11757e11	−0.429307
$120$	0	0
$121$	3.33167e11	1.16773
$122$	0	0
$123$	4.16656e11	1.33444
$124$	0	0
$125$	3.08068e11	0.902905
$126$	0	0
$127$	−2.35051e11	−0.631309	−0.315654	−	0.948874i	$-0.602224\pi$
$127$	−2.35051e11	−0.631309	−0.315654	+	0.948874i	$0.602224\pi$
$128$	0	0
$129$	5.98605e11	1.47536
$130$	0	0
$131$	5.97200e11	1.35247	0.676235	−	0.736686i	$-0.263610\pi$
$131$	5.97200e11	1.35247	0.676235	+	0.736686i	$0.263610\pi$
$132$	0	0
$133$	−2.16644e11	−0.451401
$134$	0	0
$135$	1.01513e11	0.194843
$136$	0	0
$137$	7.18124e11	1.27127	0.635633	−	0.771991i	$-0.280739\pi$
$137$	7.18124e11	1.27127	0.635633	+	0.771991i	$0.280739\pi$
$138$	0	0
$139$	5.88675e11	0.962264	0.481132	−	0.876648i	$-0.340226\pi$
$139$	5.88675e11	0.962264	0.481132	+	0.876648i	$0.340226\pi$
$140$	0	0
$141$	−4.79558e11	−0.724663
$142$	0	0
$143$	−5.74455e9	−0.00803356
$144$	0	0
$145$	−1.50631e11	−0.195159
$146$	0	0
$147$	−1.86368e11	−0.223937
$148$	0	0
$149$	−1.39301e12	−1.55392	−0.776960	−	0.629549i	$-0.783239\pi$
$149$	−1.39301e12	−1.55392	−0.776960	+	0.629549i	$0.783239\pi$
$150$	0	0
$151$	1.03117e12	1.06895	0.534477	−	0.845183i	$-0.320509\pi$
$151$	1.03117e12	1.06895	0.534477	+	0.845183i	$0.320509\pi$
$152$	0	0
$153$	3.49885e11	0.337382
$154$	0	0
$155$	7.92981e11	0.711932
$156$	0	0
$157$	−7.26716e11	−0.608018	−0.304009	−	0.952669i	$-0.598325\pi$
$157$	−7.26716e11	−0.608018	−0.304009	+	0.952669i	$0.598325\pi$
$158$	0	0
$159$	2.28564e12	1.78371
$160$	0	0
$161$	4.91200e11	0.357863
$162$	0	0
$163$	1.12891e12	0.768468	0.384234	−	0.923236i	$-0.374466\pi$
$163$	1.12891e12	0.768468	0.384234	+	0.923236i	$0.374466\pi$
$164$	0	0
$165$	−1.58431e12	−1.00850
$166$	0	0
$167$	9.81193e11	0.584539	0.292270	−	0.956336i	$-0.405590\pi$
$167$	9.81193e11	0.584539	0.292270	+	0.956336i	$0.405590\pi$
$168$	0	0
$169$	−1.79211e12	−0.999970
$170$	0	0
$171$	6.78259e11	0.354745
$172$	0	0
$173$	9.41584e11	0.461961	0.230981	−	0.972958i	$-0.425807\pi$
$173$	9.41584e11	0.461961	0.230981	+	0.972958i	$0.425807\pi$
$174$	0	0
$175$	1.43626e12	0.661491
$176$	0	0
$177$	4.50054e12	1.94721
$178$	0	0
$179$	−4.29763e12	−1.74798	−0.873991	−	0.485942i	$-0.838477\pi$
$179$	−4.29763e12	−1.74798	−0.873991	+	0.485942i	$0.838477\pi$
$180$	0	0
$181$	−2.08789e12	−0.798868	−0.399434	−	0.916762i	$-0.630793\pi$
$181$	−2.08789e12	−0.798868	−0.399434	+	0.916762i	$0.630793\pi$
$182$	0	0
$183$	1.19995e12	0.432197
$184$	0	0
$185$	−2.66481e12	−0.904111
$186$	0	0
$187$	2.17078e12	0.694202
$188$	0	0
$189$	1.12469e12	0.339230
$190$	0	0
$191$	4.60932e12	1.31206	0.656029	−	0.754736i	$-0.272235\pi$
$191$	4.60932e12	1.31206	0.656029	+	0.754736i	$0.272235\pi$
$192$	0	0
$193$	−1.69638e12	−0.455992	−0.227996	−	0.973662i	$-0.573217\pi$
$193$	−1.69638e12	−0.455992	−0.227996	+	0.973662i	$0.573217\pi$
$194$	0	0
$195$	1.47154e10	0.00373750
$196$	0	0
$197$	−4.63692e12	−1.11344	−0.556718	−	0.830701i	$-0.687940\pi$
$197$	−4.63692e12	−1.11344	−0.556718	+	0.830701i	$0.687940\pi$
$198$	0	0
$199$	−8.14553e12	−1.85024	−0.925119	−	0.379677i	$-0.876035\pi$
$199$	−8.14553e12	−1.85024	−0.925119	+	0.379677i	$0.876035\pi$
$200$	0	0
$201$	−2.62372e12	−0.564077
$202$	0	0
$203$	−1.66889e12	−0.339781
$204$	0	0
$205$	−2.76196e12	−0.532809
$206$	0	0
$207$	−1.53783e12	−0.281236
$208$	0	0
$209$	4.20809e12	0.729929
$210$	0	0
$211$	2.68527e12	0.442013	0.221006	−	0.975272i	$-0.429066\pi$
$211$	2.68527e12	0.442013	0.221006	+	0.975272i	$0.429066\pi$
$212$	0	0
$213$	8.55238e12	1.33659
$214$	0	0
$215$	−3.96808e12	−0.589073
$216$	0	0
$217$	8.78572e12	1.23951
$218$	0	0
$219$	9.07175e12	1.21688
$220$	0	0
$221$	−2.01626e10	−0.00257270
$222$	0	0
$223$	−3.26277e12	−0.396196	−0.198098	−	0.980182i	$-0.563476\pi$
$223$	−3.26277e12	−0.396196	−0.198098	+	0.980182i	$0.563476\pi$
$224$	0	0
$225$	−4.49658e12	−0.519850
$226$	0	0
$227$	−6.95158e12	−0.765493	−0.382746	−	0.923853i	$-0.625022\pi$
$227$	−6.95158e12	−0.765493	−0.382746	+	0.923853i	$0.625022\pi$
$228$	0	0
$229$	1.76851e13	1.85572	0.927859	−	0.372931i	$-0.121647\pi$
$229$	1.76851e13	1.85572	0.927859	+	0.372931i	$0.121647\pi$
$230$	0	0
$231$	−1.75531e13	−1.75585
$232$	0	0
$233$	7.22622e12	0.689372	0.344686	−	0.938718i	$-0.387985\pi$
