LMFDB - Embedded newform 8.18.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,18,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, 18, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 18 $
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:	$14.6577669876$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{114}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 114 $ x^2 - 114
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.6771$ 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)

$f(q)$	$=$	$q+18096.0 q^{3} -1.23236e6 q^{5} -2.89249e7 q^{7} +1.98325e8 q^{9} +O(q^{10})$	$q+18096.0 q^{3} -1.23236e6 q^{5} -2.89249e7 q^{7} +1.98325e8 q^{9} -3.61694e8 q^{11} -6.37402e8 q^{13} -2.23008e10 q^{15} -7.41238e9 q^{17} +3.55981e10 q^{19} -5.23425e11 q^{21} -2.46782e11 q^{23} +7.55776e11 q^{25} +1.25197e12 q^{27} +1.56479e12 q^{29} +2.13491e12 q^{31} -6.54522e12 q^{33} +3.56459e13 q^{35} -3.10238e13 q^{37} -1.15344e13 q^{39} -2.09452e13 q^{41} -8.04137e13 q^{43} -2.44408e14 q^{45} +2.25613e14 q^{47} +6.04019e14 q^{49} -1.34134e14 q^{51} -5.86376e14 q^{53} +4.45738e14 q^{55} +6.44184e14 q^{57} +5.67491e14 q^{59} +2.28400e15 q^{61} -5.73652e15 q^{63} +7.85509e14 q^{65} -3.97400e15 q^{67} -4.46577e15 q^{69} +5.20052e15 q^{71} -5.41800e15 q^{73} +1.36765e16 q^{75} +1.04620e16 q^{77} +1.80248e15 q^{79} -2.95614e15 q^{81} -6.04601e15 q^{83} +9.13474e15 q^{85} +2.83165e16 q^{87} -5.62804e16 q^{89} +1.84368e16 q^{91} +3.86333e16 q^{93} -4.38698e16 q^{95} -1.44249e17 q^{97} -7.17330e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 11592 q^{3} - 791924 q^{5} - 18932592 q^{7} + 111486618 q^{9}+O(q^{10}) $ 2 * q + 11592 * q^3 - 791924 * q^5 - 18932592 * q^7 + 111486618 * q^9	$ 2 q + 11592 q^{3} - 791924 q^{5} - 18932592 q^{7} + 111486618 q^{9} - 470771240 q^{11} - 2503523428 q^{13} - 25165411920 q^{15} - 48444688348 q^{17} - 82483300760 q^{19} - 588414407616 q^{21} - 307097031248 q^{23} + 186821492926 q^{25} + 2656687689936 q^{27} + 1989575387580 q^{29} + 10752133575232 q^{31} - 5835781169568 q^{33} + 40046902091616 q^{35} - 51174452749620 q^{37} + 602829666288 q^{39} - 114227291044524 q^{41} - 56757554203624 q^{43} - 282654737698788 q^{45} + 201553163158368 q^{47} + 471233999080306 q^{49} + 132739385855760 q^{51} + 217275463587052 q^{53} + 397696513984784 q^{55} + 14\!\cdots\!32 q^{57}+ \cdots - 62\!\cdots\!36 q^{99}+O(q^{100}) $ 2 * q + 11592 * q^3 - 791924 * q^5 - 18932592 * q^7 + 111486618 * q^9 - 470771240 * q^11 - 2503523428 * q^13 - 25165411920 * q^15 - 48444688348 * q^17 - 82483300760 * q^19 - 588414407616 * q^21 - 307097031248 * q^23 + 186821492926 * q^25 + 2656687689936 * q^27 + 1989575387580 * q^29 + 10752133575232 * q^31 - 5835781169568 * q^33 + 40046902091616 * q^35 - 51174452749620 * q^37 + 602829666288 * q^39 - 114227291044524 * q^41 - 