LMFDB - Embedded newform 8.22.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,22,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, 22, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 22 $
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:	$22.3581875430$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4963x + 96223 $ x^3 - x^2 - 4963*x + 96223
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{21}\cdot 3\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$21.2235$ 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)

$f(q)$	$=$	$q+201833. q^{3} -2.11555e7 q^{5} +7.32981e8 q^{7} +3.02761e10 q^{9} +O(q^{10})$	$q+201833. q^{3} -2.11555e7 q^{5} +7.32981e8 q^{7} +3.02761e10 q^{9} -5.59634e9 q^{11} -6.30207e10 q^{13} -4.26988e12 q^{15} +1.35490e13 q^{17} +1.39028e13 q^{19} +1.47940e14 q^{21} -2.80711e14 q^{23} -2.92803e13 q^{25} +3.99948e15 q^{27} +1.19637e15 q^{29} +3.88487e15 q^{31} -1.12953e15 q^{33} -1.55066e16 q^{35} +2.72004e16 q^{37} -1.27197e16 q^{39} -6.89865e16 q^{41} +3.24557e16 q^{43} -6.40508e17 q^{45} -2.07417e17 q^{47} -2.12844e16 q^{49} +2.73464e18 q^{51} -6.38971e17 q^{53} +1.18394e17 q^{55} +2.80603e18 q^{57} -3.04610e18 q^{59} -5.64497e18 q^{61} +2.21918e19 q^{63} +1.33324e18 q^{65} -3.96718e18 q^{67} -5.66566e19 q^{69} -2.61878e19 q^{71} +1.37659e19 q^{73} -5.90972e18 q^{75} -4.10201e18 q^{77} -1.18258e20 q^{79} +4.90526e20 q^{81} -1.60950e19 q^{83} -2.86637e20 q^{85} +2.41466e20 q^{87} -2.30103e20 q^{89} -4.61930e19 q^{91} +7.84094e20 q^{93} -2.94120e20 q^{95} +2.72050e20 q^{97} -1.69436e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 96764 q^{3} - 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9}+O(q^{10}) $ 3 * q + 96764 * q^3 - 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9	$ 3 q + 96764 q^{3} - 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9} - 40335108684 q^{11} + 133734425946 q^{13} - 1223136458200 q^{15} + 7797732274422 q^{17} + 35788199781996 q^{19} + 198539224853088 q^{21} + 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 94\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 96764 * q^3 - 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9 - 40335108684 * q^11 + 133734425946 * q^13 - 1223136458200 * q^15 + 7797732274422 * q^17 + 35788199781996 * q^19 + 198539224853088 * q^21 + 193770761479080 * q^23 + 1127564438439501 * q^25 + 5282002293508952 * q^27 + 5607343422466122 * q^29 + 11246757871503072 * q^31 + 10014149026970384 * q^33 + 5274251425350096 * q^35 - 24272499791100606 * q^37 - 52925308377862264 * q^39 - 298159108991869602 * q^41 - 33333932139754860 * q^43 - 927411477977893478 * q^45 - 120874283547603888 * q^47 - 850403331975639477 * q^49 + 3954388789815661240 * q^51 - 1138443393004854222 * q^53 + 6957552484263571704 * q^55 + 681697547133656944 * q^57 + 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 21785214559631141976 * q^63 - 20714581819561144452 * q^65 - 15793054074531629124 * q^67 - 72988310309689939168 * q^69 - 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 75231657393126995900 * q^75 + 68380237365358617888 * q^77 - 131735321299806049488 * q^79 + 500425796062282339147 * q^81 + 64013993832679681068 * q^83 + 390258202763001297252 * q^85 + 101898953973185066568 * q^87 + 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 640035369009914700160 * q^93 - 1036372406649019824696 * q^95 - 324059336514148638042 * q^97 - 949982905688477352860 * 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
$3$	201833.	1.97342	0.986708	−	0.162503i	$-0.0519569\pi$
$3$	201833.	1.97342	0.986708	+	0.162503i	$0.0519569\pi$
$4$	0	0
$5$	−2.11555e7	−0.968811	−0.484406	−	0.874844i	$-0.660964\pi$
$5$	−2.11555e7	−0.968811	−0.484406	+	0.874844i	$0.660964\pi$
$6$	0	0
$7$	7.32981e8	0.980761	0.490381	−	0.871508i	$-0.336858\pi$
$7$	7.32981e8	0.980761	0.490381	+	0.871508i	$0.336858\pi$
$8$	0	0
$9$	3.02761e10	2.89437
$10$	0	0
$11$	−5.59634e9	−0.0650551	−0.0325275	−	0.999471i	$-0.510356\pi$
$11$	−5.59634e9	−0.0650551	−0.0325275	+	0.999471i	$0.510356\pi$
$12$	0	0
$13$	−6.30207e10	−0.126788	−0.0633940	−	0.997989i	$-0.520192\pi$
$13$	−6.30207e10	−0.126788	−0.0633940	+	0.997989i	$0.520192\pi$
$14$	0	0
$15$	−4.26988e12	−1.91187
$16$	0	0
$17$	1.35490e13	1.63003	0.815013	−	0.579443i	$-0.196730\pi$
$17$	1.35490e13	1.63003	0.815013	+	0.579443i	$0.196730\pi$
$18$	0	0
$19$	1.39028e13	0.520221	0.260111	−	0.965579i	$-0.416241\pi$
$19$	1.39028e13	0.520221	0.260111	+	0.965579i	$0.416241\pi$
$20$	0	0
$21$	1.47940e14	1.93545
$22$	0	0
$23$	−2.80711e14	−1.41292	−0.706458	−	0.707754i	$-0.749708\pi$
$23$	−2.80711e14	−1.41292	−0.706458	+	0.707754i	$0.749708\pi$
$24$	0	0
$25$	−2.92803e13	−0.0614052
$26$	0	0
$27$	3.99948e15	3.73838
$28$	0	0
$29$	1.19637e15	0.528063	0.264031	−	0.964514i	$-0.414948\pi$
$29$	1.19637e15	0.528063	0.264031	+	0.964514i	$0.414948\pi$
$30$	0	0
$31$	3.88487e15	0.851292	0.425646	−	0.904890i	$-0.360047\pi$
$31$	3.88487e15	0.851292	0.425646	+	0.904890i	$0.360047\pi$
$32$	0	0
$33$	−1.12953e15	−0.128381
$34$	0	0
$35$	−1.55066e16	−0.950173
$36$	0	0
$37$	2.72004e16	0.929944	0.464972	−	0.885325i	$-0.346064\pi$
$37$	2.72004e16	0.929944	0.464972	+	0.885325i	$0.346064\pi$
$38$	0	0
$39$	−1.27197e16	−0.250205
$40$	0	0
