LMFDB - Embedded newform 8.20.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,20,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, 20, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 20 $
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:	$18.3053357245$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1453}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 363 $ x^2 - x - 363
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$19.5591$ 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)

$f(q)$	$=$	$q-50549.5 q^{3} +2.22342e6 q^{5} +1.60989e8 q^{7} +1.39299e9 q^{9} +O(q^{10})$	$q-50549.5 q^{3} +2.22342e6 q^{5} +1.60989e8 q^{7} +1.39299e9 q^{9} -1.17593e10 q^{11} -2.44117e9 q^{13} -1.12393e11 q^{15} +5.98454e11 q^{17} -2.51671e12 q^{19} -8.13789e12 q^{21} -6.35737e11 q^{23} -1.41299e13 q^{25} -1.16633e13 q^{27} -4.78062e13 q^{29} -2.03104e14 q^{31} +5.94429e14 q^{33} +3.57946e14 q^{35} +8.95266e14 q^{37} +1.23400e14 q^{39} -4.61047e14 q^{41} -2.21551e15 q^{43} +3.09721e15 q^{45} -1.04653e16 q^{47} +1.45184e16 q^{49} -3.02516e16 q^{51} -3.43716e16 q^{53} -2.61460e16 q^{55} +1.27219e17 q^{57} +9.13126e16 q^{59} -1.19474e17 q^{61} +2.24256e17 q^{63} -5.42776e15 q^{65} -1.98838e17 q^{67} +3.21362e16 q^{69} -1.38863e17 q^{71} -3.63434e16 q^{73} +7.14258e17 q^{75} -1.89312e18 q^{77} -4.36121e17 q^{79} -1.02945e18 q^{81} +4.52389e17 q^{83} +1.33062e18 q^{85} +2.41658e18 q^{87} +3.30628e18 q^{89} -3.93001e17 q^{91} +1.02668e19 q^{93} -5.59572e18 q^{95} -6.03558e18 q^{97} -1.63807e19 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9}+O(q^{10}) $ 2 * q - 27912 * q^3 + 1226620 * q^5 + 88510512 * q^7 + 743186538 * q^9	$ 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9} - 7163787608 q^{11} - 10126923604 q^{13} - 134958171120 q^{15} - 72045078940 q^{17} - 3120480472232 q^{19} - 9778613729472 q^{21} - 14759207090288 q^{23} - 32209737998450 q^{25} - 52683939132624 q^{27} - 30249539245044 q^{29} - 123389562777920 q^{31} + 698460345860448 q^{33} + 430192267170720 q^{35} + 20\!\cdots\!24 q^{37}+ \cdots - 19\!\cdots\!52 q^{99}+O(q^{100}) $ 2 * q - 27912 * q^3 + 1226620 * q^5 + 88510512 * q^7 + 743186538 * q^9 - 7163787608 * q^11 - 10126923604 * q^13 - 134958171120 * q^15 - 72045078940 * q^17 - 3120480472232 * q^19 - 9778613729472 * q^21 - 14759207090288 * q^23 - 32209737998450 * q^25 - 52683939132624 * q^27 - 30249539245044 * q^29 - 123389562777920 * q^31 + 698460345860448 * q^33 + 430192267170720 * q^35 + 2015393170174524 * q^37 - 50586287718576 * q^39 + 2540784959504244 * q^41 - 5633655093389464 * q^43 + 3744938810129580 * q^45 - 21948339587130336 * q^47 + 8372586663064786 * q^49 - 45429993200591760 * q^51 - 9418125066904676 * q^53 - 30726853722003280 * q^55 + 113550921657554592 * q^57 + 98542449590407624 * q^59 + 10292145377839820 * q^61 + 271352203396831728 * q^63 + 2233432360014760 * q^65 + 75753628003984504 * q^67 - 287583943397508672 * q^69 + 17407052566713776 * q^71 - 857508255059832268 * q^73 + 304974862137280200 * q^75 - 2226194512471875648 * q^77 - 226291921444855072 * q^79 - 1202809709767720302 * q^81 + 767515701460985048 * q^83 + 1998974022815558200 * q^85 + 2814015710341120464 * q^87 + 6092545894435174548 * q^89 + 164047654392377376 * q^91 + 12071374787086705920 * q^93 - 4993887710656339120 * q^95 + 1548148249522347076 * q^97 - 19366859521479728952 * 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
$3$	−50549.5	−1.48274	−0.741370	−	0.671096i	$-0.765824\pi$
$3$	−50549.5	−1.48274	−0.741370	+	0.671096i	$0.765824\pi$
$4$	0	0
$5$	2.22342e6	0.509105	0.254552	−	0.967059i	$-0.418072\pi$
$5$	2.22342e6	0.509105	0.254552	+	0.967059i	$0.418072\pi$
$6$	0	0
$7$	1.60989e8	1.50787	0.753934	−	0.656950i	$-0.228154\pi$
$7$	1.60989e8	1.50787	0.753934	+	0.656950i	$0.228154\pi$
$8$	0	0
$9$	1.39299e9	1.19852
$10$	0	0
$11$	−1.17593e10	−1.50367	−0.751835	−	0.659352i	$-0.770831\pi$
$11$	−1.17593e10	−1.50367	−0.751835	+	0.659352i	$0.770831\pi$
$12$	0	0
$13$	−2.44117e9	−0.0638464	−0.0319232	−	0.999490i	$-0.510163\pi$
$13$	−2.44117e9	−0.0638464	−0.0319232	+	0.999490i	$0.510163\pi$
$14$	0	0
$15$	−1.12393e11	−0.754870
$16$	0	0
$17$	5.98454e11	1.22396	0.611978	−	0.790874i	$-0.290374\pi$
$17$	5.98454e11	1.22396	0.611978	+	0.790874i	$0.290374\pi$
$18$	0	0
$19$	−2.51671e12	−1.78927	−0.894633	−	0.446802i	$-0.852563\pi$
$19$	−2.51671e12	−1.78927	−0.894633	+	0.446802i	$0.852563\pi$
$20$	0	0
$21$	−8.13789e12	−2.23578
$22$	0	0
$23$	−6.35737e11	−0.0735975	−0.0367988	−	0.999323i	$-0.511716\pi$
$23$	−6.35737e11	−0.0735975	−0.0367988	+	0.999323i	$0.511716\pi$
$24$	0	0
$25$	−1.41299e13	−0.740812
$26$	0	0
$27$	−1.16633e13	−0.294351
$28$	0	0
$29$	−4.78062e13	−0.611932	−0.305966	−	0.952043i	$-0.598979\pi$
$29$	−4.78062e13	−0.611932	−0.305966	+	0.952043i	$0.598979\pi$
$30$	0	0
$31$	−2.03104e14	−1.37970	−0.689848	−	0.723954i	$-0.742323\pi$
$31$	−2.03104e14	−1.37970	−0.689848	+	0.723954i	$0.742323\pi$
$32$	0	0
$33$	5.94429e14	2.22955
$34$	0	0
$35$	3.57946e14	0.767664
$36$	0	0
$37$	8.95266e14	1.13249	0.566246	−	0.824236i	$-0.308395\pi$
$37$	8.95266e14	1.13249	0.566246	+	0.824236i	$0.308395\pi$
$38$	0	0
$39$	1.23400e14	0.0946676
$40$	0	0
$41$	−4.61047e14	−0.219937	−0.109969	−	0.993935i	$-0.535075\pi$