56757554203624 * q^43 - 282654737698788 * q^45 + 201553163158368 * q^47 + 471233999080306 * q^49 + 132739385855760 * q^51 + 217275463587052 * q^53 + 397696513984784 * q^55 + 1412184913398432 * q^57 - 466311448890056 * q^59 + 2063817263626748 * q^61 - 6604236688046256 * q^63 - 36400955340632 * q^65 - 6637225962358328 * q^67 - 4073481953319744 * q^69 + 1543684997984528 * q^71 - 3734106800358508 * q^73 + 17376986610519480 * q^75 + 9372036167046336 * q^77 + 17755378507330976 * q^79 - 878143806358542 * q^81 + 25453597558158056 * q^83 - 8937430679761448 * q^85 + 25553707384497264 * q^87 - 67216838659203660 * q^89 - 210069989389344 * q^91 - 17413067555884800 * q^93 - 95877299737499536 * q^95 - 130979024861443388 * q^97 - 62260905496457736 * 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
$3$	18096.0	1.59240	0.796199	−	0.605034i	$-0.206841\pi$
$3$	18096.0	1.59240	0.796199	+	0.605034i	$0.206841\pi$
$4$	0	0
$5$	−1.23236e6	−1.41089	−0.705445	−	0.708765i	$-0.749253\pi$
$5$	−1.23236e6	−1.41089	−0.705445	+	0.708765i	$0.749253\pi$
$6$	0	0
$7$	−2.89249e7	−1.89644	−0.948218	−	0.317619i	$-0.897117\pi$
$7$	−2.89249e7	−1.89644	−0.948218	+	0.317619i	$0.897117\pi$
$8$	0	0
$9$	1.98325e8	1.53573
$10$	0	0
$11$	−3.61694e8	−0.508749	−0.254375	−	0.967106i	$-0.581870\pi$
$11$	−3.61694e8	−0.508749	−0.254375	+	0.967106i	$0.581870\pi$
$12$	0	0
$13$	−6.37402e8	−0.216718	−0.108359	−	0.994112i	$-0.534560\pi$
$13$	−6.37402e8	−0.216718	−0.108359	+	0.994112i	$0.534560\pi$
$14$	0	0
$15$	−2.23008e10	−2.24670
$16$	0	0
$17$	−7.41238e9	−0.257716	−0.128858	−	0.991663i	$-0.541131\pi$
$17$	−7.41238e9	−0.257716	−0.128858	+	0.991663i	$0.541131\pi$
$18$	0	0
$19$	3.55981e10	0.480863	0.240432	−	0.970666i	$-0.422711\pi$
$19$	3.55981e10	0.480863	0.240432	+	0.970666i	$0.422711\pi$
$20$	0	0
$21$	−5.23425e11	−3.01988
$22$	0	0
$23$	−2.46782e11	−0.657094	−0.328547	−	0.944488i	$-0.606559\pi$
$23$	−2.46782e11	−0.657094	−0.328547	+	0.944488i	$0.606559\pi$
$24$	0	0
$25$	7.55776e11	0.990610
$26$	0	0
$27$	1.25197e12	0.853101
$28$	0	0
$29$	1.56479e12	0.580863	0.290432	−	0.956896i	$-0.406201\pi$
$29$	1.56479e12	0.580863	0.290432	+	0.956896i	$0.406201\pi$
$30$	0	0
$31$	2.13491e12	0.449578	0.224789	−	0.974407i	$-0.427831\pi$
$31$	2.13491e12	0.449578	0.224789	+	0.974407i	$0.427831\pi$
$32$	0	0
$33$	−6.54522e12	−0.810132
$34$	0	0
$35$	3.56459e13	2.67566
$36$	0	0
$37$	−3.10238e13	−1.45205	−0.726023	−	0.687671i	$-0.758633\pi$
$37$	−3.10238e13	−1.45205	−0.726023	+	0.687671i	$0.758633\pi$
$38$	0	0
$39$	−1.15344e13	−0.345101
$40$	0	0
$41$	−2.09452e13	−0.409658	−0.204829	−	0.978798i	$-0.565664\pi$
$41$	−2.09452e13	−0.409658	−0.204829	+	0.978798i	$0.565664\pi$
$42$	0	0
$43$	−8.04137e13	−1.04918	−0.524588	−	0.851356i	$-0.675781\pi$
$43$	−8.04137e13	−1.04918	−0.524588	+	0.851356i	$0.675781\pi$
$44$	0	0
$45$	−2.44408e14	−2.16675
$46$	0	0