$233$	7.22622e12	0.689372	0.344686	+	0.938718i	$0.387985\pi$
$234$	0	0
$235$	3.17894e12	0.289340
$236$	0	0
$237$	2.12796e13	1.84862
$238$	0	0
$239$	6.24761e12	0.518233	0.259117	−	0.965846i	$-0.416569\pi$
$239$	6.24761e12	0.518233	0.259117	+	0.965846i	$0.416569\pi$
$240$	0	0
$241$	−1.87686e13	−1.48709	−0.743545	−	0.668685i	$-0.766857\pi$
$241$	−1.87686e13	−1.48709	−0.743545	+	0.668685i	$0.766857\pi$
$242$	0	0
$243$	−1.58998e13	−1.20381
$244$	0	0
$245$	1.23541e12	0.0894126
$246$	0	0
$247$	−3.90856e10	−0.00270510
$248$	0	0
$249$	−5.49533e12	−0.363829
$250$	0	0
$251$	−1.05783e13	−0.670210	−0.335105	−	0.942181i	$-0.608772\pi$
$251$	−1.05783e13	−0.670210	−0.335105	+	0.942181i	$0.608772\pi$
$252$	0	0
$253$	−9.54107e12	−0.578674
$254$	0	0
$255$	−5.56071e12	−0.322968
$256$	0	0
$257$	−2.56310e13	−1.42605	−0.713023	−	0.701141i	$-0.752674\pi$
$257$	−2.56310e13	−1.42605	−0.713023	+	0.701141i	$0.752674\pi$
$258$	0	0
$259$	−2.95244e13	−1.57410
$260$	0	0
$261$	5.22489e12	0.267026
$262$	0	0
$263$	−2.53528e13	−1.24242	−0.621211	−	0.783643i	$-0.713359\pi$
$263$	−2.53528e13	−1.24242	−0.621211	+	0.783643i	$0.713359\pi$
$264$	0	0
$265$	−1.51512e13	−0.712190
$266$	0	0
$267$	2.99212e13	1.34948
$268$	0	0
$269$	−8.98689e12	−0.389020	−0.194510	−	0.980901i	$-0.562312\pi$
$269$	−8.98689e12	−0.389020	−0.194510	+	0.980901i	$0.562312\pi$
$270$	0	0
$271$	3.94576e13	1.63983	0.819916	−	0.572484i	$-0.194020\pi$
$271$	3.94576e13	1.63983	0.819916	+	0.572484i	$0.194020\pi$
$272$	0	0
$273$	1.63037e11	0.00650717
$274$	0	0
$275$	−2.78979e13	−1.06965
$276$	0	0
$277$	1.74960e13	0.644614	0.322307	−	0.946635i	$-0.395542\pi$
$277$	1.74960e13	0.644614	0.322307	+	0.946635i	$0.395542\pi$
$278$	0	0
$279$	−2.75059e13	−0.974099
$280$	0	0
$281$	−1.03034e13	−0.350830	−0.175415	−	0.984495i	$-0.556127\pi$
$281$	−1.03034e13	−0.350830	−0.175415	+	0.984495i	$0.556127\pi$
$282$	0	0
$283$	−5.18756e12	−0.169878	−0.0849391	−	0.996386i	$-0.527070\pi$
$283$	−5.18756e12	−0.169878	−0.0849391	+	0.996386i	$0.527070\pi$
$284$	0	0
$285$	−1.07795e13	−0.339589
$286$	0	0
$287$	−3.06008e13	−0.927646
$288$	0	0
$289$	−2.66528e13	−0.777686
$290$	0	0
$291$	−7.06405e13	−1.98446
$292$	0	0
$293$	−2.39851e12	−0.0648888	−0.0324444	−	0.999474i	$-0.510329\pi$
$293$	−2.39851e12	−0.0648888	−0.0324444	+	0.999474i	$0.510329\pi$
$294$	0	0
$295$	−2.98336e13	−0.777471
$296$	0	0
$297$	−2.18461e13	−0.548545
$298$	0	0
$299$	8.86194e10	0.00214456
$300$	0	0
$301$	−4.39638e13	−1.02560
$302$	0	0
$303$	6.17207e13	1.38834
$304$	0	0
$305$	−7.95433e12	−0.172566
$306$	0	0
$307$	−2.49590e13	−0.522356	−0.261178	−	0.965291i	$-0.584111\pi$
$307$	−2.49590e13	−0.522356	−0.261178	+	0.965291i	$0.584111\pi$
$308$	0	0
$309$	3.95872e13	0.799435
$310$	0	0
$311$	3.76443e13	0.733698	0.366849	−	0.930281i	$-0.380437\pi$
$311$	3.76443e13	0.733698	0.366849	+	0.930281i	$0.380437\pi$
$312$	0	0
$313$	−3.25337e12	−0.0612125	−0.0306063	−	0.999532i	$-0.509744\pi$
$313$	−3.25337e12	−0.0612125	−0.0306063	+	0.999532i	$0.509744\pi$
$314$	0	0
$315$	1.87545e13	0.340720
$316$	0	0
$317$	6.95267e13	1.21990	0.609952	−	0.792438i	$-0.291189\pi$
$317$	6.95267e13	1.21990	0.609952	+	0.792438i	$0.291189\pi$
$318$	0	0
$319$	3.24165e13	0.549436
$320$	0	0
$321$	4.80636e13	0.787116
$322$	0	0
$323$	1.47698e13	0.233755
$324$	0	0
$325$	2.59121e11	0.00396410
$326$	0	0
$327$	−1.62688e14	−2.40626
$328$	0	0
$329$	3.52206e13	0.503755
$330$	0	0
$331$	3.29777e13	0.456212	0.228106	−	0.973636i	$-0.426747\pi$
$331$	3.29777e13	0.456212	0.228106	+	0.973636i	$0.426747\pi$
$332$	0	0
$333$	9.24337e13	1.23705
$334$	0	0
$335$	1.73923e13	0.225222
$336$	0	0
$337$	9.62604e13	1.20638	0.603189	−	0.797599i	$-0.293897\pi$
$337$	9.62604e13	1.20638	0.603189	+	0.797599i	$0.293897\pi$
$338$	0	0
$339$	1.58578e14	1.92373
$340$	0	0
$341$	−1.70654e14	−2.00432
$342$	0	0
$343$	9.37451e13	1.06618
$344$	0	0
$345$	2.44406e13	0.269220
$346$	0	0
$347$	1.78648e14	1.90627	0.953137	−	0.302540i	$-0.0978345\pi$
$347$	1.78648e14	1.90627	0.953137	+	0.302540i	$0.0978345\pi$
$348$	0	0
$349$	9.24872e13	0.956185	0.478092	−	0.878310i	$-0.341328\pi$
$349$	9.24872e13	0.956185	0.478092	+	0.878310i	$0.341328\pi$
$350$	0	0