$41$	−4.61047e14	−0.219937	−0.109969	+	0.993935i	$0.535075\pi$
$42$	0	0
$43$	−2.21551e15	−0.672238	−0.336119	−	0.941820i	$-0.609114\pi$
$43$	−2.21551e15	−0.672238	−0.336119	+	0.941820i	$0.609114\pi$
$44$	0	0
$45$	3.09721e15	0.610171
$46$	0	0
$47$	−1.04653e16	−1.36403	−0.682013	−	0.731340i	$-0.738895\pi$
$47$	−1.04653e16	−1.36403	−0.682013	+	0.731340i	$0.738895\pi$
$48$	0	0
$49$	1.45184e16	1.27367
$50$	0	0
$51$	−3.02516e16	−1.81481
$52$	0	0
$53$	−3.43716e16	−1.43080	−0.715400	−	0.698715i	$-0.753756\pi$
$53$	−3.43716e16	−1.43080	−0.715400	+	0.698715i	$0.753756\pi$
$54$	0	0
$55$	−2.61460e16	−0.765525
$56$	0	0
$57$	1.27219e17	2.65302
$58$	0	0
$59$	9.13126e16	1.37226	0.686130	−	0.727479i	$-0.259308\pi$
$59$	9.13126e16	1.37226	0.686130	+	0.727479i	$0.259308\pi$
$60$	0	0
$61$	−1.19474e17	−1.30809	−0.654047	−	0.756454i	$-0.726930\pi$
$61$	−1.19474e17	−1.30809	−0.654047	+	0.756454i	$0.726930\pi$
$62$	0	0
$63$	2.24256e17	1.80721
$64$	0	0
$65$	−5.42776e15	−0.0325045
$66$	0	0
$67$	−1.98838e17	−0.892872	−0.446436	−	0.894816i	$-0.647307\pi$
$67$	−1.98838e17	−0.892872	−0.446436	+	0.894816i	$0.647307\pi$
$68$	0	0
$69$	3.21362e16	0.109126
$70$	0	0
$71$	−1.38863e17	−0.359446	−0.179723	−	0.983717i	$-0.557520\pi$
$71$	−1.38863e17	−0.359446	−0.179723	+	0.983717i	$0.557520\pi$
$72$	0	0
$73$	−3.63434e16	−0.0722533	−0.0361267	−	0.999347i	$-0.511502\pi$
$73$	−3.63434e16	−0.0722533	−0.0361267	+	0.999347i	$0.511502\pi$
$74$	0	0
$75$	7.14258e17	1.09843
$76$	0	0
$77$	−1.89312e18	−2.26734
$78$	0	0
$79$	−4.36121e17	−0.409402	−0.204701	−	0.978825i	$-0.565622\pi$
$79$	−4.36121e17	−0.409402	−0.204701	+	0.978825i	$0.565622\pi$
$80$	0	0
$81$	−1.02945e18	−0.762073
$82$	0	0
$83$	4.52389e17	0.265626	0.132813	−	0.991141i	$-0.457599\pi$
$83$	4.52389e17	0.265626	0.132813	+	0.991141i	$0.457599\pi$
$84$	0	0
$85$	1.33062e18	0.623123
$86$	0	0
$87$	2.41658e18	0.907335
$88$	0	0
$89$	3.30628e18	1.00031	0.500154	−	0.865936i	$-0.333277\pi$
$89$	3.30628e18	1.00031	0.500154	+	0.865936i	$0.333277\pi$
$90$	0	0
$91$	−3.93001e17	−0.0962720
$92$	0	0
$93$	1.02668e19	2.04573
$94$	0	0
$95$	−5.59572e18	−0.910924
$96$	0	0
$97$	−6.03558e18	−0.806097	−0.403049	−	0.915179i	$-0.632049\pi$
$97$	−6.03558e18	−0.806097	−0.403049	+	0.915179i	$0.632049\pi$
$98$	0	0
$99$	−1.63807e19	−1.80217
$100$	0	0
$101$	9.09027e18	0.827034	0.413517	−	0.910496i	$-0.364300\pi$
$101$	9.09027e18	0.827034	0.413517	+	0.910496i	$0.364300\pi$
$102$	0	0
$103$	1.16083e19	0.876628	0.438314	−	0.898822i	$-0.355576\pi$
$103$	1.16083e19	0.876628	0.438314	+	0.898822i	$0.355576\pi$
$104$	0	0
$105$	−1.80940e19	−1.13825
$106$	0	0
$107$	9.99623e18	0.525642	0.262821	−	0.964845i	$-0.415347\pi$
$107$	9.99623e18	0.525642	0.262821	+	0.964845i	$0.415347\pi$
$108$	0	0
$109$	−4.34179e18	−0.191477	−0.0957385	−	0.995407i	$-0.530521\pi$
$109$	−4.34179e18	−0.191477	−0.0957385	+	0.995407i	$0.530521\pi$
$110$	0	0
$111$	−4.52552e19	−1.67919
$112$	0	0
$113$	−9.09237e18	−0.284729	−0.142365	−	0.989814i	$-0.545471\pi$
$113$	−9.09237e18	−0.284729	−0.142365	+	0.989814i	$0.545471\pi$
$114$	0	0
$115$	−1.41351e18	−0.0374689
$116$	0	0
$117$	−3.40053e18	−0.0765210
$118$	0	0
$119$	9.63443e19	1.84557
$120$	0	0
$121$	7.71229e19	1.26102
$122$	0	0
$123$	2.33057e19	0.326110
$124$	0	0
$125$	−7.38252e19	−0.886256
$126$	0	0
$127$	9.31085e19	0.961289	0.480645	−	0.876915i	$-0.340403\pi$
$127$	9.31085e19	0.961289	0.480645	+	0.876915i	$0.340403\pi$
$128$	0	0
$129$	1.11993e20	0.996753
$130$	0	0
$131$	1.87606e19	0.144268	0.0721340	−	0.997395i	$-0.477019\pi$
$131$	1.87606e19	0.144268	0.0721340	+	0.997395i	$0.477019\pi$
$132$	0	0
$133$	−4.05162e20	−2.69798
$134$	0	0
$135$	−2.59324e19	−0.149855
$136$	0	0
$137$	−1.84318e20	−0.926240	−0.463120	−	0.886296i	$-0.653270\pi$
$137$	−1.84318e20	−0.926240	−0.463120	+	0.886296i	$0.653270\pi$
$138$	0	0
$139$	−1.59539e20	−0.698597	−0.349299	−	0.937011i	$-0.613580\pi$
$139$	−1.59539e20	−0.698597	−0.349299	+	0.937011i	$0.613580\pi$
$140$	0	0
$141$	5.29017e20	2.02250
$142$	0	0
$143$	2.87065e19	0.0960038
$144$	0	0
$145$	−1.06293e20	−0.311537
$146$	0	0
$147$	−7.33899e20	−1.88852
$148$	0	0
$149$	3.21712e20	0.728111	0.364055	−	0.931377i	$-0.381392\pi$
$149$	3.21712e20	0.728111	0.364055	+	0.931377i	$0.381392\pi$
$150$	0	0
$151$	−1.55272e20	−0.309607	−0.154804	−	0.987945i	$-0.549474\pi$
$151$	−1.55272e20	−0.309607	−0.154804	+	0.987945i	$0.549474\pi$
$152$	0	0
$153$	8.33641e20	1.46693
$154$	0	0
$155$	−4.51587e20	−0.702410
$156$	0	0
$157$	−4.54202e20	−0.625464	−0.312732	−	0.949841i	$-0.601244\pi$
$157$	−4.54202e20	−0.625464	−0.312732	+	0.949841i	$0.601244\pi$
$158$	0	0
$159$	1.73747e21	2.12150
$160$	0	0
$161$	−1.02346e20	−0.110975
$162$	0	0
$163$	7.72395e20	0.744830	0.372415	−	0.928066i	$-0.378530\pi$
$163$	7.72395e20	0.744830	0.372415	+	0.928066i	$0.378530\pi$
$164$	0	0
$165$	1.32167e21	1.13508
$166$	0	0
$167$	1.02583e21	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$167$	1.02583e21	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$168$	0	0
$169$	−1.45596e21	−0.995924
$170$	0	0
$171$	−3.50576e21	−2.14447
$172$	0	0
$173$	6.92487e20	0.379292	0.189646	−	0.981853i	$-0.439266\pi$
$173$	6.92487e20	0.379292	0.189646	+	0.981853i	$0.439266\pi$
$174$	0	0
$175$	−2.27475e21	−1.11705
$176$	0	0
$177$	−4.61581e21	−2.03471