$47$	2.25613e14	1.38208	0.691039	−	0.722818i	$-0.257153\pi$
$47$	2.25613e14	1.38208	0.691039	+	0.722818i	$0.257153\pi$
$48$	0	0
$49$	6.04019e14	2.59647
$50$	0	0
$51$	−1.34134e14	−0.410387
$52$	0	0
$53$	−5.86376e14	−1.29369	−0.646847	−	0.762620i	$-0.723913\pi$
$53$	−5.86376e14	−1.29369	−0.646847	+	0.762620i	$0.723913\pi$
$54$	0	0
$55$	4.45738e14	0.717789
$56$	0	0
$57$	6.44184e14	0.765726
$58$	0	0
$59$	5.67491e14	0.503172	0.251586	−	0.967835i	$-0.419048\pi$
$59$	5.67491e14	0.503172	0.251586	+	0.967835i	$0.419048\pi$
$60$	0	0
$61$	2.28400e15	1.52543	0.762714	−	0.646736i	$-0.223866\pi$
$61$	2.28400e15	1.52543	0.762714	+	0.646736i	$0.223866\pi$
$62$	0	0
$63$	−5.73652e15	−2.91242
$64$	0	0
$65$	7.85509e14	0.305765
$66$	0	0
$67$	−3.97400e15	−1.19562	−0.597808	−	0.801639i	$-0.703962\pi$
$67$	−3.97400e15	−1.19562	−0.597808	+	0.801639i	$0.703962\pi$
$68$	0	0
$69$	−4.46577e15	−1.04635
$70$	0	0
$71$	5.20052e15	0.955765	0.477882	−	0.878424i	$-0.341405\pi$
$71$	5.20052e15	0.955765	0.477882	+	0.878424i	$0.341405\pi$
$72$	0	0
$73$	−5.41800e15	−0.786311	−0.393156	−	0.919472i	$-0.628617\pi$
$73$	−5.41800e15	−0.786311	−0.393156	+	0.919472i	$0.628617\pi$
$74$	0	0
$75$	1.36765e16	1.57745
$76$	0	0
$77$	1.04620e16	0.964811
$78$	0	0
$79$	1.80248e15	0.133672	0.0668358	−	0.997764i	$-0.478710\pi$
$79$	1.80248e15	0.133672	0.0668358	+	0.997764i	$0.478710\pi$
$80$	0	0
$81$	−2.95614e15	−0.177257
$82$	0	0
$83$	−6.04601e15	−0.294649	−0.147325	−	0.989088i	$-0.547066\pi$
$83$	−6.04601e15	−0.294649	−0.147325	+	0.989088i	$0.547066\pi$
$84$	0	0
$85$	9.13474e15	0.363610
$86$	0	0
$87$	2.83165e16	0.924966
$88$	0	0
$89$	−5.62804e16	−1.51545	−0.757726	−	0.652573i	$-0.773690\pi$
$89$	−5.62804e16	−1.51545	−0.757726	+	0.652573i	$0.773690\pi$
$90$	0	0
$91$	1.84368e16	0.410992
$92$	0	0
$93$	3.86333e16	0.715907
$94$	0	0
$95$	−4.38698e16	−0.678445
$96$	0	0
$97$	−1.44249e17	−1.86875	−0.934377	−	0.356287i	$-0.884043\pi$
$97$	−1.44249e17	−1.86875	−0.934377	+	0.356287i	$0.884043\pi$
$98$	0	0
$99$	−7.17330e16	−0.781303
$100$	0	0
$101$	−9.49032e16	−0.872066	−0.436033	−	0.899931i	$-0.643617\pi$
$101$	−9.49032e16	−0.872066	−0.436033	+	0.899931i	$0.643617\pi$
$102$	0	0
$103$	−6.10941e16	−0.475207	−0.237604	−	0.971362i	$-0.576362\pi$
$103$	−6.10941e16	−0.475207	−0.237604	+	0.971362i	$0.576362\pi$
$104$	0	0
$105$	6.45048e17	4.26072
$106$	0	0
$107$	2.56499e17	1.44319	0.721595	−	0.692316i	$-0.243409\pi$
$107$	2.56499e17	1.44319	0.721595	+	0.692316i	$0.243409\pi$
$108$	0	0
$109$	−7.02286e16	−0.337589	−0.168795	−	0.985651i	$-0.553987\pi$
$109$	−7.02286e16	−0.337589	−0.168795	+	0.985651i	$0.553987\pi$
$110$	0	0
$111$	−5.61406e17	−2.31224
$112$	0	0
$113$	1.78088e17	0.630183	0.315092	−	0.949061i	$-0.397965\pi$
$113$	1.78088e17	0.630183	0.315092	+	0.949061i	$0.397965\pi$
$114$	0	0
$115$	3.04125e17	0.927087
$116$	0	0
$117$	−1.26413e17	−0.332821