$41$	−6.89865e16	−0.802663	−0.401332	−	0.915933i	$-0.631453\pi$
$41$	−6.89865e16	−0.802663	−0.401332	+	0.915933i	$0.631453\pi$
$42$	0	0
$43$	3.24557e16	0.229020	0.114510	−	0.993422i	$-0.463470\pi$
$43$	3.24557e16	0.229020	0.114510	+	0.993422i	$0.463470\pi$
$44$	0	0
$45$	−6.40508e17	−2.80410
$46$	0	0
$47$	−2.07417e17	−0.575196	−0.287598	−	0.957751i	$-0.592857\pi$
$47$	−2.07417e17	−0.575196	−0.287598	+	0.957751i	$0.592857\pi$
$48$	0	0
$49$	−2.12844e16	−0.0381069
$50$	0	0
$51$	2.73464e18	3.21672
$52$	0	0
$53$	−6.38971e17	−0.501862	−0.250931	−	0.968005i	$-0.580737\pi$
$53$	−6.38971e17	−0.501862	−0.250931	+	0.968005i	$0.580737\pi$
$54$	0	0
$55$	1.18394e17	0.0630261
$56$	0	0
$57$	2.80603e18	1.02661
$58$	0	0
$59$	−3.04610e18	−0.775887	−0.387944	−	0.921683i	$-0.626814\pi$
$59$	−3.04610e18	−0.775887	−0.387944	+	0.921683i	$0.626814\pi$
$60$	0	0
$61$	−5.64497e18	−1.01321	−0.506604	−	0.862179i	$-0.669099\pi$
$61$	−5.64497e18	−1.01321	−0.506604	+	0.862179i	$0.669099\pi$
$62$	0	0
$63$	2.21918e19	2.83869
$64$	0	0
$65$	1.33324e18	0.122834
$66$	0	0
$67$	−3.96718e18	−0.265886	−0.132943	−	0.991124i	$-0.542443\pi$
$67$	−3.96718e18	−0.265886	−0.132943	+	0.991124i	$0.542443\pi$
$68$	0	0
$69$	−5.66566e19	−2.78827
$70$	0	0
$71$	−2.61878e19	−0.954743	−0.477371	−	0.878702i	$-0.658410\pi$
$71$	−2.61878e19	−0.954743	−0.477371	+	0.878702i	$0.658410\pi$
$72$	0	0
$73$	1.37659e19	0.374899	0.187449	−	0.982274i	$-0.439978\pi$
$73$	1.37659e19	0.374899	0.187449	+	0.982274i	$0.439978\pi$
$74$	0	0
$75$	−5.90972e18	−0.121178
$76$	0	0
$77$	−4.10201e18	−0.0638035
$78$	0	0
$79$	−1.18258e20	−1.40523	−0.702615	−	0.711570i	$-0.747984\pi$
$79$	−1.18258e20	−1.40523	−0.702615	+	0.711570i	$0.747984\pi$
$80$	0	0
$81$	4.90526e20	4.48301
$82$	0	0
$83$	−1.60950e19	−0.113860	−0.0569301	−	0.998378i	$-0.518131\pi$
$83$	−1.60950e19	−0.113860	−0.0569301	+	0.998378i	$0.518131\pi$
$84$	0	0
$85$	−2.86637e20	−1.57919
$86$	0	0
$87$	2.41466e20	1.04209
$88$	0	0
$89$	−2.30103e20	−0.782217	−0.391109	−	0.920345i	$-0.627908\pi$
$89$	−2.30103e20	−0.782217	−0.391109	+	0.920345i	$0.627908\pi$
$90$	0	0
$91$	−4.61930e19	−0.124349
$92$	0	0
$93$	7.84094e20	1.67995
$94$	0	0
$95$	−2.94120e20	−0.503996
$96$	0	0
$97$	2.72050e20	0.374581	0.187290	−	0.982305i	$-0.440029\pi$
$97$	2.72050e20	0.374581	0.187290	+	0.982305i	$0.440029\pi$
$98$	0	0
$99$	−1.69436e20	−0.188293
$100$	0	0
$101$	−1.18197e21	−1.06471	−0.532357	−	0.846520i	$-0.678693\pi$
$101$	−1.18197e21	−1.06471	−0.532357	+	0.846520i	$0.678693\pi$
$102$	0	0
$103$	−2.39849e21	−1.75852	−0.879258	−	0.476345i	$-0.841961\pi$
$103$	−2.39849e21	−1.75852	−0.879258	+	0.476345i	$0.841961\pi$
$104$	0	0
$105$	−3.12974e21	−1.87509
$106$	0	0
$107$	4.01203e20	0.197167	0.0985835	−	0.995129i	$-0.468569\pi$
$107$	4.01203e20	0.197167	0.0985835	+	0.995129i	$0.468569\pi$
$108$	0	0
$109$	4.25100e21	1.71994	0.859970	−	0.510345i	$-0.170482\pi$
$109$	4.25100e21	1.71994	0.859970	+	0.510345i	$0.170482\pi$
$110$	0	0
$111$	5.48993e21	1.83517
$112$	0	0
$113$	−3.36941e21	−0.933748	−0.466874	−	0.884324i	$-0.654620\pi$
$113$	−3.36941e21	−0.933748	−0.466874	+	0.884324i	$0.654620\pi$
$114$	0	0
$115$	5.93859e21	1.36885
$116$	0	0
$117$	−1.90802e21	−0.366971
$118$	0	0
$119$	9.93118e21	1.59867
$120$	0	0
$121$	−7.36893e21	−0.995768
$122$	0	0
$123$	−1.39237e22	−1.58399
$124$	0	0
$125$	1.07072e22	1.02830
$126$	0	0
$127$	−1.15394e20	−0.00938093	−0.00469046	−	0.999989i	$-0.501493\pi$
$127$	−1.15394e20	−0.00938093	−0.00469046	+	0.999989i	$0.501493\pi$
$128$	0	0
$129$	6.55063e21	0.451951
$130$	0	0
$131$	1.13322e22	0.665223	0.332611	−	0.943064i	$-0.392070\pi$
$131$	1.13322e22	0.665223	0.332611	+	0.943064i	$0.392070\pi$
$132$	0	0
$133$	1.01905e22	0.510213
$134$	0	0
$135$	−8.46111e22	−3.62178
$136$	0	0
$137$	7.18696e21	0.263621	0.131810	−	0.991275i	$-0.457921\pi$
$137$	7.18696e21	0.263621	0.131810	+	0.991275i	$0.457921\pi$
$138$	0	0
$139$	2.34620e22	0.739109	0.369555	−	0.929209i	$-0.379510\pi$
$139$	2.34620e22	0.739109	0.369555	+	0.929209i	$0.379510\pi$
$140$	0	0
$141$	−4.18635e22	−1.13510
$142$	0	0
$143$	3.52686e20	0.00824820
$144$	0	0
$145$	−2.53098e22	−0.511593
$146$	0	0
$147$	−4.29590e21	−0.0752007
$148$	0	0
$149$	−2.57122e22	−0.390557	−0.195278	−	0.980748i	$-0.562561\pi$
$149$	−2.57122e22	−0.390557	−0.195278	+	0.980748i	$0.562561\pi$
$150$	0	0
$151$	1.38339e23	1.82679	0.913393	−	0.407078i	$-0.133452\pi$
$151$	1.38339e23	1.82679	0.913393	+	0.407078i	$0.133452\pi$
$152$	0	0
$153$	4.10212e23	4.71790
$154$	0	0
$155$	−8.21865e22	−0.824741
$156$	0	0
$157$	−1.14060e23	−1.00043	−0.500215	−	0.865901i	$-0.666746\pi$
$157$	−1.14060e23	−1.00043	−0.500215	+	0.865901i	$0.666746\pi$
$158$	0	0
$159$	−1.28965e23	−0.990382
$160$	0	0
$161$	−2.05756e23	−1.38573
$162$	0	0
$163$	6.59877e22	0.390385	0.195192	−	0.980765i	$-0.437467\pi$
$163$	6.59877e22	0.390385	0.195192	+	0.980765i	$0.437467\pi$
$164$	0	0
$165$	2.38957e22	0.124377
$166$	0	0
$167$	2.09416e22	0.0960477	0.0480239	−	0.998846i	$-0.484708\pi$
$167$	2.09416e22	0.0960477	0.0480239	+	0.998846i	$0.484708\pi$
$168$	0	0
$169$	−2.43093e23	−0.983925
$170$	0	0
$171$	4.20922e23	1.50571
$172$	0	0
$173$	−1.69739e23	−0.537400	−0.268700	−	0.963224i	$-0.586594\pi$
$173$	−1.69739e23	−0.537400	−0.268700	+	0.963224i	$0.586594\pi$
$174$	0	0
$175$	−2.14619e22	−0.0602238