$178$	0	0
$179$	2.42385e21	0.960290	0.480145	−	0.877189i	$-0.340584\pi$
$179$	2.42385e21	0.960290	0.480145	+	0.877189i	$0.340584\pi$
$180$	0	0
$181$	−9.72457e20	−0.346676	−0.173338	−	0.984862i	$-0.555455\pi$
$181$	−9.72457e20	−0.346676	−0.173338	+	0.984862i	$0.555455\pi$
$182$	0	0
$183$	6.03934e21	1.93956
$184$	0	0
$185$	1.99056e21	0.576558
$186$	0	0
$187$	−7.03742e21	−1.84043
$188$	0	0
$189$	−1.87765e21	−0.443842
$190$	0	0
$191$	6.76037e21	1.44595	0.722975	−	0.690874i	$-0.242774\pi$
$191$	6.76037e21	1.44595	0.722975	+	0.690874i	$0.242774\pi$
$192$	0	0
$193$	1.53000e21	0.296413	0.148206	−	0.988956i	$-0.452650\pi$
$193$	1.53000e21	0.296413	0.148206	+	0.988956i	$0.452650\pi$
$194$	0	0
$195$	2.74371e20	0.0481957
$196$	0	0
$197$	−6.79113e21	−1.08271	−0.541356	−	0.840793i	$-0.682089\pi$
$197$	−6.79113e21	−1.08271	−0.541356	+	0.840793i	$0.682089\pi$
$198$	0	0
$199$	9.45465e21	1.36943	0.684717	−	0.728809i	$-0.259926\pi$
$199$	9.45465e21	1.36943	0.684717	+	0.728809i	$0.259926\pi$
$200$	0	0
$201$	1.00512e22	1.32390
$202$	0	0
$203$	−7.69624e21	−0.922713
$204$	0	0
$205$	−1.02510e21	−0.111971
$206$	0	0
$207$	−8.85577e20	−0.0882080
$208$	0	0
$209$	2.95949e22	2.69046
$210$	0	0
$211$	−2.14995e22	−1.78544	−0.892721	−	0.450610i	$-0.851207\pi$
$211$	−2.14995e22	−1.78544	−0.892721	+	0.450610i	$0.851207\pi$
$212$	0	0
$213$	7.01948e21	0.532965
$214$	0	0
$215$	−4.92601e21	−0.342239
$216$	0	0
$217$	−3.26975e22	−2.08040
$218$	0	0
$219$	1.83714e21	0.107133
$220$	0	0
$221$	−1.46093e21	−0.0781452
$222$	0	0
$223$	−3.89942e22	−1.91471	−0.957356	−	0.288910i	$-0.906707\pi$
$223$	−3.89942e22	−1.91471	−0.957356	+	0.288910i	$0.906707\pi$
$224$	0	0
$225$	−1.96828e22	−0.887877
$226$	0	0
$227$	1.69117e21	0.0701361	0.0350680	−	0.999385i	$-0.488835\pi$
$227$	1.69117e21	0.0701361	0.0350680	+	0.999385i	$0.488835\pi$
$228$	0	0
$229$	3.34630e21	0.127681	0.0638407	−	0.997960i	$-0.479665\pi$
$229$	3.34630e21	0.127681	0.0638407	+	0.997960i	$0.479665\pi$
$230$	0	0
$231$	9.56962e22	3.36187
$232$	0	0
$233$	1.27609e22	0.413049	0.206524	−	0.978441i	$-0.433785\pi$
$233$	1.27609e22	0.413049	0.206524	+	0.978441i	$0.433785\pi$
$234$	0	0
$235$	−2.32688e22	−0.694433
$236$	0	0
$237$	2.20457e22	0.607037
$238$	0	0
$239$	−9.15938e21	−0.232855	−0.116428	−	0.993199i	$-0.537144\pi$
$239$	−9.15938e21	−0.232855	−0.116428	+	0.993199i	$0.537144\pi$
$240$	0	0
$241$	−4.54596e22	−1.06774	−0.533868	−	0.845568i	$-0.679262\pi$
$241$	−4.54596e22	−1.06774	−0.533868	+	0.845568i	$0.679262\pi$
$242$	0	0
$243$	6.55938e22	1.42431
$244$	0	0
$245$	3.22806e22	0.648431
$246$	0	0
$247$	6.14373e21	0.114238
$248$	0	0
$249$	−2.28680e22	−0.393854
$250$	0	0
$251$	−3.68788e22	−0.588677	−0.294338	−	0.955701i	$-0.595099\pi$
$251$	−3.68788e22	−0.588677	−0.294338	+	0.955701i	$0.595099\pi$
$252$	0	0
$253$	7.47585e21	0.110666
$254$	0	0
$255$	−6.72621e22	−0.923929
$256$	0	0
$257$	4.84420e22	0.617813	0.308907	−	0.951092i	$-0.400037\pi$
$257$	4.84420e22	0.617813	0.308907	+	0.951092i	$0.400037\pi$
$258$	0	0
$259$	1.44128e23	1.70765
$260$	0	0
$261$	−6.65936e22	−0.733411
$262$	0	0
$263$	6.84645e22	0.701271	0.350635	−	0.936512i	$-0.385966\pi$
$263$	6.84645e22	0.701271	0.350635	+	0.936512i	$0.385966\pi$
$264$	0	0
$265$	−7.64226e22	−0.728427
$266$	0	0
$267$	−1.67131e23	−1.48320
$268$	0	0
$269$	−2.17729e23	−1.79999	−0.899995	−	0.435900i	$-0.856430\pi$
$269$	−2.17729e23	−1.79999	−0.899995	+	0.435900i	$0.856430\pi$
$270$	0	0
$271$	−1.88487e23	−1.45236	−0.726180	−	0.687505i	$-0.758706\pi$
$271$	−1.88487e23	−1.45236	−0.726180	+	0.687505i	$0.758706\pi$
$272$	0	0
$273$	1.98660e22	0.142746
$274$	0	0
$275$	1.66158e23	1.11394
$276$	0	0
$277$	2.07964e23	1.30146	0.650729	−	0.759310i	$-0.274463\pi$
$277$	2.07964e23	1.30146	0.650729	+	0.759310i	$0.274463\pi$
$278$	0	0
$279$	−2.82923e23	−1.65359
$280$	0	0
$281$	3.33849e23	1.82322	0.911612	−	0.411052i	$-0.134838\pi$
$281$	3.33849e23	1.82322	0.911612	+	0.411052i	$0.134838\pi$
$282$	0	0
$283$	−2.26611e22	−0.115694	−0.0578470	−	0.998325i	$-0.518424\pi$
$283$	−2.26611e22	−0.115694	−0.0578470	+	0.998325i	$0.518424\pi$
$284$	0	0
$285$	2.82861e23	1.35066
$286$	0	0
$287$	−7.42233e22	−0.331636
$288$	0	0
$289$	1.19075e23	0.498070
$290$	0	0
$291$	3.05095e23	1.19523
$292$	0	0
$293$	−1.85421e23	−0.680639	−0.340319	−	0.940310i	$-0.610535\pi$
$293$	−1.85421e23	−0.680639	−0.340319	+	0.940310i	$0.610535\pi$
$294$	0	0
$295$	2.03027e23	0.698625
$296$	0	0
$297$	1.37152e23	0.442606
$298$	0	0
$299$	1.55194e21	0.00469894
$300$	0	0
$301$	−3.56671e23	−1.01365
$302$	0	0
$303$	−4.59508e23	−1.22628
$304$	0	0
$305$	−2.65641e23	−0.665958
$306$	0	0
$307$	7.93628e23	1.86983	0.934916	−	0.354870i	$-0.115475\pi$
$307$	7.93628e23	1.86983	0.934916	+	0.354870i	$0.115475\pi$
$308$	0	0
$309$	−5.86794e23	−1.29981
$310$	0	0
$311$	−5.20440e23	−1.08429	−0.542147	−	0.840284i	$-0.682388\pi$
$311$	−5.20440e23	−1.08429	−0.542147	+	0.840284i	$0.682388\pi$
$312$	0	0
$313$	−1.65345e23	−0.324130	−0.162065	−	0.986780i	$-0.551815\pi$
$313$	−1.65345e23	−0.324130	−0.162065	+	0.986780i	$0.551815\pi$
$314$	0	0
$315$	4.98615e23	0.920059
$316$	0	0
$317$	5.50226e23	0.956044	0.478022	−	0.878348i	$-0.341354\pi$
$317$	5.50226e23	0.956044	0.478022	+	0.878348i	$0.341354\pi$
$318$	0	0
$319$	5.62169e23	0.920143
$320$	0	0