$118$	0	0
$119$	2.14402e17	0.488743
$120$	0	0
$121$	−3.74624e17	−0.741174
$122$	0	0
$123$	−3.79024e17	−0.652339
$124$	0	0
$125$	8.82838e15	0.0132479
$126$	0	0
$127$	3.23676e17	0.424404	0.212202	−	0.977226i	$-0.431937\pi$
$127$	3.23676e17	0.424404	0.212202	+	0.977226i	$0.431937\pi$
$128$	0	0
$129$	−1.45517e18	−1.67071
$130$	0	0
$131$	1.01569e18	1.02318	0.511592	−	0.859229i	$-0.329056\pi$
$131$	1.01569e18	1.02318	0.511592	+	0.859229i	$0.329056\pi$
$132$	0	0
$133$	−1.02967e18	−0.911927
$134$	0	0
$135$	−1.54287e18	−1.20363
$136$	0	0
$137$	1.05785e18	0.728282	0.364141	−	0.931344i	$-0.381363\pi$
$137$	1.05785e18	0.728282	0.364141	+	0.931344i	$0.381363\pi$
$138$	0	0
$139$	−1.96765e17	−0.119763	−0.0598813	−	0.998206i	$-0.519072\pi$
$139$	−1.96765e17	−0.119763	−0.0598813	+	0.998206i	$0.519072\pi$
$140$	0	0
$141$	4.08269e18	2.20082
$142$	0	0
$143$	2.30545e17	0.110255
$144$	0	0
$145$	−1.92839e18	−0.819534
$146$	0	0
$147$	1.09303e19	4.13462
$148$	0	0
$149$	−2.50869e18	−0.845988	−0.422994	−	0.906132i	$-0.639021\pi$
$149$	−2.50869e18	−0.845988	−0.422994	+	0.906132i	$0.639021\pi$
$150$	0	0
$151$	−6.45687e18	−1.94410	−0.972049	−	0.234777i	$-0.924564\pi$
$151$	−6.45687e18	−1.94410	−0.972049	+	0.234777i	$0.924564\pi$
$152$	0	0
$153$	−1.47006e18	−0.395784
$154$	0	0
$155$	−2.63098e18	−0.634305
$156$	0	0
$157$	2.42616e18	0.524533	0.262267	−	0.964995i	$-0.415530\pi$
$157$	2.42616e18	0.524533	0.262267	+	0.964995i	$0.415530\pi$
$158$	0	0
$159$	−1.06111e19	−2.06008
$160$	0	0
$161$	7.13815e18	1.24614
$162$	0	0
$163$	−7.59735e18	−1.19417	−0.597087	−	0.802177i	$-0.703675\pi$
$163$	−7.59735e18	−1.19417	−0.597087	+	0.802177i	$0.703675\pi$
$164$	0	0
$165$	8.06607e18	1.14301
$166$	0	0
$167$	5.87002e18	0.750844	0.375422	−	0.926854i	$-0.377498\pi$
$167$	5.87002e18	0.750844	0.375422	+	0.926854i	$0.377498\pi$
$168$	0	0
$169$	−8.24414e18	−0.953033
$170$	0	0
$171$	7.06000e18	0.738478
$172$	0	0
$173$	−1.46003e19	−1.38347	−0.691736	−	0.722151i	$-0.743154\pi$
$173$	−1.46003e19	−1.38347	−0.691736	+	0.722151i	$0.743154\pi$
$174$	0	0
$175$	−2.18607e19	−1.87863
$176$	0	0
$177$	1.02693e19	0.801250
$178$	0	0
$179$	2.58107e19	1.83041	0.915205	−	0.402988i	$-0.132028\pi$
$179$	2.58107e19	1.83041	0.915205	+	0.402988i	$0.132028\pi$
$180$	0	0
$181$	−1.10436e19	−0.712597	−0.356298	−	0.934372i	$-0.615961\pi$
$181$	−1.10436e19	−0.712597	−0.356298	+	0.934372i	$0.615961\pi$
$182$	0	0
$183$	4.13312e19	2.42909
$184$	0	0
$185$	3.82325e19	2.04868
$186$	0	0
$187$	2.68102e18	0.131113
$188$	0	0
$189$	−3.62130e19	−1.61785
$190$	0	0
$191$	−2.90525e19	−1.18686	−0.593431	−	0.804885i	$-0.702227\pi$
$191$	−2.90525e19	−1.18686	−0.593431	+	0.804885i	$0.702227\pi$
$192$	0	0
$193$	−1.92782e19	−0.720826	−0.360413	−	0.932793i	$-0.617364\pi$
$193$	−1.92782e19	−0.720826	−0.360413	+	0.932793i	$0.617364\pi$
$194$	0	0
$195$	1.42146e19	0.486900
$196$	0	0