$176$	0	0
$177$	−6.14804e23	−1.53115
$178$	0	0
$179$	5.58531e23	1.23620	0.618102	−	0.786098i	$-0.287902\pi$
$179$	5.58531e23	1.23620	0.618102	+	0.786098i	$0.287902\pi$
$180$	0	0
$181$	−2.50056e23	−0.492506	−0.246253	−	0.969206i	$-0.579199\pi$
$181$	−2.50056e23	−0.492506	−0.246253	+	0.969206i	$0.579199\pi$
$182$	0	0
$183$	−1.13934e24	−1.99948
$184$	0	0
$185$	−5.75439e23	−0.900940
$186$	0	0
$187$	−7.58250e22	−0.106041
$188$	0	0
$189$	2.93154e24	3.66646
$190$	0	0
$191$	−1.66807e23	−0.186794	−0.0933971	−	0.995629i	$-0.529773\pi$
$191$	−1.66807e23	−0.186794	−0.0933971	+	0.995629i	$0.529773\pi$
$192$	0	0
$193$	1.33244e24	1.33750	0.668751	−	0.743487i	$-0.266829\pi$
$193$	1.33244e24	1.33750	0.668751	+	0.743487i	$0.266829\pi$
$194$	0	0
$195$	2.69091e23	0.242402
$196$	0	0
$197$	−6.38870e23	−0.517032	−0.258516	−	0.966007i	$-0.583233\pi$
$197$	−6.38870e23	−0.517032	−0.258516	+	0.966007i	$0.583233\pi$
$198$	0	0
$199$	1.84623e24	1.34378	0.671889	−	0.740652i	$-0.265483\pi$
$199$	1.84623e24	1.34378	0.671889	+	0.740652i	$0.265483\pi$
$200$	0	0
$201$	−8.00707e23	−0.524705
$202$	0	0
$203$	8.76915e23	0.517904
$204$	0	0
$205$	1.45945e24	0.777629
$206$	0	0
$207$	−8.49883e24	−4.08951
$208$	0	0
$209$	−7.78046e22	−0.0338430
$210$	0	0
$211$	3.35146e24	1.31907	0.659536	−	0.751673i	$-0.270753\pi$
$211$	3.35146e24	1.31907	0.659536	+	0.751673i	$0.270753\pi$
$212$	0	0
$213$	−5.28556e24	−1.88410
$214$	0	0
$215$	−6.86619e23	−0.221877
$216$	0	0
$217$	2.84754e24	0.834915
$218$	0	0
$219$	2.77841e24	0.739831
$220$	0	0
$221$	−8.53869e23	−0.206668
$222$	0	0
$223$	4.56926e22	0.0100611	0.00503054	−	0.999987i	$-0.498399\pi$
$223$	4.56926e22	0.0100611	0.00503054	+	0.999987i	$0.498399\pi$
$224$	0	0
$225$	−8.86493e23	−0.177729
$226$	0	0
$227$	−6.05295e24	−1.10585	−0.552924	−	0.833232i	$-0.686488\pi$
$227$	−6.05295e24	−1.10585	−0.552924	+	0.833232i	$0.686488\pi$
$228$	0	0
$229$	5.16931e24	0.861311	0.430655	−	0.902516i	$-0.358282\pi$
$229$	5.16931e24	0.861311	0.430655	+	0.902516i	$0.358282\pi$
$230$	0	0
$231$	−8.27921e23	−0.125911
$232$	0	0
$233$	5.21547e24	0.724530	0.362265	−	0.932075i	$-0.382003\pi$
$233$	5.21547e24	0.724530	0.362265	+	0.932075i	$0.382003\pi$
$234$	0	0
$235$	4.38801e24	0.557257
$236$	0	0
$237$	−2.38684e25	−2.77310
$238$	0	0
$239$	1.57703e24	0.167749	0.0838747	−	0.996476i	$-0.473270\pi$
$239$	1.57703e24	0.167749	0.0838747	+	0.996476i	$0.473270\pi$
$240$	0	0
$241$	9.93088e24	0.967851	0.483926	−	0.875109i	$-0.339211\pi$
$241$	9.93088e24	0.967851	0.483926	+	0.875109i	$0.339211\pi$
$242$	0	0
$243$	5.71684e25	5.10846
$244$	0	0
$245$	4.50284e23	0.0369184
$246$	0	0
$247$	−8.76162e23	−0.0659578
$248$	0	0
$249$	−3.24850e24	−0.224693
$250$	0	0
$251$	−2.38505e25	−1.51679	−0.758393	−	0.651798i	$-0.774015\pi$
$251$	−2.38505e25	−1.51679	−0.758393	+	0.651798i	$0.774015\pi$
$252$	0	0
$253$	1.57095e24	0.0919174
$254$	0	0
$255$	−5.78528e25	−3.11639
$256$	0	0
$257$	−3.76190e25	−1.86685	−0.933425	−	0.358772i	$-0.883195\pi$
$257$	−3.76190e25	−1.86685	−0.933425	+	0.358772i	$0.883195\pi$
$258$	0	0
$259$	1.99374e25	0.912054
$260$	0	0
$261$	3.62214e25	1.52841
$262$	0	0
$263$	−2.52809e25	−0.984592	−0.492296	−	0.870428i	$-0.663842\pi$
$263$	−2.52809e25	−0.984592	−0.492296	+	0.870428i	$0.663842\pi$
$264$	0	0
$265$	1.35178e25	0.486209
$266$	0	0
$267$	−4.64424e25	−1.54364
$268$	0	0
$269$	4.38239e25	1.34683	0.673414	−	0.739266i	$-0.264827\pi$
$269$	4.38239e25	1.34683	0.673414	+	0.739266i	$0.264827\pi$
$270$	0	0
$271$	2.43576e24	0.0692560	0.0346280	−	0.999400i	$-0.488975\pi$
$271$	2.43576e24	0.0692560	0.0346280	+	0.999400i	$0.488975\pi$
$272$	0	0
$273$	−9.32326e24	−0.245392
$274$	0	0
$275$	1.63862e23	0.00399472
$276$	0	0
$277$	4.43475e25	1.00192	0.500958	−	0.865472i	$-0.332981\pi$
$277$	4.43475e25	1.00192	0.500958	+	0.865472i	$0.332981\pi$
$278$	0	0
$279$	1.17619e26	2.46396
$280$	0	0
$281$	−8.89794e25	−1.72931	−0.864655	−	0.502365i	$-0.832463\pi$
$281$	−8.89794e25	−1.72931	−0.864655	+	0.502365i	$0.832463\pi$
$282$	0	0
$283$	−2.34320e25	−0.422719	−0.211359	−	0.977408i	$-0.567789\pi$
$283$	−2.34320e25	−0.422719	−0.211359	+	0.977408i	$0.567789\pi$
$284$	0	0
$285$	−5.93631e25	−0.994594
$286$	0	0
$287$	−5.05658e25	−0.787221
$288$	0	0
$289$	1.14484e26	1.65698
$290$	0	0
$291$	5.49086e25	0.739204
$292$	0	0
$293$	5.82209e25	0.729405	0.364703	−	0.931124i	$-0.381171\pi$
$293$	5.82209e25	0.729405	0.364703	+	0.931124i	$0.381171\pi$
$294$	0	0
$295$	6.44420e25	0.751688
$296$	0	0
$297$	−2.23824e25	−0.243201
$298$	0	0
$299$	1.76906e25	0.179141
$300$	0	0
$301$	2.37894e25	0.224614
$302$	0	0
$303$	−2.38561e26	−2.10112
$304$	0	0
$305$	1.19422e26	0.981607
$306$	0	0
$307$	1.92535e26	1.47760	0.738799	−	0.673926i	$-0.235393\pi$
$307$	1.92535e26	1.47760	0.738799	+	0.673926i	$0.235393\pi$
$308$	0	0
$309$	−4.84093e26	−3.47028
$310$	0	0
$311$	1.71812e26	1.15098	0.575491	−	0.817808i	$-0.304811\pi$
$311$	1.71812e26	1.15098	0.575491	+	0.817808i	$0.304811\pi$
$312$	0	0
$313$	−3.10727e25	−0.194609	−0.0973045	−	0.995255i	$-0.531022\pi$
$313$	−3.10727e25	−0.194609	−0.0973045	+	0.995255i	$0.531022\pi$
$314$	0	0
$315$	−4.69480e26	−2.75015
$316$	0	0
$317$	−2.10055e26	−1.15136	−0.575680	−	0.817675i	$-0.695262\pi$
$317$	−2.10055e26	−1.15136	−0.575680	+	0.817675i	$0.695262\pi$
$318$	0	0