$321$	−5.05304e23	−0.779391
$322$	0	0
$323$	−1.50614e24	−2.18998
$324$	0	0
$325$	3.44934e22	0.0472982
$326$	0	0
$327$	2.19475e23	0.283911
$328$	0	0
$329$	−1.68480e24	−2.05677
$330$	0	0
$331$	5.52973e23	0.637291	0.318646	−	0.947874i	$-0.396772\pi$
$331$	5.52973e23	0.637291	0.318646	+	0.947874i	$0.396772\pi$
$332$	0	0
$333$	1.24710e24	1.35731
$334$	0	0
$335$	−4.42102e23	−0.454565
$336$	0	0
$337$	5.19323e23	0.504607	0.252303	−	0.967648i	$-0.418812\pi$
$337$	5.19323e23	0.504607	0.252303	+	0.967648i	$0.418812\pi$
$338$	0	0
$339$	4.59615e23	0.422179
$340$	0	0
$341$	2.38837e24	2.07461
$342$	0	0
$343$	5.02207e23	0.412657
$344$	0	0
$345$	7.14524e22	0.0555566
$346$	0	0
$347$	1.43821e24	1.05850	0.529250	−	0.848466i	$-0.322473\pi$
$347$	1.43821e24	1.05850	0.529250	+	0.848466i	$0.322473\pi$
$348$	0	0
$349$	−8.46494e23	−0.589905	−0.294952	−	0.955512i	$-0.595304\pi$
$349$	−8.46494e23	−0.589905	−0.294952	+	0.955512i	$0.595304\pi$
$350$	0	0
$351$	2.84721e22	0.0187932
$352$	0	0
$353$	1.34329e24	0.840063	0.420031	−	0.907510i	$-0.362019\pi$
$353$	1.34329e24	0.840063	0.420031	+	0.907510i	$0.362019\pi$
$354$	0	0
$355$	−3.08752e23	−0.182996
$356$	0	0
$357$	−4.87015e24	−2.73650
$358$	0	0
$359$	−1.45409e24	−0.774805	−0.387403	−	0.921911i	$-0.626628\pi$
$359$	−1.45409e24	−0.774805	−0.387403	+	0.921911i	$0.626628\pi$
$360$	0	0
$361$	4.35543e24	2.20147
$362$	0	0
$363$	−3.89852e24	−1.86977
$364$	0	0
$365$	−8.08067e22	−0.0367845
$366$	0	0
$367$	−1.21387e24	−0.524620	−0.262310	−	0.964984i	$-0.584484\pi$
$367$	−1.21387e24	−0.524620	−0.262310	+	0.964984i	$0.584484\pi$
$368$	0	0
$369$	−6.42234e23	−0.263599
$370$	0	0
$371$	−5.53343e24	−2.15746
$372$	0	0
$373$	3.39082e24	1.25624	0.628118	−	0.778118i	$-0.283826\pi$
$373$	3.39082e24	1.25624	0.628118	+	0.778118i	$0.283826\pi$
$374$	0	0
$375$	3.73183e24	1.31409
$376$	0	0
$377$	1.16703e23	0.0390696
$378$	0	0
$379$	4.13450e24	1.31629	0.658144	−	0.752892i	$-0.271342\pi$
$379$	4.13450e24	1.31629	0.658144	+	0.752892i	$0.271342\pi$
$380$	0	0
$381$	−4.70659e24	−1.42534
$382$	0	0
$383$	3.88317e24	1.11892	0.559459	−	0.828858i	$-0.311009\pi$
$383$	3.88317e24	1.11892	0.559459	+	0.828858i	$0.311009\pi$
$384$	0	0
$385$	−4.20921e24	−1.15431
$386$	0	0
$387$	−3.08618e24	−0.805689
$388$	0	0
$389$	2.58080e24	0.641553	0.320777	−	0.947155i	$-0.396056\pi$
$389$	2.58080e24	0.641553	0.320777	+	0.947155i	$0.396056\pi$
$390$	0	0
$391$	−3.80460e23	−0.0900802
$392$	0	0
$393$	−9.48340e23	−0.213912
$394$	0	0
$395$	−9.69683e23	−0.208429
$396$	0	0
$397$	−4.11532e24	−0.843129	−0.421564	−	0.906799i	$-0.638519\pi$
$397$	−4.11532e24	−0.843129	−0.421564	+	0.906799i	$0.638519\pi$
$398$	0	0
$399$	2.04807e25	4.00040
$400$	0	0
$401$	6.60175e24	1.22967	0.614833	−	0.788657i	$-0.289223\pi$
$401$	6.60175e24	1.22967	0.614833	+	0.788657i	$0.289223\pi$
$402$	0	0
$403$	4.95813e23	0.0880886
$404$	0	0
$405$	−2.28890e24	−0.387975
$406$	0	0
$407$	−1.05277e25	−1.70289
$408$	0	0
$409$	−1.13278e25	−1.74893	−0.874467	−	0.485084i	$-0.838789\pi$
$409$	−1.13278e25	−1.74893	−0.874467	+	0.485084i	$0.838789\pi$
$410$	0	0
$411$	9.31721e24	1.37337
$412$	0	0
$413$	1.47003e25	2.06919
$414$	0	0
$415$	1.00585e24	0.135231
$416$	0	0
$417$	8.06463e24	1.03584
$418$	0	0
$419$	−1.26882e25	−1.55729	−0.778643	−	0.627468i	$-0.784092\pi$
$419$	−1.26882e25	−1.55729	−0.778643	+	0.627468i	$0.784092\pi$
$420$	0	0
$421$	−2.67974e24	−0.314349	−0.157175	−	0.987571i	$-0.550239\pi$
$421$	−2.67974e24	−0.314349	−0.157175	+	0.987571i	$0.550239\pi$
$422$	0	0
$423$	−1.45781e25	−1.63481
$424$	0	0
$425$	−8.45608e24	−0.906722
$426$	0	0
$427$	−1.92339e25	−1.97244
$428$	0	0
$429$	−1.45110e24	−0.142349
$430$	0	0
$431$	7.37367e24	0.692069	0.346035	−	0.938222i	$-0.387528\pi$
$431$	7.37367e24	0.692069	0.346035	+	0.938222i	$0.387528\pi$
$432$	0	0
$433$	−1.67233e25	−1.50206	−0.751028	−	0.660271i	$-0.770442\pi$
$433$	−1.67233e25	−1.50206	−0.751028	+	0.660271i	$0.770442\pi$
$434$	0	0
$435$	5.37308e24	0.461929
$436$	0	0
$437$	1.59997e24	0.131686
$438$	0	0
$439$	1.48622e25	1.17130	0.585651	−	0.810563i	$-0.300839\pi$
$439$	1.48622e25	1.17130	0.585651	+	0.810563i	$0.300839\pi$
$440$	0	0
$441$	2.02240e25	1.52651
$442$	0	0
$443$	7.05393e24	0.510030	0.255015	−	0.966937i	$-0.417920\pi$
$443$	7.05393e24	0.510030	0.255015	+	0.966937i	$0.417920\pi$
$444$	0	0
$445$	7.35126e24	0.509262
$446$	0	0
$447$	−1.62624e25	−1.07960
$448$	0	0
$449$	6.86136e24	0.436586	0.218293	−	0.975883i	$-0.429951\pi$
$449$	6.86136e24	0.436586	0.218293	+	0.975883i	$0.429951\pi$
$450$	0	0
$451$	5.42161e24	0.330713
$452$	0	0
$453$	7.84891e24	0.459067
$454$	0	0
$455$	−8.73807e23	−0.0490125
$456$	0	0
$457$	−1.33077e25	−0.715975	−0.357988	−	0.933726i	$-0.616537\pi$
$457$	−1.33077e25	−0.715975	−0.357988	+	0.933726i	$0.616537\pi$
$458$	0	0
$459$	−6.97994e24	−0.360272
$460$	0	0
$461$	2.20176e25	1.09046	0.545231	−	0.838286i	$-0.316442\pi$
$461$	2.20176e25	1.09046	0.545231	+	0.838286i	$0.316442\pi$
$462$	0	0
$463$	−2.34544e25	−1.11482	−0.557410	−	0.830237i	$-0.688205\pi$
$463$	−2.34544e25	−1.11482	−0.557410	+	0.830237i	$0.688205\pi$
$464$	0	0
$465$	2.28275e25	1.04149
$466$	0	0
$467$	1.76282e25	0.772143	0.386071	−	0.922469i	$-0.373832\pi$
$467$	1.76282e25	0.772143	0.386071	+	0.922469i	$0.373832\pi$
$468$	0	0
$469$	−3.20107e25	−1.34633