$197$	2.85325e19	0.896143	0.448071	−	0.893998i	$-0.352111\pi$
$197$	2.85325e19	0.896143	0.448071	+	0.893998i	$0.352111\pi$
$198$	0	0
$199$	−9.19463e18	−0.265023	−0.132511	−	0.991181i	$-0.542304\pi$
$199$	−9.19463e18	−0.265023	−0.132511	+	0.991181i	$0.542304\pi$
$200$	0	0
$201$	−7.19135e19	−1.90390
$202$	0	0
$203$	−4.52615e19	−1.10157
$204$	0	0
$205$	2.58121e19	0.577983
$206$	0	0
$207$	−4.89430e19	−1.00912
$208$	0	0
$209$	−1.28756e19	−0.244639
$210$	0	0
$211$	6.65585e19	1.16628	0.583140	−	0.812372i	$-0.301824\pi$
$211$	6.65585e19	1.16628	0.583140	+	0.812372i	$0.301824\pi$
$212$	0	0
$213$	9.41086e19	1.52196
$214$	0	0
$215$	9.90988e19	1.48027
$216$	0	0
$217$	−6.17520e19	−0.852596
$218$	0	0
$219$	−9.80441e19	−1.25212
$220$	0	0
$221$	4.72467e18	0.0558518
$222$	0	0
$223$	−8.04446e19	−0.880857	−0.440428	−	0.897788i	$-0.645173\pi$
$223$	−8.04446e19	−0.880857	−0.440428	+	0.897788i	$0.645173\pi$
$224$	0	0
$225$	1.49889e20	1.52131
$226$	0	0
$227$	−2.93685e19	−0.276479	−0.138240	−	0.990399i	$-0.544144\pi$
$227$	−2.93685e19	−0.276479	−0.138240	+	0.990399i	$0.544144\pi$
$228$	0	0
$229$	7.88248e19	0.688749	0.344375	−	0.938832i	$-0.388091\pi$
$229$	7.88248e19	0.688749	0.344375	+	0.938832i	$0.388091\pi$
$230$	0	0
$231$	1.89320e20	1.53636
$232$	0	0
$233$	9.16048e19	0.690864	0.345432	−	0.938444i	$-0.387732\pi$
$233$	9.16048e19	0.690864	0.345432	+	0.938444i	$0.387732\pi$
$234$	0	0
$235$	−2.78037e20	−1.94996
$236$	0	0
$237$	3.26176e19	0.212859
$238$	0	0
$239$	2.61549e20	1.58917	0.794586	−	0.607152i	$-0.207688\pi$
$239$	2.61549e20	1.58917	0.794586	+	0.607152i	$0.207688\pi$
$240$	0	0
$241$	3.65469e18	0.0206874	0.0103437	−	0.999947i	$-0.496707\pi$
$241$	3.65469e18	0.0206874	0.0103437	+	0.999947i	$0.496707\pi$
$242$	0	0
$243$	−2.15173e20	−1.13536
$244$	0	0
$245$	−7.44369e20	−3.66334
$246$	0	0
$247$	−2.26903e19	−0.104212
$248$	0	0
$249$	−1.09409e20	−0.469199
$250$	0	0
$251$	6.70886e19	0.268796	0.134398	−	0.990927i	$-0.457090\pi$
$251$	6.70886e19	0.268796	0.134398	+	0.990927i	$0.457090\pi$
$252$	0	0
$253$	8.92597e19	0.334296
$254$	0	0
$255$	1.65302e20	0.579011
$256$	0	0
$257$	−4.57087e20	−1.49819	−0.749095	−	0.662462i	$-0.769512\pi$
$257$	−4.57087e20	−1.49819	−0.749095	+	0.662462i	$0.769512\pi$
$258$	0	0
$259$	8.97360e20	2.75371
$260$	0	0
$261$	3.10337e20	0.892051
$262$	0	0
$263$	−5.07629e19	−0.136749	−0.0683743	−	0.997660i	$-0.521781\pi$
$263$	−5.07629e19	−0.136749	−0.0683743	+	0.997660i	$0.521781\pi$
$264$	0	0
$265$	7.22628e20	1.82526
$266$	0	0
$267$	−1.01845e21	−2.41320
$268$	0	0
$269$	1.44913e20	0.322265	0.161132	−	0.986933i	$-0.448485\pi$
$269$	1.44913e20	0.322265	0.161132	+	0.986933i	$0.448485\pi$
$270$	0	0
$271$	1.80251e19	0.0376390	0.0188195	−	0.999823i	$-0.494009\pi$
$271$	1.80251e19	0.0376390	0.0188195	+	0.999823i	$0.494009\pi$
$272$	0	0
$273$	3.33632e20	0.654463
$274$	0	0