$351$	2.02911e11	0.00203290
$352$	0	0
$353$	7.79284e13	0.756719	0.378360	−	0.925659i	$-0.376488\pi$
$353$	7.79284e13	0.756719	0.378360	+	0.925659i	$0.376488\pi$
$354$	0	0
$355$	−5.66927e13	−0.533668
$356$	0	0
$357$	−6.16091e13	−0.562302
$358$	0	0
$359$	−1.85589e14	−1.64260	−0.821302	−	0.570494i	$-0.806752\pi$
$359$	−1.85589e14	−1.64260	−0.821302	+	0.570494i	$0.806752\pi$
$360$	0	0
$361$	−8.78587e13	−0.754215
$362$	0	0
$363$	1.83666e14	1.52948
$364$	0	0
$365$	−6.01356e13	−0.485871
$366$	0	0
$367$	1.51794e14	1.19012	0.595062	−	0.803680i	$-0.297128\pi$
$367$	1.51794e14	1.19012	0.595062	+	0.803680i	$0.297128\pi$
$368$	0	0
$369$	9.58035e13	0.729015
$370$	0	0
$371$	−1.67866e14	−1.23996
$372$	0	0
$373$	−1.01362e14	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$373$	−1.01362e14	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$374$	0	0
$375$	1.69830e14	1.18262
$376$	0	0
$377$	−3.01091e11	−0.00203620
$378$	0	0
$379$	−1.53470e14	−1.00811	−0.504055	−	0.863672i	$-0.668159\pi$
$379$	−1.53470e14	−1.00811	−0.504055	+	0.863672i	$0.668159\pi$
$380$	0	0
$381$	−1.29578e14	−0.826881
$382$	0	0
$383$	2.10193e14	1.30324	0.651622	−	0.758544i	$-0.274089\pi$
$383$	2.10193e14	1.30324	0.651622	+	0.758544i	$0.274089\pi$
$384$	0	0
$385$	1.16358e14	0.701069
$386$	0	0
$387$	1.37640e14	0.805999
$388$	0	0
$389$	−2.05132e13	−0.116765	−0.0583823	−	0.998294i	$-0.518594\pi$
$389$	−2.05132e13	−0.116765	−0.0583823	+	0.998294i	$0.518594\pi$
$390$	0	0
$391$	−3.34879e13	−0.185317
$392$	0	0
$393$	3.29221e14	1.77145
$394$	0	0
$395$	−1.41060e14	−0.738108
$396$	0	0
$397$	−2.03637e14	−1.03636	−0.518178	−	0.855273i	$-0.673389\pi$
$397$	−2.03637e14	−1.03636	−0.518178	+	0.855273i	$0.673389\pi$
$398$	0	0
$399$	−1.19430e14	−0.591240
$400$	0	0
$401$	−3.43647e14	−1.65508	−0.827538	−	0.561409i	$-0.810259\pi$
$401$	−3.43647e14	−1.65508	−0.827538	+	0.561409i	$0.810259\pi$
$402$	0	0
$403$	1.58507e12	0.00742797
$404$	0	0
$405$	1.38018e14	0.629411
$406$	0	0
$407$	5.73482e14	2.54537
$408$	0	0
$409$	−5.75859e13	−0.248793	−0.124396	−	0.992233i	$-0.539699\pi$
$409$	−5.75859e13	−0.248793	−0.124396	+	0.992233i	$0.539699\pi$
$410$	0	0
$411$	3.95884e14	1.66509
$412$	0	0
$413$	−3.30537e14	−1.35361
$414$	0	0
$415$	3.64279e13	0.145268
$416$	0	0
$417$	3.24522e14	1.26036
$418$	0	0
$419$	4.77562e13	0.180656	0.0903280	−	0.995912i	$-0.471208\pi$
$419$	4.77562e13	0.180656	0.0903280	+	0.995912i	$0.471208\pi$
$420$	0	0
$421$	−1.98821e14	−0.732676	−0.366338	−	0.930482i	$-0.619389\pi$
$421$	−1.98821e14	−0.732676	−0.366338	+	0.930482i	$0.619389\pi$
$422$	0	0
$423$	−1.10267e14	−0.395889
$424$	0	0
$425$	−9.79179e13	−0.342549
$426$	0	0
$427$	−8.81290e13	−0.300445
$428$	0	0
$429$	−3.16683e12	−0.0105223
$430$	0	0
$431$	2.95924e14	0.958419	0.479210	−	0.877701i	$-0.340923\pi$
$431$	2.95924e14	0.958419	0.479210	+	0.877701i	$0.340923\pi$
$432$	0	0
$433$	−1.46462e14	−0.462425	−0.231212	−	0.972903i	$-0.574269\pi$
$433$	−1.46462e14	−0.462425	−0.231212	+	0.972903i	$0.574269\pi$
$434$	0	0
$435$	−8.30389e13	−0.255617
$436$	0	0
$437$	−6.49169e13	−0.194854
$438$	0	0
$439$	2.74004e14	0.802052	0.401026	−	0.916067i	$-0.368654\pi$
$439$	2.74004e14	0.802052	0.401026	+	0.916067i	$0.368654\pi$
$440$	0	0
$441$	−4.28525e13	−0.122339
$442$	0	0
$443$	−3.76008e14	−1.04707	−0.523536	−	0.852004i	$-0.675387\pi$
$443$	−3.76008e14	−1.04707	−0.523536	+	0.852004i	$0.675387\pi$
$444$	0	0
$445$	−1.98344e14	−0.538815
$446$	0	0
$447$	−7.67930e14	−2.03531
$448$	0	0
$449$	−6.84249e14	−1.76954	−0.884768	−	0.466033i	$-0.845683\pi$
$449$	−6.84249e14	−1.76954	−0.884768	+	0.466033i	$0.845683\pi$
$450$	0	0
$451$	5.94389e14	1.50003
$452$	0	0
$453$	5.68461e14	1.40010
$454$	0	0
$455$	−1.08075e12	−0.00259815
$456$	0	0
$457$	1.10262e12	0.00258753	0.00129377	−	0.999999i	$-0.499588\pi$
$457$	1.10262e12	0.00258753	0.00129377	+	0.999999i	$0.499588\pi$
$458$	0	0
$459$	−7.66767e13	−0.175668
$460$	0	0
$461$	3.06439e14	0.685471	0.342735	−	0.939432i	$-0.388647\pi$
$461$	3.06439e14	0.685471	0.342735	+	0.939432i	$0.388647\pi$
$462$	0	0
$463$	7.60027e13	0.166010	0.0830049	−	0.996549i	$-0.473548\pi$
$463$	7.60027e13	0.166010	0.0830049	+	0.996549i	$0.473548\pi$