$470$	0	0
$471$	2.29597e25	0.927400
$472$	0	0
$473$	2.60529e25	1.01082
$474$	0	0
$475$	3.55609e25	1.32551
$476$	0	0
$477$	−4.78793e25	−1.71484
$478$	0	0
$479$	−2.55873e25	−0.880719	−0.440359	−	0.897822i	$-0.645149\pi$
$479$	−2.55873e25	−0.880719	−0.440359	+	0.897822i	$0.645149\pi$
$480$	0	0
$481$	−2.18550e24	−0.0723056
$482$	0	0
$483$	5.17356e24	0.164548
$484$	0	0
$485$	−1.34197e25	−0.410388
$486$	0	0
$487$	1.39237e25	0.409478	0.204739	−	0.978817i	$-0.434365\pi$
$487$	1.39237e25	0.409478	0.204739	+	0.978817i	$0.434365\pi$
$488$	0	0
$489$	−3.90442e25	−1.10439
$490$	0	0
$491$	−5.82993e24	−0.158631	−0.0793157	−	0.996850i	$-0.525274\pi$
$491$	−5.82993e24	−0.158631	−0.0793157	+	0.996850i	$0.525274\pi$
$492$	0	0
$493$	−2.86098e25	−0.748978
$494$	0	0
$495$	−3.64211e25	−0.917496
$496$	0	0
$497$	−2.23554e25	−0.541997
$498$	0	0
$499$	1.49503e25	0.348895	0.174447	−	0.984666i	$-0.444186\pi$
$499$	1.49503e25	0.348895	0.174447	+	0.984666i	$0.444186\pi$
$500$	0	0
$501$	−5.18554e25	−1.16503
$502$	0	0
$503$	−1.92509e25	−0.416443	−0.208222	−	0.978082i	$-0.566768\pi$
$503$	−1.92509e25	−0.416443	−0.208222	+	0.978082i	$0.566768\pi$
$504$	0	0
$505$	2.02115e25	0.421047
$506$	0	0
$507$	7.35981e25	1.47670
$508$	0	0
$509$	5.27794e25	1.02011	0.510053	−	0.860143i	$-0.329626\pi$
$509$	5.27794e25	1.02011	0.510053	+	0.860143i	$0.329626\pi$
$510$	0	0
$511$	−5.85086e24	−0.108949
$512$	0	0
$513$	2.93531e25	0.526671
$514$	0	0
$515$	2.58102e25	0.446296
$516$	0	0
$517$	1.23065e26	2.05104
$518$	0	0
$519$	−3.50049e25	−0.562391
$520$	0	0
$521$	4.00961e25	0.621074	0.310537	−	0.950561i	$-0.399491\pi$
$521$	4.00961e25	0.621074	0.310537	+	0.950561i	$0.399491\pi$
$522$	0	0
$523$	−5.30249e25	−0.791979	−0.395989	−	0.918255i	$-0.629598\pi$
$523$	−5.30249e25	−0.791979	−0.395989	+	0.918255i	$0.629598\pi$
$524$	0	0
$525$	1.14987e26	1.65629
$526$	0	0
$527$	−1.21549e26	−1.68869
$528$	0	0
$529$	−7.42113e25	−0.994583
$530$	0	0
$531$	1.27198e26	1.64468
$532$	0	0
$533$	1.12549e24	0.0140422
$534$	0	0
$535$	2.22259e25	0.267607
$536$	0	0
$537$	−1.22525e26	−1.42386
$538$	0	0
$539$	−1.70727e26	−1.91518
$540$	0	0
$541$	1.31942e26	1.42892	0.714461	−	0.699676i	$-0.246672\pi$
$541$	1.31942e26	1.42892	0.714461	+	0.699676i	$0.246672\pi$
$542$	0	0
$543$	4.91572e25	0.514031
$544$	0	0
$545$	−9.65363e24	−0.0974819
$546$	0	0
$547$	3.18692e25	0.310807	0.155404	−	0.987851i	$-0.450332\pi$
$547$	3.18692e25	0.310807	0.155404	+	0.987851i	$0.450332\pi$
$548$	0	0
$549$	−1.66426e26	−1.56778
$550$	0	0
$551$	1.20314e26	1.09491
$552$	0	0
$553$	−7.02106e25	−0.617325
$554$	0	0
$555$	−1.00622e26	−0.854885
$556$	0	0
$557$	−1.30222e26	−1.06920	−0.534600	−	0.845105i	$-0.679538\pi$
$557$	−1.30222e26	−1.06920	−0.534600	+	0.845105i	$0.679538\pi$
$558$	0	0
$559$	5.40843e24	0.0429199
$560$	0	0
$561$	3.55738e26	2.72887
$562$	0	0
$563$	−8.14771e23	−0.00604235	−0.00302118	−	0.999995i	$-0.500962\pi$
$563$	−8.14771e23	−0.00604235	−0.00302118	+	0.999995i	$0.500962\pi$
$564$	0	0
$565$	−2.02162e25	−0.144957
$566$	0	0
$567$	−1.65729e26	−1.14911
$568$	0	0
$569$	−1.13811e26	−0.763161	−0.381580	−	0.924336i	$-0.624620\pi$
$569$	−1.13811e26	−0.763161	−0.381580	+	0.924336i	$0.624620\pi$
$570$	0	0
$571$	−6.33817e25	−0.411075	−0.205537	−	0.978649i	$-0.565894\pi$
$571$	−6.33817e25	−0.411075	−0.205537	+	0.978649i	$0.565894\pi$
$572$	0	0
$573$	−3.41733e26	−2.14397
$574$	0	0
$575$	8.98289e24	0.0545220
$576$	0	0
$577$	1.26802e26	0.744654	0.372327	−	0.928102i	$-0.378560\pi$
$577$	1.26802e26	0.744654	0.372327	+	0.928102i	$0.378560\pi$
$578$	0	0
$579$	−7.73408e25	−0.439503
$580$	0	0
$581$	7.28295e25	0.400529
$582$	0	0
$583$	4.04187e26	2.15145
$584$	0	0
$585$	−7.56082e24	−0.0389572
$586$	0	0
$587$	2.28792e26	1.14125	0.570623	−	0.821212i	$-0.306702\pi$
$587$	2.28792e26	1.14125	0.570623	+	0.821212i	$0.306702\pi$
$588$	0	0
$589$	5.11156e26	2.46864
$590$	0	0
$591$	3.43288e26	1.60538
$592$	0	0
$593$	−1.87324e26	−0.848348	−0.424174	−	0.905581i	$-0.639435\pi$
$593$	−1.87324e26	−0.848348	−0.424174	+	0.905581i	$0.639435\pi$
$594$	0	0
$595$	2.14214e26	0.939587
$596$	0	0
$597$	−4.77928e26	−2.03051
$598$	0	0
$599$	−8.85467e25	−0.364433	−0.182217	−	0.983258i	$-0.558327\pi$
$599$	−8.85467e25	−0.364433	−0.182217	+	0.983258i	$0.558327\pi$
$600$	0	0
$601$	−3.61951e25	−0.144325	−0.0721625	−	0.997393i	$-0.522990\pi$
$601$	−3.61951e25	−0.144325	−0.0721625	+	0.997393i	$0.522990\pi$
$602$	0	0
$603$	−2.76980e26	−1.07012
$604$	0	0
$605$	1.71477e26	0.641992
$606$	0	0
$607$	−2.25964e26	−0.819873	−0.409936	−	0.912114i	$-0.634449\pi$
$607$	−2.25964e26	−0.819873	−0.409936	+	0.912114i	$0.634449\pi$
$608$	0	0
$609$	3.89041e26	1.36814
$610$	0	0
$611$	2.55476e25	0.0870882
$612$	0	0
$613$	−3.22922e26	−1.06714	−0.533572	−	0.845755i	$-0.679151\pi$
$613$	−3.22922e26	−1.06714	−0.533572	+	0.845755i	$0.679151\pi$
$614$	0	0
$615$	5.18184e25	0.166024
$616$	0	0
$617$	−5.57559e26	−1.73213	−0.866067	−	0.499927i	$-0.833360\pi$
$617$	−5.57559e26	−1.73213	−0.866067	+	0.499927i	$0.833360\pi$
$618$	0	0
$619$	−3.69876e26	−1.11428	−0.557141	−	0.830418i	$-0.688102\pi$
$619$	−3.69876e26	−1.11428	−0.557141	+	0.830418i	$0.688102\pi$
$620$	0	0
$621$	7.41478e24	0.0216635
$622$	0	0
$623$	5.32273e26	1.50833
$624$	0	0
$625$	1.05361e26	0.289615
$626$	0	0