$275$	−2.73360e20	−0.503972
$276$	0	0
$277$	−1.22325e20	−0.212050	−0.106025	−	0.994363i	$-0.533812\pi$
$277$	−1.22325e20	−0.212050	−0.106025	+	0.994363i	$0.533812\pi$
$278$	0	0
$279$	4.23406e20	0.690432
$280$	0	0
$281$	1.19889e20	0.183982	0.0919910	−	0.995760i	$-0.470677\pi$
$281$	1.19889e20	0.183982	0.0919910	+	0.995760i	$0.470677\pi$
$282$	0	0
$283$	9.21025e20	1.33072	0.665360	−	0.746523i	$-0.268278\pi$
$283$	9.21025e20	1.33072	0.665360	+	0.746523i	$0.268278\pi$
$284$	0	0
$285$	−7.93867e20	−1.08036
$286$	0	0
$287$	6.05837e20	0.776891
$288$	0	0
$289$	−7.72297e20	−0.933582
$290$	0	0
$291$	−2.61032e21	−2.97580
$292$	0	0
$293$	1.54477e21	1.66146	0.830729	−	0.556677i	$-0.187924\pi$
$293$	1.54477e21	1.66146	0.830729	+	0.556677i	$0.187924\pi$
$294$	0	0
$295$	−6.99354e20	−0.709920
$296$	0	0
$297$	−4.52829e20	−0.434014
$298$	0	0
$299$	1.57299e20	0.142404
$300$	0	0
$301$	2.32596e21	1.98970
$302$	0	0
$303$	−1.71737e21	−1.38868
$304$	0	0
$305$	−2.81471e21	−2.15221
$306$	0	0
$307$	−7.43532e19	−0.0537804	−0.0268902	−	0.999638i	$-0.508560\pi$
$307$	−7.43532e19	−0.0537804	−0.0268902	+	0.999638i	$0.508560\pi$
$308$	0	0
$309$	−1.10556e21	−0.756719
$310$	0	0
$311$	3.18377e19	0.0206290	0.0103145	−	0.999947i	$-0.496717\pi$
$311$	3.18377e19	0.0206290	0.0103145	+	0.999947i	$0.496717\pi$
$312$	0	0
$313$	1.82416e21	1.11927	0.559637	−	0.828738i	$-0.310941\pi$
$313$	1.82416e21	1.11927	0.559637	+	0.828738i	$0.310941\pi$
$314$	0	0
$315$	7.06947e21	4.10910
$316$	0	0
$317$	−3.70538e20	−0.204093	−0.102047	−	0.994780i	$-0.532539\pi$
$317$	−3.70538e20	−0.204093	−0.102047	+	0.994780i	$0.532539\pi$
$318$	0	0
$319$	−5.65977e20	−0.295514
$320$	0	0
$321$	4.64160e21	2.29813
$322$	0	0
$323$	−2.63867e20	−0.123926
$324$	0	0
$325$	−4.81733e20	−0.214683
$326$	0	0
$327$	−1.27086e21	−0.537577
$328$	0	0
$329$	−6.52583e21	−2.62102
$330$	0	0
$331$	2.21102e19	0.00843442	0.00421721	−	0.999991i	$-0.498658\pi$
$331$	2.21102e19	0.00843442	0.00421721	+	0.999991i	$0.498658\pi$
$332$	0	0
$333$	−6.15279e21	−2.22995
$334$	0	0
$335$	4.89740e21	1.68688
$336$	0	0
$337$	3.35669e21	1.09915	0.549576	−	0.835444i	$-0.314789\pi$
$337$	3.35669e21	1.09915	0.549576	+	0.835444i	$0.314789\pi$
$338$	0	0
$339$	3.22267e21	1.00350
$340$	0	0
$341$	−7.72184e20	−0.228722
$342$	0	0
$343$	−1.07424e22	−3.02761
$344$	0	0
$345$	5.50344e21	1.47629
$346$	0	0
$347$	1.52263e21	0.388861	0.194431	−	0.980916i	$-0.437714\pi$
$347$	1.52263e21	0.388861	0.194431	+	0.980916i	$0.437714\pi$
$348$	0	0
$349$	−2.87454e21	−0.699121	−0.349561	−	0.936914i	$-0.613669\pi$
$349$	−2.87454e21	−0.699121	−0.349561	+	0.936914i	$0.613669\pi$
$350$	0	0
$351$	−7.98005e20	−0.184882
$352$	0	0
$353$	2.36637e21	0.522393	0.261196	−	0.965286i	$-0.415883\pi$
$353$	2.36637e21	0.522393	0.261196	+	0.965286i	$0.415883\pi$
$354$	0	0
$355$	−6.40892e21	−1.34848
$356$	0	0
$357$	3.87982e21	0.778274
$358$	0	0