$464$	0	0
$465$	4.37150e14	0.932480
$466$	0	0
$467$	9.91065e13	0.206471	0.103236	−	0.994657i	$-0.467080\pi$
$467$	9.91065e13	0.206471	0.103236	+	0.994657i	$0.467080\pi$
$468$	0	0
$469$	1.92696e14	0.392122
$470$	0	0
$471$	−4.00620e14	−0.796376
$472$	0	0
$473$	8.53953e14	1.65843
$474$	0	0
$475$	−1.89816e14	−0.360178
$476$	0	0
$477$	5.25548e14	0.974453
$478$	0	0
$479$	−6.81262e14	−1.23444	−0.617219	−	0.786792i	$-0.711741\pi$
$479$	−6.81262e14	−1.23444	−0.617219	+	0.786792i	$0.711741\pi$
$480$	0	0
$481$	−5.32662e12	−0.00943308
$482$	0	0
$483$	2.70787e14	0.468725
$484$	0	0
$485$	4.68268e14	0.792346
$486$	0	0
$487$	−6.88411e14	−1.13878	−0.569388	−	0.822069i	$-0.692820\pi$
$487$	−6.88411e14	−1.13878	−0.569388	+	0.822069i	$0.692820\pi$
$488$	0	0
$489$	6.22337e14	1.00653
$490$	0	0
$491$	−1.86593e14	−0.295085	−0.147542	−	0.989056i	$-0.547136\pi$
$491$	−1.86593e14	−0.295085	−0.147542	+	0.989056i	$0.547136\pi$
$492$	0	0
$493$	1.13778e14	0.175954
$494$	0	0
$495$	−3.64287e14	−0.550954
$496$	0	0
$497$	−6.28120e14	−0.929142
$498$	0	0
$499$	−1.23594e14	−0.178831	−0.0894156	−	0.995994i	$-0.528500\pi$
$499$	−1.23594e14	−0.178831	−0.0894156	+	0.995994i	$0.528500\pi$
$500$	0	0
$501$	5.40907e14	0.765624
$502$	0	0
$503$	7.47743e14	1.03545	0.517724	−	0.855548i	$-0.326779\pi$
$503$	7.47743e14	1.03545	0.517724	+	0.855548i	$0.326779\pi$
$504$	0	0
$505$	−4.09139e14	−0.554330
$506$	0	0
$507$	−9.87944e14	−1.30975
$508$	0	0
$509$	−4.13938e13	−0.0537017	−0.0268508	−	0.999639i	$-0.508548\pi$
$509$	−4.13938e13	−0.0537017	−0.0268508	+	0.999639i	$0.508548\pi$
$510$	0	0
$511$	−6.66264e14	−0.845925
$512$	0	0
$513$	−1.48639e14	−0.184709
$514$	0	0
$515$	−2.62419e14	−0.319194
$516$	0	0
$517$	−6.84124e14	−0.814586
$518$	0	0
$519$	5.19072e14	0.605072
$520$	0	0
$521$	−1.04645e15	−1.19430	−0.597149	−	0.802130i	$-0.703700\pi$
$521$	−1.04645e15	−1.19430	−0.597149	+	0.802130i	$0.703700\pi$
$522$	0	0
$523$	−1.23919e15	−1.38478	−0.692388	−	0.721525i	$-0.743442\pi$
$523$	−1.23919e15	−1.38478	−0.692388	+	0.721525i	$0.743442\pi$
$524$	0	0
$525$	7.91774e14	0.866414
$526$	0	0
$527$	−5.98972e14	−0.641871
$528$	0	0
$529$	−8.05623e14	−0.845523
$530$	0	0
$531$	1.03483e15	1.06377
$532$	0	0
$533$	−5.52081e12	−0.00555909
$534$	0	0
$535$	−3.18608e14	−0.314276
$536$	0	0
$537$	−2.36917e15	−2.28949
$538$	0	0
$539$	−2.65867e14	−0.251725
$540$	0	0
$541$	−4.83405e14	−0.448462	−0.224231	−	0.974536i	$-0.571987\pi$
$541$	−4.83405e14	−0.448462	−0.224231	+	0.974536i	$0.571987\pi$
$542$	0	0
$543$	−1.15100e15	−1.04635
$544$	0	0
$545$	1.07844e15	0.960761
$546$	0	0
$547$	−4.12315e14	−0.359997	−0.179998	−	0.983667i	$-0.557609\pi$
$547$	−4.12315e14	−0.359997	−0.179998	+	0.983667i	$0.557609\pi$
$548$	0	0
$549$	2.75910e14	0.236113
$550$	0	0
$551$	2.20560e14	0.185009
$552$	0	0
$553$	−1.56285e15	−1.28508
$554$	0	0
$555$	−1.46904e15	−1.18420
$556$	0	0
$557$	6.38861e14	0.504897	0.252449	−	0.967610i	$-0.418764\pi$
$557$	6.38861e14	0.504897	0.252449	+	0.967610i	$0.418764\pi$
$558$	0	0
$559$	−7.93169e12	−0.00614612
$560$	0	0
$561$	1.19669e15	0.909259
$562$	0	0
$563$	5.82711e13	0.0434167	0.0217083	−	0.999764i	$-0.493089\pi$
$563$	5.82711e13	0.0434167	0.0217083	+	0.999764i	$0.493089\pi$
$564$	0	0
$565$	−1.05120e15	−0.768099
$566$	0	0
$567$	1.52916e15	1.09583
$568$	0	0
$569$	−7.62456e14	−0.535917	−0.267959	−	0.963430i	$-0.586349\pi$
$569$	−7.62456e14	−0.535917	−0.267959	+	0.963430i	$0.586349\pi$
$570$	0	0
$571$	−4.96039e14	−0.341993	−0.170996	−	0.985272i	$-0.554699\pi$
$571$	−4.96039e14	−0.341993	−0.170996	+	0.985272i	$0.554699\pi$
$572$	0	0
$573$	2.54100e15	1.71852
$574$	0	0
$575$	4.30372e14	0.285543
$576$	0	0
$577$	−1.21340e15	−0.789835	−0.394917	−	0.918717i	$-0.629227\pi$
$577$	−1.21340e15	−0.789835	−0.394917	+	0.918717i	$0.629227\pi$
$578$	0	0
$579$	−9.35171e14	−0.597254
$580$	0	0
$581$	4.03599e14	0.252919
$582$	0	0
$583$	3.26063e15	2.00505
$584$	0	0
$585$	3.38358e12	0.00204183
$586$	0	0
$587$	2.52382e14	0.149468	0.0747341	−	0.997203i	$-0.476189\pi$
$587$	2.52382e14	0.149468	0.0747341	+	0.997203i	$0.476189\pi$
$588$	0	0
$589$	−1.16112e15	−0.674904
$590$	0	0
$591$	−2.55622e15	−1.45837
$592$	0	0