$319$	−6.69528e24	−0.0343532
$320$	0	0
$321$	8.09759e25	0.389093
$322$	0	0
$323$	1.88369e26	0.847974
$324$	0	0
$325$	1.84526e24	0.00778544
$326$	0	0
$327$	8.57992e26	3.39416
$328$	0	0
$329$	−1.52033e26	−0.564130
$330$	0	0
$331$	−2.36914e26	−0.824891	−0.412446	−	0.910982i	$-0.635325\pi$
$331$	−2.36914e26	−0.824891	−0.412446	+	0.910982i	$0.635325\pi$
$332$	0	0
$333$	8.23523e26	2.69160
$334$	0	0
$335$	8.39278e25	0.257594
$336$	0	0
$337$	−6.34353e25	−0.182902	−0.0914508	−	0.995810i	$-0.529150\pi$
$337$	−6.34353e25	−0.182902	−0.0914508	+	0.995810i	$0.529150\pi$
$338$	0	0
$339$	−6.80057e26	−1.84267
$340$	0	0
$341$	−2.17411e25	−0.0553809
$342$	0	0
$343$	−4.25005e26	−1.01814
$344$	0	0
$345$	1.19860e27	2.70131
$346$	0	0
$347$	−2.30760e26	−0.489442	−0.244721	−	0.969593i	$-0.578696\pi$
$347$	−2.30760e26	−0.489442	−0.244721	+	0.969593i	$0.578696\pi$
$348$	0	0
$349$	−6.64125e26	−1.32612	−0.663060	−	0.748566i	$-0.730742\pi$
$349$	−6.64125e26	−1.32612	−0.663060	+	0.748566i	$0.730742\pi$
$350$	0	0
$351$	−2.52050e26	−0.473982
$352$	0	0
$353$	−7.72081e26	−1.36782	−0.683909	−	0.729567i	$-0.739722\pi$
$353$	−7.72081e26	−1.36782	−0.683909	+	0.729567i	$0.739722\pi$
$354$	0	0
$355$	5.54017e26	0.924965
$356$	0	0
$357$	2.00444e27	3.15483
$358$	0	0
$359$	8.35001e26	1.23935	0.619677	−	0.784857i	$-0.287264\pi$
$359$	8.35001e26	1.23935	0.619677	+	0.784857i	$0.287264\pi$
$360$	0	0
$361$	−5.20923e26	−0.729370
$362$	0	0
$363$	−1.48729e27	−1.96506
$364$	0	0
$365$	−2.91225e26	−0.363206
$366$	0	0
$367$	−4.88336e26	−0.575075	−0.287538	−	0.957769i	$-0.592837\pi$
$367$	−4.88336e26	−0.575075	−0.287538	+	0.957769i	$0.592837\pi$
$368$	0	0
$369$	−2.08864e27	−2.32321
$370$	0	0
$371$	−4.68353e26	−0.492207
$372$	0	0
$373$	−1.33813e27	−1.32909	−0.664547	−	0.747247i	$-0.731375\pi$
$373$	−1.33813e27	−1.32909	−0.664547	+	0.747247i	$0.731375\pi$
$374$	0	0
$375$	2.16106e27	2.02927
$376$	0	0
$377$	−7.53960e25	−0.0669520
$378$	0	0
$379$	7.26568e25	0.0610329	0.0305165	−	0.999534i	$-0.490285\pi$
$379$	7.26568e25	0.0610329	0.0305165	+	0.999534i	$0.490285\pi$
$380$	0	0
$381$	−2.32904e25	−0.0185125
$382$	0	0
$383$	1.24472e27	0.936449	0.468225	−	0.883610i	$-0.344894\pi$
$383$	1.24472e27	0.936449	0.468225	+	0.883610i	$0.344894\pi$
$384$	0	0
$385$	8.67803e25	0.0618135
$386$	0	0
$387$	9.82635e26	0.662868
$388$	0	0
$389$	2.14241e27	1.36909	0.684544	−	0.728972i	$-0.260002\pi$
$389$	2.14241e27	1.36909	0.684544	+	0.728972i	$0.260002\pi$
$390$	0	0
$391$	−3.80336e27	−2.30309
$392$	0	0
$393$	2.28722e27	1.31276
$394$	0	0
$395$	2.50182e27	1.36140
$396$	0	0
$397$	2.44762e27	1.26312	0.631560	−	0.775327i	$-0.282415\pi$
$397$	2.44762e27	1.26312	0.631560	+	0.775327i	$0.282415\pi$
$398$	0	0
$399$	2.05677e27	1.00686
$400$	0	0
$401$	2.80970e27	1.30510	0.652550	−	0.757746i	$-0.273699\pi$
$401$	2.80970e27	1.30510	0.652550	+	0.757746i	$0.273699\pi$
$402$	0	0
$403$	−2.44827e26	−0.107934
$404$	0	0
$405$	−1.03774e28	−4.34319
$406$	0	0
$407$	−1.52223e26	−0.0604976
$408$	0	0
$409$	−3.81600e27	−1.44050	−0.720250	−	0.693714i	$-0.755973\pi$
$409$	−3.81600e27	−1.44050	−0.720250	+	0.693714i	$0.755973\pi$
$410$	0	0
$411$	1.45057e27	0.520233
$412$	0	0
$413$	−2.23274e27	−0.760960
$414$	0	0
$415$	3.40499e26	0.110309
$416$	0	0
$417$	4.73539e27	1.45857
$418$	0	0
$419$	−4.13782e26	−0.121206	−0.0606031	−	0.998162i	$-0.519302\pi$
$419$	−4.13782e26	−0.121206	−0.0606031	+	0.998162i	$0.519302\pi$
$420$	0	0
$421$	4.16773e27	1.16128	0.580641	−	0.814160i	$-0.302802\pi$
$421$	4.16773e27	1.16128	0.580641	+	0.814160i	$0.302802\pi$
$422$	0	0
$423$	−6.27978e27	−1.66483
$424$	0	0
$425$	−3.96719e26	−0.100092
$426$	0	0
$427$	−4.13766e27	−0.993715
$428$	0	0
$429$	7.11835e25	0.0162771
$430$	0	0
$431$	3.31148e27	0.721125	0.360563	−	0.932735i	$-0.382585\pi$
$431$	3.31148e27	0.721125	0.360563	+	0.932735i	$0.382585\pi$
$432$	0	0
$433$	7.71199e27	1.59972	0.799859	−	0.600188i	$-0.204908\pi$
$433$	7.71199e27	1.59972	0.799859	+	0.600188i	$0.204908\pi$
$434$	0	0
$435$	−5.10835e27	−1.00959
$436$	0	0
$437$	−3.90265e27	−0.735029
$438$	0	0
$439$	−7.40474e27	−1.32933	−0.664665	−	0.747142i	$-0.731426\pi$
$439$	−7.40474e27	−1.32933	−0.664665	+	0.747142i	$0.731426\pi$
$440$	0	0
$441$	−6.44411e26	−0.110295
$442$	0	0
$443$	2.59541e27	0.423611	0.211805	−	0.977312i	$-0.432066\pi$
$443$	2.59541e27	0.423611	0.211805	+	0.977312i	$0.432066\pi$
$444$	0	0
$445$	4.86795e27	0.757821
$446$	0	0
$447$	−5.18958e27	−0.770731
$448$	0	0
$449$	−8.63721e27	−1.22402	−0.612008	−	0.790852i	$-0.709638\pi$
$449$	−8.63721e27	−1.22402	−0.612008	+	0.790852i	$0.709638\pi$
$450$	0	0
$451$	3.86072e26	0.0522173
$452$	0	0
$453$	2.79214e28	3.60501
$454$	0	0
$455$	9.77238e26	0.120470
$456$	0	0
$457$	−7.22330e27	−0.850385	−0.425193	−	0.905103i	$-0.639794\pi$
$457$	−7.22330e27	−0.850385	−0.425193	+	0.905103i	$0.639794\pi$
$458$	0	0
$459$	5.41890e28	6.09366
$460$	0	0
$461$	−6.67275e27	−0.716878	−0.358439	−	0.933553i	$-0.616691\pi$
$461$	−6.67275e27	−0.716878	−0.358439	+	0.933553i	$0.616691\pi$
$462$	0	0
$463$	−7.22059e27	−0.741263	−0.370632	−	0.928780i	$-0.620859\pi$
$463$	−7.22059e27	−0.741263	−0.370632	+	0.928780i	$0.620859\pi$
$464$	0	0
$465$	−1.65879e28	−1.62756
$466$	0	0
$467$	−3.14336e27	−0.294827	−0.147413	−	0.989075i	$-0.547095\pi$
$467$	−3.14336e27	−0.294827	−0.147413	+	0.989075i	$0.547095\pi$