$359$	−2.93438e21	−0.561324	−0.280662	−	0.959807i	$-0.590554\pi$
$359$	−2.93438e21	−0.561324	−0.280662	+	0.959807i	$0.590554\pi$
$360$	0	0
$361$	−4.21316e21	−0.768770
$362$	0	0
$363$	−6.77920e21	−1.18024
$364$	0	0
$365$	6.67693e21	1.10940
$366$	0	0
$367$	4.10268e21	0.650738	0.325369	−	0.945587i	$-0.394512\pi$
$367$	4.10268e21	0.650738	0.325369	+	0.945587i	$0.394512\pi$
$368$	0	0
$369$	−4.15395e21	−0.629126
$370$	0	0
$371$	1.69609e22	2.45341
$372$	0	0
$373$	1.13317e22	1.56592	0.782959	−	0.622073i	$-0.213709\pi$
$373$	1.13317e22	1.56592	0.782959	+	0.622073i	$0.213709\pi$
$374$	0	0
$375$	1.59758e20	0.0210959
$376$	0	0
$377$	−9.97402e20	−0.125883
$378$	0	0
$379$	3.98618e21	0.480976	0.240488	−	0.970652i	$-0.422692\pi$
$379$	3.98618e21	0.480976	0.240488	+	0.970652i	$0.422692\pi$
$380$	0	0
$381$	5.85725e21	0.675820
$382$	0	0
$383$	4.81687e21	0.531588	0.265794	−	0.964030i	$-0.414366\pi$
$383$	4.81687e21	0.531588	0.265794	+	0.964030i	$0.414366\pi$
$384$	0	0
$385$	−1.28929e22	−1.36124
$386$	0	0
$387$	−1.59480e22	−1.61125
$388$	0	0
$389$	−6.01398e21	−0.581555	−0.290777	−	0.956791i	$-0.593914\pi$
$389$	−6.01398e21	−0.581555	−0.290777	+	0.956791i	$0.593914\pi$
$390$	0	0
$391$	1.82924e21	0.169344
$392$	0	0
$393$	1.83799e22	1.62932
$394$	0	0
$395$	−2.22130e21	−0.188596
$396$	0	0
$397$	−1.72178e22	−1.40042	−0.700208	−	0.713938i	$-0.746910\pi$
$397$	−1.72178e22	−1.40042	−0.700208	+	0.713938i	$0.746910\pi$
$398$	0	0
$399$	−1.86329e22	−1.45215
$400$	0	0
$401$	−1.12923e22	−0.843441	−0.421720	−	0.906726i	$-0.638574\pi$
$401$	−1.12923e22	−0.843441	−0.421720	+	0.906726i	$0.638574\pi$
$402$	0	0
$403$	−1.36079e21	−0.0974316
$404$	0	0
$405$	3.64304e21	0.250090
$406$	0	0
$407$	1.12211e22	0.738727
$408$	0	0
$409$	−8.24389e21	−0.520576	−0.260288	−	0.965531i	$-0.583817\pi$
$409$	−8.24389e21	−0.520576	−0.260288	+	0.965531i	$0.583817\pi$
$410$	0	0
$411$	1.91429e22	1.15972
$412$	0	0
$413$	−1.64146e22	−0.954234
$414$	0	0
$415$	7.45087e21	0.415717
$416$	0	0
$417$	−3.56065e21	−0.190710
$418$	0	0
$419$	1.52574e22	0.784621	0.392311	−	0.919833i	$-0.371676\pi$
$419$	1.52574e22	0.784621	0.392311	+	0.919833i	$0.371676\pi$
$420$	0	0
$421$	8.43945e21	0.416789	0.208395	−	0.978045i	$-0.433176\pi$
$421$	8.43945e21	0.416789	0.208395	+	0.978045i	$0.433176\pi$
$422$	0	0
$423$	4.47447e22	2.12250
$424$	0	0
$425$	−5.60210e21	−0.255297
$426$	0	0
$427$	−6.60643e22	−2.89288
$428$	0	0
$429$	4.17193e21	0.175570
$430$	0	0
$431$	1.88889e22	0.764098	0.382049	−	0.924142i	$-0.375219\pi$
$431$	1.88889e22	0.764098	0.382049	+	0.924142i	$0.375219\pi$
$432$	0	0
$433$	−1.76820e21	−0.0687676	−0.0343838	−	0.999409i	$-0.510947\pi$
$433$	−1.76820e21	−0.0687676	−0.0343838	+	0.999409i	$0.510947\pi$
$434$	0	0
$435$	−3.48962e22	−1.30503
$436$	0	0
$437$	−8.78499e21	−0.315972
$438$	0	0
$439$	4.02315e22	1.39193	0.695965	−	0.718076i	$-0.254977\pi$