$593$	1.27472e15	0.713862	0.356931	−	0.934131i	$-0.383823\pi$
$593$	1.27472e15	0.713862	0.356931	+	0.934131i	$0.383823\pi$
$594$	0	0
$595$	4.08400e14	0.224513
$596$	0	0
$597$	−4.49043e15	−2.42342
$598$	0	0
$599$	−1.41976e15	−0.752258	−0.376129	−	0.926567i	$-0.622745\pi$
$599$	−1.41976e15	−0.752258	−0.376129	+	0.926567i	$0.622745\pi$
$600$	0	0
$601$	3.50205e15	1.82185	0.910924	−	0.412573i	$-0.135370\pi$
$601$	3.50205e15	1.82185	0.910924	+	0.412573i	$0.135370\pi$
$602$	0	0
$603$	−6.03283e14	−0.308160
$604$	0	0
$605$	−1.21750e15	−0.610683
$606$	0	0
$607$	3.12148e15	1.53753	0.768764	−	0.639533i	$-0.220872\pi$
$607$	3.12148e15	1.53753	0.768764	+	0.639533i	$0.220872\pi$
$608$	0	0
$609$	−9.20018e14	−0.445042
$610$	0	0
$611$	6.35429e12	0.00301884
$612$	0	0
$613$	3.38076e15	1.57755	0.788773	−	0.614685i	$-0.210717\pi$
$613$	3.38076e15	1.57755	0.788773	+	0.614685i	$0.210717\pi$
$614$	0	0
$615$	−1.52260e15	−0.697868
$616$	0	0
$617$	−3.27266e14	−0.147344	−0.0736720	−	0.997283i	$-0.523472\pi$
$617$	−3.27266e14	−0.147344	−0.0736720	+	0.997283i	$0.523472\pi$
$618$	0	0
$619$	2.13718e15	0.945241	0.472621	−	0.881266i	$-0.343308\pi$
$619$	2.13718e15	0.945241	0.472621	+	0.881266i	$0.343308\pi$
$620$	0	0
$621$	3.37012e14	0.146434
$622$	0	0
$623$	−2.19753e15	−0.938103
$624$	0	0
$625$	6.06339e14	0.254317
$626$	0	0
$627$	2.31981e15	0.956053
$628$	0	0
$629$	2.01284e15	0.815139
$630$	0	0
$631$	−3.36186e15	−1.33788	−0.668941	−	0.743316i	$-0.733252\pi$
$631$	−3.36186e15	−1.33788	−0.668941	+	0.743316i	$0.733252\pi$
$632$	0	0
$633$	1.48032e15	0.578943
$634$	0	0
$635$	8.58956e14	0.330153
$636$	0	0
$637$	2.46943e12	0.000932890 0
$638$	0	0
$639$	1.96649e15	0.730191
$640$	0	0
$641$	−6.56033e13	−0.0239445	−0.0119723	−	0.999928i	$-0.503811\pi$
$641$	−6.56033e13	−0.0239445	−0.0119723	+	0.999928i	$0.503811\pi$
$642$	0	0
$643$	2.63738e15	0.946263	0.473131	−	0.880992i	$-0.343124\pi$
$643$	2.63738e15	0.946263	0.473131	+	0.880992i	$0.343124\pi$
$644$	0	0
$645$	−2.18751e15	−0.771562
$646$	0	0
$647$	2.09388e15	0.726069	0.363035	−	0.931776i	$-0.381741\pi$
$647$	2.09388e15	0.726069	0.363035	+	0.931776i	$0.381741\pi$
$648$	0	0
$649$	6.42035e15	2.18883
$650$	0	0
$651$	4.84335e15	1.62349
$652$	0	0
$653$	−1.49359e15	−0.492276	−0.246138	−	0.969235i	$-0.579162\pi$
$653$	−1.49359e15	−0.492276	−0.246138	+	0.969235i	$0.579162\pi$
$654$	0	0
$655$	−2.18237e15	−0.707296
$656$	0	0
$657$	2.08591e15	0.664792
$658$	0	0
$659$	1.48461e15	0.465310	0.232655	−	0.972559i	$-0.425259\pi$
$659$	1.48461e15	0.465310	0.232655	+	0.972559i	$0.425259\pi$
$660$	0	0
$661$	2.19787e15	0.677475	0.338737	−	0.940881i	$-0.390000\pi$
$661$	2.19787e15	0.677475	0.338737	+	0.940881i	$0.390000\pi$
$662$	0	0
$663$	−1.11151e13	−0.00336970
$664$	0	0
$665$	7.91691e14	0.236068
$666$	0	0
$667$	−5.00079e14	−0.146672
$668$	0	0
$669$	−1.79868e15	−0.518933
$670$	0	0
$671$	1.71182e15	0.485829
$672$	0	0
$673$	1.64475e14	0.0459216	0.0229608	−	0.999736i	$-0.492691\pi$
$673$	1.64475e14	0.0459216	0.0229608	+	0.999736i	$0.492691\pi$
$674$	0	0
$675$	9.85417e14	0.270676
$676$	0	0
$677$	−9.91084e14	−0.267838	−0.133919	−	0.990992i	$-0.542756\pi$
$677$	−9.91084e14	−0.267838	−0.133919	+	0.990992i	$0.542756\pi$
$678$	0	0
$679$	5.18811e15	1.37951
$680$	0	0
$681$	−3.83223e15	−1.00263
$682$	0	0
$683$	−1.73762e15	−0.447344	−0.223672	−	0.974664i	$-0.571804\pi$
$683$	−1.73762e15	−0.447344	−0.223672	+	0.974664i	$0.571804\pi$
$684$	0	0
$685$	−2.62427e15	−0.664830
$686$	0	0
$687$	9.74935e15	2.43060
$688$	0	0
$689$	−3.02854e13	−0.00743067
$690$	0	0
$691$	1.27029e15	0.306743	0.153371	−	0.988169i	$-0.450987\pi$
$691$	1.27029e15	0.306743	0.153371	+	0.988169i	$0.450987\pi$
$692$	0	0
$693$	−4.03607e15	−0.959237
$694$	0	0
$695$	−2.15122e15	−0.503232
$696$	0	0
$697$	2.08623e15	0.480376
$698$	0	0
$699$	3.98364e15	0.902933
$700$	0	0
$701$	−6.91240e15	−1.54234	−0.771169	−	0.636631i	$-0.780328\pi$
$701$	−6.91240e15	−1.54234	−0.771169	+	0.636631i	$0.780328\pi$
$702$	0	0
$703$	3.90194e15	0.857089
$704$	0	0
$705$	1.75247e15	0.378974
$706$	0	0
$707$	−4.53300e15	−0.965116
$708$	0	0
$709$	5.79605e15	1.21500	0.607502	−	0.794318i	$-0.292172\pi$
$709$	5.79605e15	1.21500	0.607502	+	0.794318i	$0.292172\pi$