$468$	0	0
$469$	−2.90787e27	−0.260771
$470$	0	0
$471$	−2.30210e28	−1.97427
$472$	0	0
$473$	−1.81634e26	−0.0148989
$474$	0	0
$475$	−4.07076e26	−0.0319443
$476$	0	0
$477$	−1.93456e28	−1.45257
$478$	0	0
$479$	1.98918e28	1.42939	0.714696	−	0.699435i	$-0.246565\pi$
$479$	1.98918e28	1.42939	0.714696	+	0.699435i	$0.246565\pi$
$480$	0	0
$481$	−1.71419e27	−0.117906
$482$	0	0
$483$	−4.15282e28	−2.73463
$484$	0	0
$485$	−5.75536e27	−0.362898
$486$	0	0
$487$	1.49710e28	0.904058	0.452029	−	0.892003i	$-0.350700\pi$
$487$	1.49710e28	0.904058	0.452029	+	0.892003i	$0.350700\pi$
$488$	0	0
$489$	1.33185e28	0.770392
$490$	0	0
$491$	−5.87261e27	−0.325444	−0.162722	−	0.986672i	$-0.552027\pi$
$491$	−5.87261e27	−0.325444	−0.162722	+	0.986672i	$0.552027\pi$
$492$	0	0
$493$	1.62096e28	0.860756
$494$	0	0
$495$	3.58450e27	0.182421
$496$	0	0
$497$	−1.91952e28	−0.936375
$498$	0	0
$499$	3.56403e28	1.66681	0.833404	−	0.552665i	$-0.186389\pi$
$499$	3.56403e28	1.66681	0.833404	+	0.552665i	$0.186389\pi$
$500$	0	0
$501$	4.22670e27	0.189542
$502$	0	0
$503$	−1.66833e28	−0.717495	−0.358748	−	0.933435i	$-0.616796\pi$
$503$	−1.66833e28	−0.717495	−0.358748	+	0.933435i	$0.616796\pi$
$504$	0	0
$505$	2.50053e28	1.03151
$506$	0	0
$507$	−4.90641e28	−1.94169
$508$	0	0
$509$	−3.66847e28	−1.39299	−0.696495	−	0.717562i	$-0.745258\pi$
$509$	−3.66847e28	−1.39299	−0.696495	+	0.717562i	$0.745258\pi$
$510$	0	0
$511$	1.00901e28	0.367686
$512$	0	0
$513$	5.56037e28	1.94478
$514$	0	0
$515$	5.07413e28	1.70367
$516$	0	0
$517$	1.16078e27	0.0374194
$518$	0	0
$519$	−3.42589e28	−1.06051
$520$	0	0
$521$	−3.16512e28	−0.941009	−0.470505	−	0.882398i	$-0.655928\pi$
$521$	−3.16512e28	−0.941009	−0.470505	+	0.882398i	$0.655928\pi$
$522$	0	0
$523$	4.41639e28	1.26125	0.630623	−	0.776090i	$-0.282800\pi$
$523$	4.41639e28	1.26125	0.630623	+	0.776090i	$0.282800\pi$
$524$	0	0
$525$	−4.33171e27	−0.118847
$526$	0	0
$527$	5.26362e28	1.38763
$528$	0	0
$529$	3.93269e28	0.996334
$530$	0	0
$531$	−9.22243e28	−2.24571
$532$	0	0
$533$	4.34758e27	0.101768
$534$	0	0
$535$	−8.48766e27	−0.191018
$536$	0	0
$537$	1.12730e29	2.43954
$538$	0	0
$539$	1.19115e26	0.00247905
$540$	0	0
$541$	8.65965e28	1.73352	0.866760	−	0.498725i	$-0.166198\pi$
$541$	8.65965e28	1.73352	0.866760	+	0.498725i	$0.166198\pi$
$542$	0	0
$543$	−5.04694e28	−0.971919
$544$	0	0
$545$	−8.99322e28	−1.66630
$546$	0	0
$547$	−6.71643e28	−1.19749	−0.598745	−	0.800940i	$-0.704334\pi$
$547$	−6.71643e28	−1.19749	−0.598745	+	0.800940i	$0.704334\pi$
$548$	0	0
$549$	−1.70908e29	−2.93260
$550$	0	0
$551$	1.66328e28	0.274710
$552$	0	0
$553$	−8.66811e28	−1.37820
$554$	0	0
$555$	−1.16142e29	−1.77793
$556$	0	0
$557$	6.28272e28	0.926122	0.463061	−	0.886326i	$-0.346751\pi$
$557$	6.28272e28	0.926122	0.463061	+	0.886326i	$0.346751\pi$
$558$	0	0
$559$	−2.04538e27	−0.0290369
$560$	0	0
$561$	−1.53040e28	−0.209264
$562$	0	0
$563$	9.00019e28	1.18553	0.592766	−	0.805375i	$-0.298036\pi$
$563$	9.00019e28	1.18553	0.592766	+	0.805375i	$0.298036\pi$
$564$	0	0
$565$	7.12816e28	0.904625
$566$	0	0
$567$	3.59547e29	4.39676
$568$	0	0
$569$	−1.19277e29	−1.40565	−0.702824	−	0.711364i	$-0.748078\pi$
$569$	−1.19277e29	−1.40565	−0.702824	+	0.711364i	$0.748078\pi$
$570$	0	0
$571$	−1.14674e29	−1.30253	−0.651265	−	0.758851i	$-0.725761\pi$
$571$	−1.14674e29	−1.30253	−0.651265	+	0.758851i	$0.725761\pi$
$572$	0	0
$573$	−3.36671e28	−0.368623
$574$	0	0
$575$	8.21928e27	0.0867604
$576$	0	0
$577$	−2.51949e28	−0.256429	−0.128214	−	0.991746i	$-0.540925\pi$
$577$	−2.51949e28	−0.256429	−0.128214	+	0.991746i	$0.540925\pi$
$578$	0	0
$579$	2.68929e29	2.63945
$580$	0	0
$581$	−1.17974e28	−0.111670
$582$	0	0
$583$	3.57590e27	0.0326487
$584$	0	0
$585$	4.03653e28	0.355526
$586$	0	0
$587$	−4.22050e28	−0.358644	−0.179322	−	0.983790i	$-0.557390\pi$
$587$	−4.22050e28	−0.358644	−0.179322	+	0.983790i	$0.557390\pi$
$588$	0	0
$589$	5.40104e28	0.442860
$590$	0	0
$591$	−1.28945e29	−1.02032
$592$	0	0
$593$	−1.61710e29	−1.23499	−0.617495	−	0.786575i	$-0.711852\pi$
$593$	−1.61710e29	−1.23499	−0.617495	+	0.786575i	$0.711852\pi$
$594$	0	0
$595$	−2.10100e29	−1.54881
$596$	0	0
$597$	3.72629e29	2.65183
$598$	0	0
$599$	−8.18099e28	−0.562113	−0.281057	−	0.959691i	$-0.590685\pi$
$599$	−8.18099e28	−0.562113	−0.281057	+	0.959691i	$0.590685\pi$
$600$	0	0
$601$	2.09704e28	0.139131	0.0695657	−	0.997577i	$-0.477839\pi$
$601$	2.09704e28	0.139131	0.0695657	+	0.997577i	$0.477839\pi$
$602$	0	0
$603$	−1.20111e29	−0.769574
$604$	0	0
$605$	1.55894e29	0.964711
$606$	0	0
$607$	1.39211e29	0.832131	0.416066	−	0.909335i	$-0.363409\pi$
$607$	1.39211e29	0.832131	0.416066	+	0.909335i	$0.363409\pi$
$608$	0	0
$609$	1.76990e29	1.02204
$610$	0	0
$611$	1.30715e28	0.0729280
$612$	0	0
$613$	−1.57634e29	−0.849795	−0.424898	−	0.905241i	$-0.639690\pi$
$613$	−1.57634e29	−0.849795	−0.424898	+	0.905241i	$0.639690\pi$
$614$	0	0
$615$	2.94564e29	1.53459
$616$	0	0
$617$	1.96390e29	0.988836	0.494418	−	0.869224i	$-0.335381\pi$
$617$	1.96390e29	0.988836	0.494418	+	0.869224i	$0.335381\pi$
$618$	0	0
$619$	−2.32870e29	−1.13334	−0.566671	−	0.823944i	$-0.691769\pi$
$619$	−2.32870e29	−1.13334	−0.566671	+	0.823944i	$0.691769\pi$
$620$	0	0
$621$	−1.12270e30	−5.28202
$622$	0	0
$623$	−1.68661e29	−0.767168
$624$	0	0
$625$	−2.12554e29	−0.934824