$627$	−1.49601e27	−3.98926
$628$	0	0
$629$	5.35775e26	1.38612
$630$	0	0
$631$	−1.48829e26	−0.373603	−0.186801	−	0.982398i	$-0.559812\pi$
$631$	−1.48829e26	−0.373603	−0.186801	+	0.982398i	$0.559812\pi$
$632$	0	0
$633$	1.08679e27	2.64735
$634$	0	0
$635$	2.07020e26	0.489397
$636$	0	0
$637$	−3.54419e25	−0.0813192
$638$	0	0
$639$	−1.93435e26	−0.430802
$640$	0	0
$641$	−2.74129e26	−0.592657	−0.296329	−	0.955086i	$-0.595762\pi$
$641$	−2.74129e26	−0.592657	−0.296329	+	0.955086i	$0.595762\pi$
$642$	0	0
$643$	−2.47762e26	−0.520032	−0.260016	−	0.965604i	$-0.583728\pi$
$643$	−2.47762e26	−0.520032	−0.260016	+	0.965604i	$0.583728\pi$
$644$	0	0
$645$	2.49007e26	0.507452
$646$	0	0
$647$	−1.36261e26	−0.269638	−0.134819	−	0.990870i	$-0.543045\pi$
$647$	−1.36261e26	−0.269638	−0.134819	+	0.990870i	$0.543045\pi$
$648$	0	0
$649$	−1.07378e27	−2.06343
$650$	0	0
$651$	1.65284e27	3.08469
$652$	0	0
$653$	−6.13257e25	−0.111165	−0.0555824	−	0.998454i	$-0.517702\pi$
$653$	−6.13257e25	−0.111165	−0.0555824	+	0.998454i	$0.517702\pi$
$654$	0	0
$655$	4.17128e25	0.0734475
$656$	0	0
$657$	−5.06260e25	−0.0865969
$658$	0	0
$659$	−2.67926e26	−0.445250	−0.222625	−	0.974904i	$-0.571463\pi$
$659$	−2.67926e26	−0.445250	−0.222625	+	0.974904i	$0.571463\pi$
$660$	0	0
$661$	3.12853e26	0.505157	0.252578	−	0.967576i	$-0.418721\pi$
$661$	3.12853e26	0.505157	0.252578	+	0.967576i	$0.418721\pi$
$662$	0	0
$663$	7.38492e25	0.115869
$664$	0	0
$665$	−9.00848e26	−1.37355
$666$	0	0
$667$	3.03922e25	0.0450367
$668$	0	0
$669$	1.97114e27	2.83902
$670$	0	0
$671$	1.40493e27	1.96694
$672$	0	0
$673$	7.60450e26	1.03497	0.517485	−	0.855692i	$-0.326868\pi$
$673$	7.60450e26	1.03497	0.517485	+	0.855692i	$0.326868\pi$
$674$	0	0
$675$	1.64801e26	0.218058
$676$	0	0
$677$	−1.44198e27	−1.85510	−0.927550	−	0.373700i	$-0.878089\pi$
$677$	−1.44198e27	−1.85510	−0.927550	+	0.373700i	$0.878089\pi$
$678$	0	0
$679$	−9.71659e26	−1.21549
$680$	0	0
$681$	−8.54878e25	−0.103994
$682$	0	0
$683$	8.31240e26	0.983399	0.491700	−	0.870765i	$-0.336376\pi$
$683$	8.31240e26	0.983399	0.491700	+	0.870765i	$0.336376\pi$
$684$	0	0
$685$	−4.09818e26	−0.471553
$686$	0	0
$687$	−1.69154e26	−0.189318
$688$	0	0
$689$	8.39069e25	0.0913514
$690$	0	0
$691$	−2.10819e26	−0.223290	−0.111645	−	0.993748i	$-0.535612\pi$
$691$	−2.10819e26	−0.223290	−0.111645	+	0.993748i	$0.535612\pi$
$692$	0	0
$693$	−2.63710e27	−2.71744
$694$	0	0
$695$	−3.54723e26	−0.355659
$696$	0	0
$697$	−2.75915e26	−0.269194
$698$	0	0
$699$	−6.45059e26	−0.612444
$700$	0	0
$701$	9.41456e26	0.869919	0.434959	−	0.900450i	$-0.356763\pi$
$701$	9.41456e26	0.869919	0.434959	+	0.900450i	$0.356763\pi$
$702$	0	0
$703$	−2.25313e27	−2.02633
$704$	0	0
$705$	1.17623e27	1.02966
$706$	0	0
$707$	1.46343e27	1.24706
$708$	0	0
$709$	−1.83074e27	−1.51875	−0.759376	−	0.650652i	$-0.774495\pi$
$709$	−1.83074e27	−1.51875	−0.759376	+	0.650652i	$0.774495\pi$
$710$	0	0
$711$	−6.07513e26	−0.490676
$712$	0	0
$713$	1.29121e26	0.101542
$714$	0	0
$715$	6.38268e25	0.0488760
$716$	0	0
$717$	4.63002e26	0.345264
$718$	0	0
$719$	1.73532e27	1.26025	0.630124	−	0.776495i	$-0.283004\pi$
$719$	1.73532e27	1.26025	0.630124	+	0.776495i	$0.283004\pi$
$720$	0	0
$721$	1.86880e27	1.32184
$722$	0	0
$723$	2.29796e27	1.58317
$724$	0	0
$725$	6.75495e26	0.453326
$726$	0	0
$727$	5.33232e26	0.348610	0.174305	−	0.984692i	$-0.444232\pi$
$727$	5.33232e26	0.348610	0.174305	+	0.984692i	$0.444232\pi$
$728$	0	0
$729$	−2.11925e27	−1.34980
$730$	0	0
$731$	−1.32588e27	−0.822790
$732$	0	0
$733$	3.90931e26	0.236381	0.118190	−	0.992991i	$-0.462291\pi$
$733$	3.90931e26	0.236381	0.118190	+	0.992991i	$0.462291\pi$
$734$	0	0
$735$	−1.63177e27	−0.961455
$736$	0	0
$737$	2.33820e27	1.34258
$738$	0	0
$739$	7.87032e26	0.440423	0.220212	−	0.975452i	$-0.429325\pi$
$739$	7.87032e26	0.440423	0.220212	+	0.975452i	$0.429325\pi$
$740$	0	0
$741$	−3.10563e26	−0.169385
$742$	0	0
$743$	−3.68492e27	−1.95900	−0.979500	−	0.201445i	$-0.935436\pi$
$743$	−3.68492e27	−1.95900	−0.979500	+	0.201445i	$0.935436\pi$
$744$	0	0
$745$	7.15302e26	0.370685
$746$	0	0
$747$	6.30174e26	0.318357
$748$	0	0
$749$	1.60928e27	0.792600
$750$	0	0
$751$	3.86208e26	0.185457	0.0927283	−	0.995691i	$-0.470441\pi$
$751$	3.86208e26	0.185457	0.0927283	+	0.995691i	$0.470441\pi$
$752$	0	0
$753$	1.86421e27	0.872855
$754$	0	0
$755$	−3.45235e26	−0.157623
$756$	0	0
$757$	3.93310e27	1.75116	0.875578	−	0.483078i	$-0.160481\pi$
$757$	3.93310e27	1.75116	0.875578	+	0.483078i	$0.160481\pi$
$758$	0	0
$759$	−3.77900e26	−0.164089
$760$	0	0
$761$	−1.44215e27	−0.610742	−0.305371	−	0.952234i	$-0.598780\pi$
$761$	−1.44215e27	−0.610742	−0.305371	+	0.952234i	$0.598780\pi$
$762$	0	0
$763$	−6.98978e26	−0.288722
$764$	0	0
$765$	1.85354e27	0.746824
$766$	0	0
$767$	−2.22910e26	−0.0876139
$768$	0	0
$769$	−4.77621e27	−1.83140	−0.915701	−	0.401861i	$-0.868364\pi$
$769$	−4.77621e27	−1.83140	−0.915701	+	0.401861i	$0.868364\pi$
$770$	0	0
$771$	−2.44872e27	−0.916056
$772$	0	0
$773$	−3.06095e27	−1.11725	−0.558625	−	0.829420i	$-0.688671\pi$
$773$	−3.06095e27	−1.11725	−0.558625	+	0.829420i	$0.688671\pi$
$774$	0	0
$775$	2.86984e27	1.02210
$776$	0	0
$777$	−7.28557e27	−2.53200
$778$	0	0
$779$	1.16032e27	0.393526
$780$	0	0
$781$	1.63294e27	0.540488
$782$	0	0
$783$	5.57576e26	0.180122
$784$	0	0
$785$	−1.00988e27	−0.318427