$710$	0	0
$711$	4.89291e15	1.00991
$712$	0	0
$713$	2.63262e15	0.535052
$714$	0	0
$715$	2.09926e13	0.00420128
$716$	0	0
$717$	3.44415e15	0.678776
$718$	0	0
$719$	2.64053e14	0.0512486	0.0256243	−	0.999672i	$-0.491843\pi$
$719$	2.64053e14	0.0512486	0.0256243	+	0.999672i	$0.491843\pi$
$720$	0	0
$721$	−2.90744e15	−0.555733
$722$	0	0
$723$	−1.03466e16	−1.94778
$724$	0	0
$725$	−1.46222e15	−0.271115
$726$	0	0
$727$	−9.04171e15	−1.65124	−0.825622	−	0.564224i	$-0.809176\pi$
$727$	−9.04171e15	−1.65124	−0.825622	+	0.564224i	$0.809176\pi$
$728$	0	0
$729$	−2.07464e15	−0.373200
$730$	0	0
$731$	2.99726e15	0.531103
$732$	0	0
$733$	−7.18071e15	−1.25342	−0.626709	−	0.779254i	$-0.715598\pi$
$733$	−7.18071e15	−1.25342	−0.626709	+	0.779254i	$0.715598\pi$
$734$	0	0
$735$	6.81052e14	0.117112
$736$	0	0
$737$	−3.74292e15	−0.634073
$738$	0	0
$739$	4.88569e15	0.815419	0.407710	−	0.913112i	$-0.366328\pi$
$739$	4.88569e15	0.815419	0.407710	+	0.913112i	$0.366328\pi$
$740$	0	0
$741$	−2.15469e13	−0.00354311
$742$	0	0
$743$	−2.69418e15	−0.436504	−0.218252	−	0.975892i	$-0.570035\pi$
$743$	−2.69418e15	−0.436504	−0.218252	+	0.975892i	$0.570035\pi$
$744$	0	0
$745$	5.09052e15	0.812649
$746$	0	0
$747$	−1.26357e15	−0.198763
$748$	0	0
$749$	−3.52998e15	−0.547169
$750$	0	0
$751$	5.17450e15	0.790403	0.395201	−	0.918594i	$-0.370675\pi$
$751$	5.17450e15	0.790403	0.395201	+	0.918594i	$0.370675\pi$
$752$	0	0
$753$	−5.83156e15	−0.877834
$754$	0	0
$755$	−3.76826e15	−0.559027
$756$	0	0
$757$	−7.96286e15	−1.16424	−0.582119	−	0.813104i	$-0.697776\pi$
$757$	−7.96286e15	−1.16424	−0.582119	+	0.813104i	$0.697776\pi$
$758$	0	0
$759$	−5.25975e15	−0.757941
$760$	0	0
$761$	−1.12233e15	−0.159406	−0.0797032	−	0.996819i	$-0.525397\pi$
$761$	−1.12233e15	−0.159406	−0.0797032	+	0.996819i	$0.525397\pi$
$762$	0	0
$763$	1.19484e16	1.67273
$764$	0	0
$765$	−1.27860e15	−0.176440
$766$	0	0
$767$	−5.96335e13	−0.00811178
$768$	0	0
$769$	4.36417e15	0.585204	0.292602	−	0.956234i	$-0.405479\pi$
$769$	4.36417e15	0.585204	0.292602	+	0.956234i	$0.405479\pi$
$770$	0	0
$771$	−1.41297e16	−1.86782
$772$	0	0
$773$	9.18355e15	1.19680	0.598402	−	0.801196i	$-0.295802\pi$
$773$	9.18355e15	1.19680	0.598402	+	0.801196i	$0.295802\pi$
$774$	0	0
$775$	7.69774e15	0.989017
$776$	0	0
$777$	−1.62761e16	−2.06174
$778$	0	0
$779$	4.04419e15	0.505098
$780$	0	0
$781$	1.22006e16	1.50245
$782$	0	0
$783$	−1.14502e15	−0.139035
$784$	0	0
$785$	2.65567e15	0.317973
$786$	0	0
$787$	−9.76151e15	−1.15254	−0.576270	−	0.817260i	$-0.695492\pi$
$787$	−9.76151e15	−1.15254	−0.576270	+	0.817260i	$0.695492\pi$
$788$	0	0
$789$	−1.39764e16	−1.62731
$790$	0	0
$791$	−1.16466e16	−1.33730
$792$	0	0
$793$	−1.58997e13	−0.00180047
$794$	0	0
$795$	−8.35250e15	−0.932819
$796$	0	0
$797$	1.38967e16	1.53070	0.765349	−	0.643615i	$-0.222566\pi$
$797$	1.38967e16	1.53070	0.765349	+	0.643615i	$0.222566\pi$
$798$	0	0
$799$	−2.40118e15	−0.260866
$800$	0	0
$801$	6.87992e15	0.737233
$802$	0	0
$803$	1.29415e16	1.36788
$804$	0	0
$805$	−1.79501e15	−0.187150
$806$	0	0
$807$	−4.95425e15	−0.509534
$808$	0	0
$809$	1.32141e16	1.34067	0.670334	−	0.742060i	$-0.266151\pi$
$809$	1.32141e16	1.34067	0.670334	+	0.742060i	$0.266151\pi$
$810$	0	0
$811$	5.53890e15	0.554381	0.277191	−	0.960815i	$-0.410597\pi$
$811$	5.53890e15	0.554381	0.277191	+	0.960815i	$0.410597\pi$
$812$	0	0
$813$	2.17520e16	2.14783
$814$	0	0
$815$	−4.12540e15	−0.401883
$816$	0	0
$817$	5.81025e15	0.558436
$818$	0	0
$819$	3.74879e13	0.00355492
$820$	0	0
$821$	1.63030e16	1.52538	0.762692	−	0.646761i	$-0.223877\pi$
$821$	1.63030e16	1.52538	0.762692	+	0.646761i	$0.223877\pi$
$822$	0	0
$823$	−1.00504e16	−0.927864	−0.463932	−	0.885871i	$-0.653562\pi$
$823$	−1.00504e16	−0.927864	−0.463932	+	0.885871i	$0.653562\pi$
$824$	0	0
$825$	−1.53794e16	−1.40102
$826$	0	0
$827$	2.30840e15	0.207506	0.103753	−	0.994603i	$-0.466915\pi$
$827$	2.30840e15	0.207506	0.103753	+	0.994603i	$0.466915\pi$
$828$	0	0
$829$	1.49111e16	1.32270	0.661348	−	0.750079i	$-0.269985\pi$
$829$	1.49111e16	1.32270	0.661348	+	0.750079i	$0.269985\pi$
$830$	0	0
$831$	9.64511e15	0.844309
$832$	0	0
$833$	−9.33159e14	−0.0806136
$834$	0	0
$835$	−3.58561e15	−0.305695