$626$	0	0
$627$	−1.57035e28	−0.0667864
$628$	0	0
$629$	3.68539e29	1.51583
$630$	0	0
$631$	−2.44143e29	−0.971261	−0.485631	−	0.874164i	$-0.661410\pi$
$631$	−2.44143e29	−0.971261	−0.485631	+	0.874164i	$0.661410\pi$
$632$	0	0
$633$	6.76435e29	2.60308
$634$	0	0
$635$	2.44123e27	0.00908835
$636$	0	0
$637$	1.34136e27	0.00483150
$638$	0	0
$639$	−7.92865e29	−2.76338
$640$	0	0
$641$	−3.44014e28	−0.116029	−0.0580145	−	0.998316i	$-0.518477\pi$
$641$	−3.44014e28	−0.116029	−0.0580145	+	0.998316i	$0.518477\pi$
$642$	0	0
$643$	4.55690e29	1.48749	0.743745	−	0.668463i	$-0.233048\pi$
$643$	4.55690e29	1.48749	0.743745	+	0.668463i	$0.233048\pi$
$644$	0	0
$645$	−1.38582e29	−0.437855
$646$	0	0
$647$	4.99700e29	1.52832	0.764161	−	0.645026i	$-0.223153\pi$
$647$	4.99700e29	1.52832	0.764161	+	0.645026i	$0.223153\pi$
$648$	0	0
$649$	1.70470e28	0.0504754
$650$	0	0
$651$	5.74726e29	1.64763
$652$	0	0
$653$	2.87688e29	0.798607	0.399303	−	0.916819i	$-0.369252\pi$
$653$	2.87688e29	0.798607	0.399303	+	0.916819i	$0.369252\pi$
$654$	0	0
$655$	−2.39740e29	−0.644475
$656$	0	0
$657$	4.16778e29	1.08510
$658$	0	0
$659$	4.60817e29	1.16207	0.581034	−	0.813879i	$-0.302648\pi$
$659$	4.60817e29	1.16207	0.581034	+	0.813879i	$0.302648\pi$
$660$	0	0
$661$	2.94237e29	0.718756	0.359378	−	0.933192i	$-0.382989\pi$
$661$	2.94237e29	0.718756	0.359378	+	0.933192i	$0.382989\pi$
$662$	0	0
$663$	−1.72339e29	−0.407841
$664$	0	0
$665$	−2.15585e29	−0.494300
$666$	0	0
$667$	−3.35833e29	−0.746109
$668$	0	0
$669$	9.22226e27	0.0198547
$670$	0	0
$671$	3.15912e28	0.0659143
$672$	0	0
$673$	−4.64032e29	−0.938403	−0.469202	−	0.883091i	$-0.655458\pi$
$673$	−4.64032e29	−0.938403	−0.469202	+	0.883091i	$0.655458\pi$
$674$	0	0
$675$	−1.17106e29	−0.229556
$676$	0	0
$677$	−3.90684e29	−0.742411	−0.371206	−	0.928551i	$-0.621056\pi$
$677$	−3.90684e29	−0.742411	−0.371206	+	0.928551i	$0.621056\pi$
$678$	0	0
$679$	1.99408e29	0.367374
$680$	0	0
$681$	−1.22168e30	−2.18230
$682$	0	0
$683$	1.80255e29	0.312227	0.156113	−	0.987739i	$-0.450103\pi$
$683$	1.80255e29	0.312227	0.156113	+	0.987739i	$0.450103\pi$
$684$	0	0
$685$	−1.52044e29	−0.255399
$686$	0	0
$687$	1.04334e30	1.69972
$688$	0	0
$689$	4.02684e28	0.0636301
$690$	0	0
$691$	−9.69163e28	−0.148552	−0.0742758	−	0.997238i	$-0.523665\pi$
$691$	−9.69163e28	−0.148552	−0.0742758	+	0.997238i	$0.523665\pi$
$692$	0	0
$693$	−1.24193e29	−0.184671
$694$	0	0
$695$	−4.96350e29	−0.716057
$696$	0	0
$697$	−9.34700e29	−1.30836
$698$	0	0
$699$	1.05265e30	1.42980
$700$	0	0
$701$	5.44452e29	0.717663	0.358831	−	0.933402i	$-0.383175\pi$
$701$	5.44452e29	0.717663	0.358831	+	0.933402i	$0.383175\pi$
$702$	0	0
$703$	3.78160e29	0.483777
$704$	0	0
$705$	8.85645e29	1.09970
$706$	0	0
$707$	−8.66363e29	−1.04423
$708$	0	0
$709$	1.34223e30	1.57052	0.785258	−	0.619168i	$-0.212530\pi$
$709$	1.34223e30	1.57052	0.785258	+	0.619168i	$0.212530\pi$
$710$	0	0
$711$	−3.58041e30	−4.06725
$712$	0	0
$713$	−1.09052e30	−1.20281
$714$	0	0
$715$	−7.46126e27	−0.00799095
$716$	0	0
$717$	3.18296e29	0.331039
$718$	0	0
$719$	−9.27979e29	−0.937313	−0.468657	−	0.883380i	$-0.655262\pi$
$719$	−9.27979e29	−0.937313	−0.468657	+	0.883380i	$0.655262\pi$
$720$	0	0
$721$	−1.75805e30	−1.72469
$722$	0	0
$723$	2.00438e30	1.90997
$724$	0	0
$725$	−3.50300e28	−0.0324258
$726$	0	0
$727$	1.97796e29	0.177871	0.0889357	−	0.996037i	$-0.471653\pi$
$727$	1.97796e29	0.177871	0.0889357	+	0.996037i	$0.471653\pi$
$728$	0	0
$729$	6.40738e30	5.59811
$730$	0	0
$731$	4.39744e29	0.373308
$732$	0	0
$733$	−8.95031e28	−0.0738323	−0.0369161	−	0.999318i	$-0.511753\pi$
$733$	−8.95031e28	−0.0738323	−0.0369161	+	0.999318i	$0.511753\pi$
$734$	0	0
$735$	9.08821e28	0.0728553
$736$	0	0
$737$	2.22017e28	0.0172973
$738$	0	0
$739$	1.59180e29	0.120537	0.0602687	−	0.998182i	$-0.480804\pi$
$739$	1.59180e29	0.120537	0.0602687	+	0.998182i	$0.480804\pi$
$740$	0	0
$741$	−1.76838e29	−0.130162
$742$	0	0
$743$	9.50012e28	0.0679746	0.0339873	−	0.999422i	$-0.489179\pi$
$743$	9.50012e28	0.0679746	0.0339873	+	0.999422i	$0.489179\pi$
$744$	0	0
$745$	5.43956e29	0.378376
$746$	0	0
$747$	−4.87295e29	−0.329553
$748$	0	0
$749$	2.94074e29	0.193374
$750$	0	0
$751$	2.97935e30	1.90503	0.952516	−	0.304488i	$-0.0984855\pi$
$751$	2.97935e30	1.90503	0.952516	+	0.304488i	$0.0984855\pi$
$752$	0	0
$753$	−4.81382e30	−2.99325
$754$	0	0
$755$	−2.92664e30	−1.76981
$756$	0	0
$757$	8.10128e29	0.476482	0.238241	−	0.971206i	$-0.423429\pi$
$757$	8.10128e29	0.476482	0.238241	+	0.971206i	$0.423429\pi$
$758$	0	0
$759$	3.17070e29	0.181391
$760$	0	0
$761$	8.84872e29	0.492426	0.246213	−	0.969216i	$-0.420814\pi$
$761$	8.84872e29	0.492426	0.246213	+	0.969216i	$0.420814\pi$
$762$	0	0
$763$	3.11590e30	1.68685
$764$	0	0
$765$	−8.67826e30	−4.57075
$766$	0	0
$767$	1.91968e29	0.0983732
$768$	0	0
$769$	2.59763e29	0.129524	0.0647620	−	0.997901i	$-0.479371\pi$
$769$	2.59763e29	0.129524	0.0647620	+	0.997901i	$0.479371\pi$
$770$	0	0
$771$	−7.59275e30	−3.68407
$772$	0	0
$773$	3.97566e30	1.87726	0.938630	−	0.344925i	$-0.112096\pi$
$773$	3.97566e30	1.87726	0.938630	+	0.344925i	$0.112096\pi$
$774$	0	0
$775$	−1.13750e29	−0.0522738
$776$	0	0
$777$	4.02402e30	1.79986
$778$	0	0
$779$	−9.59102e29	−0.417562
$780$	0	0
$781$	1.46556e29	0.0621109
$782$	0	0
$783$	4.78484e30	1.97410
$784$	0	0
$785$	2.41300e30	0.969228