$786$	0	0
$787$	2.46755e27	0.759461	0.379731	−	0.925097i	$-0.376017\pi$
$787$	2.46755e27	0.759461	0.379731	+	0.925097i	$0.376017\pi$
$788$	0	0
$789$	−3.46084e27	−1.03980
$790$	0	0
$791$	−1.46377e27	−0.429334
$792$	0	0
$793$	2.91656e26	0.0835171
$794$	0	0
$795$	3.86313e27	1.08007
$796$	0	0
$797$	−4.72784e27	−1.29065	−0.645326	−	0.763907i	$-0.723278\pi$
$797$	−4.72784e27	−1.29065	−0.645326	+	0.763907i	$0.723278\pi$
$798$	0	0
$799$	−6.26301e27	−1.66951
$800$	0	0
$801$	4.60561e27	1.19889
$802$	0	0
$803$	4.27374e26	0.108645
$804$	0	0
$805$	−2.27560e26	−0.0564981
$806$	0	0
$807$	1.10061e28	2.66892
$808$	0	0
$809$	4.73872e27	1.12241	0.561203	−	0.827678i	$-0.310339\pi$
$809$	4.73872e27	1.12241	0.561203	+	0.827678i	$0.310339\pi$
$810$	0	0
$811$	−1.55266e27	−0.359234	−0.179617	−	0.983737i	$-0.557486\pi$
$811$	−1.55266e27	−0.359234	−0.179617	+	0.983737i	$0.557486\pi$
$812$	0	0
$813$	9.52795e27	2.15347
$814$	0	0
$815$	1.71736e27	0.379197
$816$	0	0
$817$	5.57580e27	1.20281
$818$	0	0
$819$	−5.47446e26	−0.115384
$820$	0	0
$821$	−7.90129e27	−1.62719	−0.813595	−	0.581432i	$-0.802493\pi$
$821$	−7.90129e27	−1.62719	−0.813595	+	0.581432i	$0.802493\pi$
$822$	0	0
$823$	7.55379e27	1.52008	0.760040	−	0.649877i	$-0.225180\pi$
$823$	7.55379e27	1.52008	0.760040	+	0.649877i	$0.225180\pi$
$824$	0	0
$825$	−8.39920e27	−1.65168
$826$	0	0
$827$	−9.52722e27	−1.83090	−0.915448	−	0.402436i	$-0.868164\pi$
$827$	−9.52722e27	−1.83090	−0.915448	+	0.402436i	$0.868164\pi$
$828$	0	0
$829$	1.29069e27	0.242412	0.121206	−	0.992627i	$-0.461324\pi$
$829$	1.29069e27	0.242412	0.121206	+	0.992627i	$0.461324\pi$
$830$	0	0
$831$	−1.05125e28	−1.92973
$832$	0	0
$833$	8.68861e27	1.55892
$834$	0	0
$835$	2.28086e27	0.400016
$836$	0	0
$837$	2.36886e27	0.406114
$838$	0	0
$839$	9.22538e27	1.54613	0.773065	−	0.634327i	$-0.218723\pi$
$839$	9.22538e27	1.54613	0.773065	+	0.634327i	$0.218723\pi$
$840$	0	0
$841$	−3.81783e27	−0.625540
$842$	0	0
$843$	−1.68759e28	−2.70337
$844$	0	0
$845$	−3.23722e27	−0.507030
$846$	0	0
$847$	1.24159e28	1.90145
$848$	0	0
$849$	1.14551e27	0.171544
$850$	0	0
$851$	−5.69154e26	−0.0833487
$852$	0	0
$853$	−6.63470e27	−0.950180	−0.475090	−	0.879937i	$-0.657584\pi$
$853$	−6.63470e27	−0.950180	−0.475090	+	0.879937i	$0.657584\pi$
$854$	0	0
$855$	−7.79479e27	−1.09176
$856$	0	0
$857$	2.92147e27	0.400206	0.200103	−	0.979775i	$-0.435872\pi$
$857$	2.92147e27	0.400206	0.200103	+	0.979775i	$0.435872\pi$
$858$	0	0
$859$	8.47684e27	1.13579	0.567896	−	0.823100i	$-0.307757\pi$
$859$	8.47684e27	1.13579	0.567896	+	0.823100i	$0.307757\pi$
$860$	0	0
$861$	3.75195e27	0.491730
$862$	0	0
$863$	−9.99055e27	−1.28082	−0.640408	−	0.768035i	$-0.721235\pi$
$863$	−9.99055e27	−1.28082	−0.640408	+	0.768035i	$0.721235\pi$
$864$	0	0
$865$	1.53969e27	0.193099
$866$	0	0
$867$	−6.01918e27	−0.738509
$868$	0	0
$869$	5.12850e27	0.615605
$870$	0	0
$871$	4.85398e26	0.0570066
$872$	0	0
$873$	−8.40751e27	−0.966122
$874$	0	0
$875$	−1.18850e28	−1.33636
$876$	0	0
$877$	−3.66387e27	−0.403129	−0.201564	−	0.979475i	$-0.564603\pi$
$877$	−3.66387e27	−0.403129	−0.201564	+	0.979475i	$0.564603\pi$
$878$	0	0
$879$	9.37294e27	1.00921
$880$	0	0
$881$	3.90178e27	0.411141	0.205571	−	0.978642i	$-0.434095\pi$
$881$	3.90178e27	0.411141	0.205571	+	0.978642i	$0.434095\pi$
$882$	0	0
$883$	−6.02824e27	−0.621676	−0.310838	−	0.950463i	$-0.600610\pi$
$883$	−6.02824e27	−0.621676	−0.310838	+	0.950463i	$0.600610\pi$
$884$	0	0
$885$	−1.02629e28	−1.03588
$886$	0	0
$887$	−1.43600e28	−1.41867	−0.709333	−	0.704874i	$-0.751004\pi$
$887$	−1.43600e28	−1.41867	−0.709333	+	0.704874i	$0.751004\pi$
$888$	0	0
$889$	1.49894e28	1.44950
$890$	0	0
$891$	1.21056e28	1.14591
$892$	0	0
$893$	2.63382e28	2.44061
$894$	0	0
$895$	5.38925e27	0.488888
$896$	0	0
$897$	−7.84500e25	−0.00696730
$898$	0	0
$899$	9.70964e27	0.844279
$900$	0	0
$901$	−2.05698e28	−1.75124
$902$	0	0
$903$	1.80296e28	1.50297
$904$	0	0
$905$	−2.16218e27	−0.176495
$906$	0	0
$907$	8.84058e27	0.706662	0.353331	−	0.935498i	$-0.385049\pi$
$907$	8.84058e27	0.706662	0.353331	+	0.935498i	$0.385049\pi$
$908$	0	0
$909$	1.26627e28	0.991215
$910$	0	0
$911$	1.77289e28	1.35912	0.679561	−	0.733619i	$-0.262170\pi$
$911$	1.77289e28	1.35912	0.679561	+	0.733619i	$0.262170\pi$
$912$	0	0
$913$	−5.31979e27	−0.399413
$914$	0	0
$915$	1.34280e28	0.987442
$916$	0	0
$917$	3.02024e27	0.217537
$918$	0	0
$919$	−1.67320e28	−1.18046	−0.590230	−	0.807235i	$-0.700963\pi$
$919$	−1.67320e28	−1.18046	−0.590230	+	0.807235i	$0.700963\pi$
$920$	0	0
$921$	−4.01175e28	−2.77247
$922$	0	0
$923$	3.38989e26	0.0229493
$924$	0	0
$925$	−1.26500e28	−0.838965
$926$	0	0
$927$	1.61703e28	1.05065
$928$	0	0
$929$	−2.04557e28	−1.30216	−0.651082	−	0.759008i	$-0.725685\pi$
$929$	−2.04557e28	−1.30216	−0.651082	+	0.759008i	$0.725685\pi$
$930$	0	0
$931$	−3.65387e28	−2.27893
$932$	0	0
$933$	2.63080e28	1.60772
$934$	0	0
$935$	−1.56472e28	−0.936970
$936$	0	0
$937$	−2.17132e27	−0.127408	−0.0637042	−	0.997969i	$-0.520291\pi$
$937$	−2.17132e27	−0.127408	−0.0637042	+	0.997969i	$0.520291\pi$
$938$	0	0
$939$	8.35809e27	0.480600
$940$	0	0
$941$	8.37859e27	0.472138	0.236069	−	0.971736i	$-0.424141\pi$
$941$	8.37859e27	0.472138	0.236069	+	0.971736i	$0.424141\pi$
$942$	0	0
$943$	2.93105e26	0.0161868
$944$	0	0