$836$	0	0
$837$	6.02788e15	0.507194
$838$	0	0
$839$	−1.93037e16	−1.60306	−0.801529	−	0.597956i	$-0.795980\pi$
$839$	−1.93037e16	−1.60306	−0.801529	+	0.597956i	$0.795980\pi$
$840$	0	0
$841$	−1.05015e16	−0.860739
$842$	0	0
$843$	−5.68002e15	−0.459513
$844$	0	0
$845$	6.54897e15	0.522951
$846$	0	0
$847$	−1.34892e16	−1.06323
$848$	0	0
$849$	−2.85977e15	−0.222505
$850$	0	0
$851$	−8.84692e15	−0.679485
$852$	0	0
$853$	−1.28155e16	−0.971667	−0.485833	−	0.874051i	$-0.661484\pi$
$853$	−1.28155e16	−0.971667	−0.485833	+	0.874051i	$0.661484\pi$
$854$	0	0
$855$	−2.47859e15	−0.185520
$856$	0	0
$857$	−4.41075e15	−0.325926	−0.162963	−	0.986632i	$-0.552105\pi$
$857$	−4.41075e15	−0.325926	−0.162963	+	0.986632i	$0.552105\pi$
$858$	0	0
$859$	9.04065e15	0.659534	0.329767	−	0.944062i	$-0.393030\pi$
$859$	9.04065e15	0.659534	0.329767	+	0.944062i	$0.393030\pi$
$860$	0	0
$861$	−1.68695e16	−1.21502
$862$	0	0
$863$	−9.65353e15	−0.686478	−0.343239	−	0.939248i	$-0.611524\pi$
$863$	−9.65353e15	−0.686478	−0.343239	+	0.939248i	$0.611524\pi$
$864$	0	0
$865$	−3.44087e15	−0.241590
$866$	0	0
$867$	−1.46930e16	−1.01860
$868$	0	0
$869$	3.03569e16	2.07801
$870$	0	0
$871$	3.47650e13	0.00234986
$872$	0	0
$873$	−1.62427e16	−1.08413
$874$	0	0
$875$	−1.24730e16	−0.822104
$876$	0	0
$877$	−2.78709e16	−1.81406	−0.907032	−	0.421061i	$-0.861658\pi$
$877$	−2.78709e16	−1.81406	−0.907032	+	0.421061i	$0.861658\pi$
$878$	0	0
$879$	−1.32224e15	−0.0849906
$880$	0	0
$881$	−1.05187e16	−0.667717	−0.333859	−	0.942623i	$-0.608351\pi$
$881$	−1.05187e16	−0.667717	−0.333859	+	0.942623i	$0.608351\pi$
$882$	0	0
$883$	1.36308e16	0.854547	0.427274	−	0.904122i	$-0.359474\pi$
$883$	1.36308e16	0.854547	0.427274	+	0.904122i	$0.359474\pi$
$884$	0	0
$885$	−1.64465e16	−1.01832
$886$	0	0
$887$	2.26455e16	1.38485	0.692425	−	0.721490i	$-0.256542\pi$
$887$	2.26455e16	1.38485	0.692425	+	0.721490i	$0.256542\pi$
$888$	0	0
$889$	9.51669e15	0.574813
$890$	0	0
$891$	−2.97023e16	−1.77200
$892$	0	0
$893$	−4.65474e15	−0.274292
$894$	0	0
$895$	1.57050e16	0.914136
$896$	0	0
$897$	4.88537e13	0.00280892
$898$	0	0
$899$	−8.94453e15	−0.508018
$900$	0	0
$901$	1.14444e16	0.642104
$902$	0	0
$903$	−2.42362e16	−1.34333
$904$	0	0
$905$	7.62986e15	0.417781
$906$	0	0
$907$	1.24222e16	0.671984	0.335992	−	0.941865i	$-0.390929\pi$
$907$	1.24222e16	0.671984	0.335992	+	0.941865i	$0.390929\pi$
$908$	0	0
$909$	1.41917e16	0.758462
$910$	0	0
$911$	1.30030e16	0.686582	0.343291	−	0.939229i	$-0.388458\pi$
$911$	1.30030e16	0.686582	0.343291	+	0.939229i	$0.388458\pi$
$912$	0	0
$913$	−7.83949e15	−0.408977
$914$	0	0
$915$	−4.38502e15	−0.226025
$916$	0	0
$917$	−2.41793e16	−1.23144
$918$	0	0
$919$	6.28559e15	0.316309	0.158154	−	0.987414i	$-0.449446\pi$
$919$	6.28559e15	0.316309	0.158154	+	0.987414i	$0.449446\pi$
$920$	0	0
$921$	−1.37593e16	−0.684176
$922$	0	0
$923$	−1.13322e14	−0.00556805
$924$	0	0
$925$	−2.58682e16	−1.25599
$926$	0	0
$927$	9.10246e15	0.436737
$928$	0	0
$929$	−1.96785e16	−0.933052	−0.466526	−	0.884507i	$-0.654495\pi$
$929$	−1.96785e16	−0.933052	−0.466526	+	0.884507i	$0.654495\pi$
$930$	0	0
$931$	−1.80895e15	−0.0847623
$932$	0	0
$933$	2.07524e16	0.960989
$934$	0	0
$935$	−7.93275e15	−0.363045
$936$	0	0
$937$	2.38475e16	1.07864	0.539319	−	0.842102i	$-0.318682\pi$
$937$	2.38475e16	1.07864	0.539319	+	0.842102i	$0.318682\pi$
$938$	0	0
$939$	−1.79350e15	−0.0801755
$940$	0	0
$941$	6.08299e15	0.268766	0.134383	−	0.990929i	$-0.457095\pi$
$941$	6.08299e15	0.268766	0.134383	+	0.990929i	$0.457095\pi$
$942$	0	0
$943$	−9.16946e15	−0.400433
$944$	0	0
$945$	−4.11002e15	−0.177406
$946$	0	0
$947$	−6.47876e15	−0.276418	−0.138209	−	0.990403i	$-0.544135\pi$
$947$	−6.47876e15	−0.276418	−0.138209	+	0.990403i	$0.544135\pi$
$948$	0	0
$949$	−1.20203e14	−0.00506936
$950$	0	0
$951$	3.83283e16	1.59782
$952$	0	0
$953$	1.56847e16	0.646344	0.323172	−	0.946340i	$-0.395251\pi$
$953$	1.56847e16	0.646344	0.323172	+	0.946340i	$0.395251\pi$
$954$	0	0
$955$	−1.68440e16	−0.686163
$956$	0	0
$957$	1.78704e16	0.719646
$958$	0	0
$959$	−2.90753e16	−1.15750
$960$	0	0
$961$	2.16792e16	0.853228
$962$	0	0
$963$	1.10515e16	0.430007
$964$	0	0
$965$	6.19914e15	0.238469
$966$	0	0