$786$	0	0
$787$	1.54080e30	0.602577	0.301288	−	0.953533i	$-0.402583\pi$
$787$	1.54080e30	0.602577	0.301288	+	0.953533i	$0.402583\pi$
$788$	0	0
$789$	−5.10251e30	−1.94301
$790$	0	0
$791$	−2.46971e30	−0.915784
$792$	0	0
$793$	3.55750e29	0.128463
$794$	0	0
$795$	2.72833e30	0.959493
$796$	0	0
$797$	1.29587e29	0.0443862	0.0221931	−	0.999754i	$-0.492935\pi$
$797$	1.29587e29	0.0443862	0.0221931	+	0.999754i	$0.492935\pi$
$798$	0	0
$799$	−2.81029e30	−0.937585
$800$	0	0
$801$	−6.96663e30	−2.26403
$802$	0	0
$803$	−7.70386e28	−0.0243891
$804$	0	0
$805$	4.35287e30	1.34251
$806$	0	0
$807$	8.84510e30	2.65785
$808$	0	0
$809$	−2.25920e30	−0.661449	−0.330724	−	0.943727i	$-0.607293\pi$
$809$	−2.25920e30	−0.661449	−0.330724	+	0.943727i	$0.607293\pi$
$810$	0	0
$811$	4.17856e30	1.19209	0.596043	−	0.802953i	$-0.296739\pi$
$811$	4.17856e30	1.19209	0.596043	+	0.802953i	$0.296739\pi$
$812$	0	0
$813$	4.91617e29	0.136671
$814$	0	0
$815$	−1.39600e30	−0.378209
$816$	0	0
$817$	4.51224e29	0.119141
$818$	0	0
$819$	−1.39855e30	−0.359911
$820$	0	0
$821$	−1.32971e29	−0.0333543	−0.0166772	−	0.999861i	$-0.505309\pi$
$821$	−1.32971e29	−0.0333543	−0.0166772	+	0.999861i	$0.505309\pi$
$822$	0	0
$823$	−1.12434e30	−0.274915	−0.137457	−	0.990508i	$-0.543893\pi$
$823$	−1.12434e30	−0.274915	−0.137457	+	0.990508i	$0.543893\pi$
$824$	0	0
$825$	3.30728e28	0.00788324
$826$	0	0
$827$	−3.53339e30	−0.821077	−0.410539	−	0.911843i	$-0.634659\pi$
$827$	−3.53339e30	−0.821077	−0.410539	+	0.911843i	$0.634659\pi$
$828$	0	0
$829$	3.22988e30	0.731751	0.365875	−	0.930664i	$-0.380770\pi$
$829$	3.22988e30	0.731751	0.365875	+	0.930664i	$0.380770\pi$
$830$	0	0
$831$	8.95077e30	1.97720
$832$	0	0
$833$	−2.88384e29	−0.0621152
$834$	0	0
$835$	−4.43031e29	−0.0930521
$836$	0	0
$837$	1.55374e31	3.18245
$838$	0	0
$839$	1.24571e30	0.248838	0.124419	−	0.992230i	$-0.460293\pi$
$839$	1.24571e30	0.248838	0.124419	+	0.992230i	$0.460293\pi$
$840$	0	0
$841$	−3.70155e30	−0.721150
$842$	0	0
$843$	−1.79590e31	−3.41265
$844$	0	0
$845$	5.14276e30	0.953237
$846$	0	0
$847$	−5.40129e30	−0.976611
$848$	0	0
$849$	−4.72934e30	−0.834200
$850$	0	0
$851$	−7.63544e30	−1.31393
$852$	0	0
$853$	−6.54885e30	−1.09951	−0.549757	−	0.835325i	$-0.685280\pi$
$853$	−6.54885e30	−1.09951	−0.549757	+	0.835325i	$0.685280\pi$
$854$	0	0
$855$	−8.90483e30	−1.45875
$856$	0	0
$857$	1.91673e30	0.306381	0.153191	−	0.988197i	$-0.451045\pi$
$857$	1.91673e30	0.306381	0.153191	+	0.988197i	$0.451045\pi$
$858$	0	0
$859$	−2.28266e30	−0.356051	−0.178026	−	0.984026i	$-0.556971\pi$
$859$	−2.28266e30	−0.356051	−0.178026	+	0.984026i	$0.556971\pi$
$860$	0	0
$861$	−1.02058e31	−1.55352
$862$	0	0
$863$	−7.87662e30	−1.17011	−0.585055	−	0.810994i	$-0.698927\pi$
$863$	−7.87662e30	−1.17011	−0.585055	+	0.810994i	$0.698927\pi$
$864$	0	0
$865$	3.59092e30	0.520639
$866$	0	0
$867$	2.31067e31	3.26992
$868$	0	0
$869$	6.61814e29	0.0914173
$870$	0	0
$871$	2.50014e29	0.0337112
$872$	0	0
$873$	8.23662e30	1.08418
$874$	0	0
$875$	7.84817e30	1.00852
$876$	0	0
$877$	−1.25742e31	−1.57756	−0.788778	−	0.614678i	$-0.789286\pi$
$877$	−1.25742e31	−1.57756	−0.788778	+	0.614678i	$0.789286\pi$
$878$	0	0
$879$	1.17509e31	1.43942
$880$	0	0
$881$	−5.18000e29	−0.0619560	−0.0309780	−	0.999520i	$-0.509862\pi$
$881$	−5.18000e29	−0.0619560	−0.0309780	+	0.999520i	$0.509862\pi$
$882$	0	0
$883$	−4.26502e30	−0.498120	−0.249060	−	0.968488i	$-0.580122\pi$
$883$	−4.26502e30	−0.498120	−0.249060	+	0.968488i	$0.580122\pi$
$884$	0	0
$885$	1.30065e31	1.48339
$886$	0	0
$887$	1.39563e31	1.55444	0.777218	−	0.629232i	$-0.216630\pi$
$887$	1.39563e31	1.55444	0.777218	+	0.629232i	$0.216630\pi$
$888$	0	0
$889$	−8.45819e28	−0.00920045
$890$	0	0
$891$	−2.74515e30	−0.291643
$892$	0	0
$893$	−2.88366e30	−0.299229
$894$	0	0
$895$	−1.18160e31	−1.19765
$896$	0	0
$897$	3.57054e30	0.353520
$898$	0	0
$899$	4.64773e30	0.449536
$900$	0	0
$901$	−8.65743e30	−0.818048
$902$	0	0
$903$	4.80149e30	0.443256
$904$	0	0
$905$	5.29006e30	0.477145
$906$	0	0
$907$	−7.99799e30	−0.704862	−0.352431	−	0.935838i	$-0.614645\pi$
$907$	−7.99799e30	−0.704862	−0.352431	+	0.935838i	$0.614645\pi$
$908$	0	0
$909$	−3.57855e31	−3.08168
$910$	0	0
$911$	6.04758e30	0.508907	0.254454	−	0.967085i	$-0.418104\pi$
$911$	6.04758e30	0.508907	0.254454	+	0.967085i	$0.418104\pi$
$912$	0	0
$913$	9.00733e28	0.00740718
$914$	0	0
$915$	2.41034e31	1.93712
$916$	0	0
$917$	8.30632e30	0.652425
$918$	0	0
$919$	2.36741e31	1.81744	0.908719	−	0.417408i	$-0.137061\pi$
$919$	2.36741e31	1.81744	0.908719	+	0.417408i	$0.137061\pi$
$920$	0	0
$921$	3.88598e31	2.91591
$922$	0	0
$923$	1.65037e30	0.121050
$924$	0	0
$925$	−7.96435e29	−0.0571034
$926$	0	0
$927$	−7.26169e31	−5.08980
$928$	0	0
$929$	−2.41968e31	−1.65803	−0.829016	−	0.559226i	$-0.811099\pi$
$929$	−2.41968e31	−1.65803	−0.829016	+	0.559226i	$0.811099\pi$
$930$	0	0
$931$	−2.95912e29	−0.0198240
$932$	0	0
$933$	3.46773e31	2.27137
$934$	0	0
$935$	1.60412e30	0.102734
$936$	0	0
$937$	−8.58736e30	−0.537767	−0.268883	−	0.963173i	$-0.586655\pi$
$937$	−8.58736e30	−0.537767	−0.268883	+	0.963173i	$0.586655\pi$
$938$	0	0
$939$	−6.27148e30	−0.384044
$940$	0	0
$941$	−5.47302e30	−0.327745	−0.163872	−	0.986482i	$-0.552399\pi$
$941$	−5.47302e30	−0.327745	−0.163872	+	0.986482i	$0.552399\pi$
$942$	0	0
$943$	1.93652e31	1.13410