$945$	−4.17482e27	−0.225962
$946$	0	0
$947$	1.69589e28	0.899648	0.449824	−	0.893117i	$-0.351487\pi$
$947$	1.69589e28	0.899648	0.449824	+	0.893117i	$0.351487\pi$
$948$	0	0
$949$	8.87204e25	0.00461311
$950$	0	0
$951$	−2.78136e28	−1.41756
$952$	0	0
$953$	3.53298e27	0.176506	0.0882529	−	0.996098i	$-0.471872\pi$
$953$	3.53298e27	0.176506	0.0882529	+	0.996098i	$0.471872\pi$
$954$	0	0
$955$	1.50312e28	0.736140
$956$	0	0
$957$	−2.84173e28	−1.36433
$958$	0	0
$959$	−2.96732e28	−1.39665
$960$	0	0
$961$	1.95807e28	0.903560
$962$	0	0
$963$	1.39247e28	0.629992
$964$	0	0
$965$	3.40184e27	0.150905
$966$	0	0
$967$	−9.62859e27	−0.418805	−0.209402	−	0.977830i	$-0.567152\pi$
$967$	−9.62859e27	−0.418805	−0.209402	+	0.977830i	$0.567152\pi$
$968$	0	0
$969$	7.61345e28	3.24718
$970$	0	0
$971$	−2.59014e28	−1.08328	−0.541640	−	0.840611i	$-0.682196\pi$
$971$	−2.59014e28	−1.08328	−0.541640	+	0.840611i	$0.682196\pi$
$972$	0	0
$973$	−2.56840e28	−1.05339
$974$	0	0
$975$	−1.74363e27	−0.0701309
$976$	0	0
$977$	3.99288e28	1.57503	0.787514	−	0.616297i	$-0.211368\pi$
$977$	3.99288e28	1.57503	0.787514	+	0.616297i	$0.211368\pi$
$978$	0	0
$979$	−3.88796e28	−1.50413
$980$	0	0
$981$	−6.04807e27	−0.229489
$982$	0	0
$983$	3.74542e27	0.139393	0.0696966	−	0.997568i	$-0.477797\pi$
$983$	3.74542e27	0.139393	0.0696966	+	0.997568i	$0.477797\pi$
$984$	0	0
$985$	−1.50996e28	−0.551214
$986$	0	0
$987$	8.51656e28	3.04966
$988$	0	0
$989$	1.40848e27	0.0494750
$990$	0	0
$991$	−3.40811e28	−1.17440	−0.587198	−	0.809443i	$-0.699769\pi$
$991$	−3.40811e28	−1.17440	−0.587198	+	0.809443i	$0.699769\pi$
$992$	0	0
$993$	−2.79525e28	−0.944937
$994$	0	0
$995$	2.10217e28	0.697186
$996$	0	0
$997$	−4.93862e27	−0.160695	−0.0803473	−	0.996767i	$-0.525603\pi$
$997$	−4.93862e27	−0.160695	−0.0803473	+	0.996767i	$0.525603\pi$
$998$	0	0
$999$	−1.04417e28	−0.333350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.20.a.a.1.1	✓	2
3.2	odd	2		72.20.a.a.1.1		2
4.3	odd	2		16.20.a.e.1.2		2
8.3	odd	2		64.20.a.j.1.1		2
8.5	even	2		64.20.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.20.a.a.1.1	✓	2	1.1	even	1	trivial
16.20.a.e.1.2		2	4.3	odd	2
64.20.a.j.1.1		2	8.3	odd	2
64.20.a.k.1.2		2	8.5	even	2
72.20.a.a.1.1		2	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:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q-50549.5 q^{3} +2.22342e6 q^{5} +1.60989e8 q^{7} +1.39299e9 q^{9} +O(q^{10})\)	\(q-50549.5 q^{3} +2.22342e6 q^{5} +1.60989e8 q^{7} +1.39299e9 q^{9} -1.17593e10 q^{11} -2.44117e9 q^{13} -1.12393e11 q^{15} +5.98454e11 q^{17} -2.51671e12 q^{19} -8.13789e12 q^{21} -6.35737e11 q^{23} -1.41299e13 q^{25} -1.16633e13 q^{27} -4.78062e13 q^{29} -2.03104e14 q^{31} +5.94429e14 q^{33} +3.57946e14 q^{35} +8.95266e14 q^{37} +1.23400e14 q^{39} -4.61047e14 q^{41} -2.21551e15 q^{43} +3.09721e15 q^{45} -1.04653e16 q^{47} +1.45184e16 q^{49} -3.02516e16 q^{51} -3.43716e16 q^{53} -2.61460e16 q^{55} +1.27219e17 q^{57} +9.13126e16 q^{59} -1.19474e17 q^{61} +2.24256e17 q^{63} -5.42776e15 q^{65} -1.98838e17 q^{67} +3.21362e16 q^{69} -1.38863e17 q^{71} -3.63434e16 q^{73} +7.14258e17 q^{75} -1.89312e18 q^{77} -4.36121e17 q^{79} -1.02945e18 q^{81} +4.52389e17 q^{83} +1.33062e18 q^{85} +2.41658e18 q^{87} +3.30628e18 q^{89} -3.93001e17 q^{91} +1.02668e19 q^{93} -5.59572e18 q^{95} -6.03558e18 q^{97} -1.63807e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9}+O(q^{10}) \) 2 * q - 27912 * q^3 + 1226620 * q^5 + 88510512 * q^7 + 743186538 * q^9	\( 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9} - 7163787608 q^{11} - 10126923604 q^{13} - 134958171120 q^{15} - 72045078940 q^{17} - 3120480472232 q^{19} - 9778613729472 q^{21} - 14759207090288 q^{23} - 32209737998450 q^{25} - 52683939132624 q^{27} - 30249539245044 q^{29} - 123389562777920 q^{31} + 698460345860448 q^{33} + 430192267170720 q^{35} + 20\!\cdots\!24 q^{37}+ \cdots - 19\!\cdots\!52 q^{99}+O(q^{100}) \) 2 * q - 27912 * q^3 + 1226620 * q^5 + 88510512 * q^7 + 743186538 * q^9 - 7163787608 * q^11 - 10126923604 * q^13 - 134958171120 * q^15 - 72045078940 * q^17 - 3120480472232 * q^19 - 9778613729472 * q^21 - 14759207090288 * q^23 - 32209737998450 * q^25 - 52683939132624 * q^27 - 30249539245044 * q^29 - 123389562777920 * q^31 + 698460345860448 * q^33 + 430192267170720 * q^35 + 2015393170174524 * q^37 - 50586287718576 * q^39 + 2540784959504244 * q^41 - 5633655093389464 * q^43 + 3744938810129580 * q^45 - 21948339587130336 * q^47 + 8372586663064786 * q^49 - 45429993200591760 * q^51 - 9418125066904676 * q^53 - 30726853722003280 * q^55 + 113550921657554592 * q^57 + 98542449590407624 * q^59 + 10292145377839820 * q^61 + 271352203396831728 * q^63 + 2233432360014760 * q^65 + 75753628003984504 * q^67 - 287583943397508672 * q^69 + 17407052566713776 * q^71 - 857508255059832268 * q^73 + 304974862137280200 * q^75 - 2226194512471875648 * q^77 - 226291921444855072 * q^79 - 1202809709767720302 * q^81 + 767515701460985048 * q^83 + 1998974022815558200 * q^85 + 2814015710341120464 * q^87 + 6092545894435174548 * q^89 + 164047654392377376 * q^91 + 12071374787086705920 * q^93 - 4993887710656339120 * q^95 + 1548148249522347076 * q^97 - 19366859521479728952 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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