$967$	4.09978e16	1.55925	0.779623	−	0.626249i	$-0.215411\pi$
$967$	4.09978e16	1.55925	0.779623	+	0.626249i	$0.215411\pi$
$968$	0	0
$969$	8.14224e15	0.306170
$970$	0	0
$971$	−3.41053e15	−0.126799	−0.0633996	−	0.997988i	$-0.520194\pi$
$971$	−3.41053e15	−0.126799	−0.0633996	+	0.997988i	$0.520194\pi$
$972$	0	0
$973$	−2.38341e16	−0.876150
$974$	0	0
$975$	1.42847e14	0.00519214
$976$	0	0
$977$	−4.94757e16	−1.77816	−0.889082	−	0.457748i	$-0.848656\pi$
$977$	−4.94757e16	−1.77816	−0.889082	+	0.457748i	$0.848656\pi$
$978$	0	0
$979$	4.26848e16	1.51694
$980$	0	0
$981$	−3.74076e16	−1.31456
$982$	0	0
$983$	4.70999e16	1.63673	0.818363	−	0.574702i	$-0.194882\pi$
$983$	4.70999e16	1.63673	0.818363	+	0.574702i	$0.194882\pi$
$984$	0	0
$985$	1.69449e16	0.582290
$986$	0	0
$987$	1.94162e16	0.659813
$988$	0	0
$989$	−1.31737e16	−0.442718
$990$	0	0
$991$	−7.35285e15	−0.244372	−0.122186	−	0.992507i	$-0.538990\pi$
$991$	−7.35285e15	−0.244372	−0.122186	+	0.992507i	$0.538990\pi$
$992$	0	0
$993$	1.81798e16	0.597542
$994$	0	0
$995$	2.97665e16	0.967612
$996$	0	0
$997$	2.19052e16	0.704244	0.352122	−	0.935954i	$-0.385460\pi$
$997$	2.19052e16	0.704244	0.352122	+	0.935954i	$0.385460\pi$
$998$	0	0
$999$	−2.02567e16	−0.644107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.12.a.e.1.5	✓	6
4.3	odd	2		128.12.a.g.1.2	yes	6
8.3	odd	2		128.12.a.f.1.5	yes	6
8.5	even	2		128.12.a.h.1.2	yes	6
16.3	odd	4		256.12.b.q.129.3		12
16.5	even	4		256.12.b.p.129.3		12
16.11	odd	4		256.12.b.q.129.10		12
16.13	even	4		256.12.b.p.129.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.a.e.1.5	✓	6	1.1	even	1	trivial
128.12.a.f.1.5	yes	6	8.3	odd	2
128.12.a.g.1.2	yes	6	4.3	odd	2
128.12.a.h.1.2	yes	6	8.5	even	2
256.12.b.p.129.3		12	16.5	even	4
256.12.b.p.129.10		12	16.13	even	4
256.12.b.q.129.3		12	16.3	odd	4
256.12.b.q.129.10		12	16.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	128.a (trivial)

\(f(q)\)	\(=\)	\(q+551.275 q^{3} -3654.34 q^{5} -40487.8 q^{7} +126757. q^{9} +786434. q^{11} -7304.56 q^{13} -2.01455e6 q^{15} +2.76028e6 q^{17} +5.35085e6 q^{19} -2.23199e7 q^{21} -1.21321e7 q^{23} -3.54739e7 q^{25} -2.77786e7 q^{27} +4.12196e7 q^{29} -2.16997e8 q^{31} +4.33541e8 q^{33} +1.47956e8 q^{35} +7.29218e8 q^{37} -4.02682e6 q^{39} +7.55803e8 q^{41} +1.08585e9 q^{43} -4.63214e8 q^{45} -8.69907e8 q^{47} -3.38067e8 q^{49} +1.52167e9 q^{51} +4.14610e9 q^{53} -2.87390e9 q^{55} +2.94979e9 q^{57} +8.16388e9 q^{59} +2.17668e9 q^{61} -5.13212e9 q^{63} +2.66933e7 q^{65} -4.75936e9 q^{67} -6.68811e9 q^{69} +1.55138e10 q^{71} +1.64559e10 q^{73} -1.95559e10 q^{75} -3.18410e10 q^{77} +3.86007e10 q^{79} -3.77683e10 q^{81} -9.96841e9 q^{83} -1.00870e10 q^{85} +2.27234e10 q^{87} +5.42764e10 q^{89} +2.95745e8 q^{91} -1.19625e11 q^{93} -1.95538e10 q^{95} -1.28140e11 q^{97} +9.96862e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 20 q^{3} - 1804 q^{5} + 49368 q^{7} + 313814 q^{9} - 688460 q^{11} - 2290348 q^{13} + 4828264 q^{15} + 4127636 q^{17} + 9936364 q^{19} + 20325616 q^{21} + 9921320 q^{23} + 51633002 q^{25} - 132503384 q^{27}+ \cdots + 240482467988 q^{99}+O(q^{100}) \) 6 * q - 20 * q^3 - 1804 * q^5 + 49368 * q^7 + 313814 * q^9 - 688460 * q^11 - 2290348 * q^13 + 4828264 * q^15 + 4127636 * q^17 + 9936364 * q^19 + 20325616 * q^21 + 9921320 * q^23 + 51633002 * q^25 - 132503384 * q^27 + 25222532 * q^29 - 246133120 * q^31 - 337991288 * q^33 + 227061200 * q^35 - 660146172 * q^37 + 648086056 * q^39 + 486915836 * q^41 + 1147261348 * q^43 + 626323028 * q^45 + 603975088 * q^47 + 1846879062 * q^49 - 1556268056 * q^51 - 428043996 * q^53 - 1355415976 * q^55 - 783352648 * q^57 + 809510676 * q^59 - 2822052332 * q^61 + 14526601496 * q^63 + 1376905688 * q^65 + 9742675036 * q^67 - 8441748208 * q^69 + 16181067512 * q^71 + 35963668692 * q^73 + 51500880116 * q^75 - 72754679408 * q^77 - 6103946192 * q^79 + 81240620990 * q^81 + 67929536124 * q^83 - 76741413928 * q^85 + 132686631816 * q^87 + 22721416116 * q^89 + 44438236496 * q^91 - 37482073344 * q^93 + 282717822568 * q^95 + 218896625492 * q^97 + 240482467988 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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