$944$	0	0
$945$	−6.20183e31	−3.55211
$946$	0	0
$947$	2.44243e31	1.36819	0.684096	−	0.729392i	$-0.260197\pi$
$947$	2.44243e31	1.36819	0.684096	+	0.729392i	$0.260197\pi$
$948$	0	0
$949$	−8.67536e29	−0.0475327
$950$	0	0
$951$	−4.23960e31	−2.27211
$952$	0	0
$953$	1.04469e31	0.547661	0.273831	−	0.961778i	$-0.411709\pi$
$953$	1.04469e31	0.547661	0.273831	+	0.961778i	$0.411709\pi$
$954$	0	0
$955$	3.52889e30	0.180968
$956$	0	0
$957$	−1.35133e30	−0.0677931
$958$	0	0
$959$	5.26791e30	0.258549
$960$	0	0
$961$	−5.73329e30	−0.275301
$962$	0	0
$963$	1.21469e31	0.570675
$964$	0	0
$965$	−2.81884e31	−1.29579
$966$	0	0
$967$	−3.41008e31	−1.53386	−0.766931	−	0.641729i	$-0.778217\pi$
$967$	−3.41008e31	−1.53386	−0.766931	+	0.641729i	$0.778217\pi$
$968$	0	0
$969$	3.80190e31	1.67340
$970$	0	0
$971$	3.17245e31	1.36645	0.683224	−	0.730209i	$-0.260577\pi$
$971$	3.17245e31	1.36645	0.683224	+	0.730209i	$0.260577\pi$
$972$	0	0
$973$	1.71972e31	0.724890
$974$	0	0
$975$	3.72435e29	0.0153639
$976$	0	0
$977$	2.33987e31	0.944710	0.472355	−	0.881408i	$-0.343404\pi$
$977$	2.33987e31	0.944710	0.472355	+	0.881408i	$0.343404\pi$
$978$	0	0
$979$	1.28774e30	0.0508872
$980$	0	0
$981$	1.28704e32	4.97814
$982$	0	0
$983$	1.15166e28	0.000436024 0	0.000218012	−	1.00000i	$-0.499931\pi$
$983$	1.15166e28	0.000436024 0	0.000218012		1.00000i	$0.499931\pi$
$984$	0	0
$985$	1.35156e31	0.500906
$986$	0	0
$987$	−3.06852e31	−1.11326
$988$	0	0
$989$	−9.11067e30	−0.323586
$990$	0	0
$991$	−1.52057e31	−0.528730	−0.264365	−	0.964423i	$-0.585162\pi$
$991$	−1.52057e31	−0.528730	−0.264365	+	0.964423i	$0.585162\pi$
$992$	0	0
$993$	−4.78170e31	−1.62785
$994$	0	0
$995$	−3.90579e31	−1.30187
$996$	0	0
$997$	−4.15395e31	−1.35569	−0.677847	−	0.735203i	$-0.737087\pi$
$997$	−4.15395e31	−1.35569	−0.677847	+	0.735203i	$0.737087\pi$
$998$	0	0
$999$	1.08787e32	3.47649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.22.a.b.1.3	✓	3
3.2	odd	2		72.22.a.f.1.2		3
4.3	odd	2		16.22.a.f.1.1		3
8.3	odd	2		64.22.a.m.1.3		3
8.5	even	2		64.22.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.22.a.b.1.3	✓	3	1.1	even	1	trivial
16.22.a.f.1.1		3	4.3	odd	2
64.22.a.l.1.1		3	8.5	even	2
64.22.a.m.1.3		3	8.3	odd	2
72.22.a.f.1.2		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q+201833. q^{3} -2.11555e7 q^{5} +7.32981e8 q^{7} +3.02761e10 q^{9} +O(q^{10})\)	\(q+201833. q^{3} -2.11555e7 q^{5} +7.32981e8 q^{7} +3.02761e10 q^{9} -5.59634e9 q^{11} -6.30207e10 q^{13} -4.26988e12 q^{15} +1.35490e13 q^{17} +1.39028e13 q^{19} +1.47940e14 q^{21} -2.80711e14 q^{23} -2.92803e13 q^{25} +3.99948e15 q^{27} +1.19637e15 q^{29} +3.88487e15 q^{31} -1.12953e15 q^{33} -1.55066e16 q^{35} +2.72004e16 q^{37} -1.27197e16 q^{39} -6.89865e16 q^{41} +3.24557e16 q^{43} -6.40508e17 q^{45} -2.07417e17 q^{47} -2.12844e16 q^{49} +2.73464e18 q^{51} -6.38971e17 q^{53} +1.18394e17 q^{55} +2.80603e18 q^{57} -3.04610e18 q^{59} -5.64497e18 q^{61} +2.21918e19 q^{63} +1.33324e18 q^{65} -3.96718e18 q^{67} -5.66566e19 q^{69} -2.61878e19 q^{71} +1.37659e19 q^{73} -5.90972e18 q^{75} -4.10201e18 q^{77} -1.18258e20 q^{79} +4.90526e20 q^{81} -1.60950e19 q^{83} -2.86637e20 q^{85} +2.41466e20 q^{87} -2.30103e20 q^{89} -4.61930e19 q^{91} +7.84094e20 q^{93} -2.94120e20 q^{95} +2.72050e20 q^{97} -1.69436e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 96764 q^{3} - 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9}+O(q^{10}) \) 3 * q + 96764 * q^3 - 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9	\( 3 q + 96764 q^{3} - 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9} - 40335108684 q^{11} + 133734425946 q^{13} - 1223136458200 q^{15} + 7797732274422 q^{17} + 35788199781996 q^{19} + 198539224853088 q^{21} + 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 94\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 96764 * q^3 - 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9 - 40335108684 * q^11 + 133734425946 * q^13 - 1223136458200 * q^15 + 7797732274422 * q^17 + 35788199781996 * q^19 + 198539224853088 * q^21 + 193770761479080 * q^23 + 1127564438439501 * q^25 + 5282002293508952 * q^27 + 5607343422466122 * q^29 + 11246757871503072 * q^31 + 10014149026970384 * q^33 + 5274251425350096 * q^35 - 24272499791100606 * q^37 - 52925308377862264 * q^39 - 298159108991869602 * q^41 - 33333932139754860 * q^43 - 927411477977893478 * q^45 - 120874283547603888 * q^47 - 850403331975639477 * q^49 + 3954388789815661240 * q^51 - 1138443393004854222 * q^53 + 6957552484263571704 * q^55 + 681697547133656944 * q^57 + 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 21785214559631141976 * q^63 - 20714581819561144452 * q^65 - 15793054074531629124 * q^67 - 72988310309689939168 * q^69 - 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 75231657393126995900 * q^75 + 68380237365358617888 * q^77 - 131735321299806049488 * q^79 + 500425796062282339147 * q^81 + 64013993832679681068 * q^83 + 390258202763001297252 * q^85 + 101898953973185066568 * q^87 + 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 640035369009914700160 * q^93 - 1036372406649019824696 * q^95 - 324059336514148638042 * q^97 - 949982905688477352860 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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