$439$	4.02315e22	1.39193	0.695965	+	0.718076i	$0.254977\pi$
$440$	0	0
$441$	1.19792e23	3.98749
$442$	0	0
$443$	−2.43504e22	−0.779963	−0.389981	−	0.920823i	$-0.627519\pi$
$443$	−2.43504e22	−0.779963	−0.389981	+	0.920823i	$0.627519\pi$
$444$	0	0
$445$	6.93578e22	2.13814
$446$	0	0
$447$	−4.53973e22	−1.34715
$448$	0	0
$449$	1.07884e22	0.308221	0.154111	−	0.988054i	$-0.450749\pi$
$449$	1.07884e22	0.308221	0.154111	+	0.988054i	$0.450749\pi$
$450$	0	0
$451$	7.57576e21	0.208413
$452$	0	0
$453$	−1.16844e23	−3.09578
$454$	0	0
$455$	−2.27208e22	−0.579864
$456$	0	0
$457$	−6.83731e22	−1.68112	−0.840558	−	0.541722i	$-0.817773\pi$
$457$	−6.83731e22	−1.68112	−0.840558	+	0.541722i	$0.817773\pi$
$458$	0	0
$459$	−9.28005e21	−0.219858
$460$	0	0
$461$	−2.33712e22	−0.533610	−0.266805	−	0.963751i	$-0.585968\pi$
$461$	−2.33712e22	−0.533610	−0.266805	+	0.963751i	$0.585968\pi$
$462$	0	0
$463$	−2.28585e22	−0.503049	−0.251524	−	0.967851i	$-0.580932\pi$
$463$	−2.28585e22	−0.503049	−0.251524	+	0.967851i	$0.580932\pi$
$464$	0	0
$465$	−4.76102e22	−1.01007
$466$	0	0
$467$	−5.39107e21	−0.110276	−0.0551381	−	0.998479i	$-0.517560\pi$
$467$	−5.39107e21	−0.110276	−0.0551381	+	0.998479i	$0.517560\pi$
$468$	0	0
$469$	1.14947e23	2.26741
$470$	0	0
$471$	4.39039e22	0.835266
$472$	0	0
$473$	2.90852e22	0.533767
$474$	0	0
$475$	2.69042e22	0.476348
$476$	0	0
$477$	−1.16293e23	−1.98677
$478$	0	0
$479$	−9.99522e22	−1.64794	−0.823968	−	0.566636i	$-0.808245\pi$
$479$	−9.99522e22	−1.64794	−0.823968	+	0.566636i	$0.808245\pi$
$480$	0	0
$481$	1.97746e22	0.314684
$482$	0	0
$483$	1.29172e23	1.98435
$484$	0	0
$485$	1.77766e23	2.63660
$486$	0	0
$487$	8.24172e22	1.18038	0.590190	−	0.807265i	$-0.299053\pi$
$487$	8.24172e22	1.18038	0.590190	+	0.807265i	$0.299053\pi$
$488$	0	0
$489$	−1.37482e23	−1.90160
$490$	0	0
$491$	−1.24257e22	−0.166008	−0.0830038	−	0.996549i	$-0.526451\pi$
$491$	−1.24257e22	−0.166008	−0.0830038	+	0.996549i	$0.526451\pi$
$492$	0	0
$493$	−1.15988e22	−0.149698
$494$	0	0
$495$	8.84009e22	1.10233
$496$	0	0
$497$	−1.50425e23	−1.81255
$498$	0	0
$499$	5.90767e21	0.0687957	0.0343979	−	0.999408i	$-0.489049\pi$
$499$	5.90767e21	0.0687957	0.0343979	+	0.999408i	$0.489049\pi$
$500$	0	0
$501$	1.06224e23	1.19564
$502$	0	0
$503$	−6.76058e22	−0.735624	−0.367812	−	0.929900i	$-0.619893\pi$
$503$	−6.76058e22	−0.735624	−0.367812	+	0.929900i	$0.619893\pi$
$504$	0	0
$505$	1.16955e23	1.23039
$506$	0	0
$507$	−1.49186e23	−1.51761
$508$	0	0
$509$	1.23744e21	0.0121737	0.00608686	−	0.999981i	$-0.498062\pi$
$509$	1.23744e21	0.0121737	0.00608686	+	0.999981i	$0.498062\pi$
$510$	0	0
$511$	1.56715e23	1.49119
$512$	0	0
$513$	4.45677e22	0.410225
$514$	0	0
$515$	7.52901e22	0.670465
$516$	0	0
$517$	−8.16030e22	−0.703131
$518$	0	0
$519$	−2.64207e23	−2.20304
$520$	0	0
$521$	3.73793e22	0.301655	0.150827	−	0.988560i	$-0.451806\pi$
$521$	3.73793e22	0.301655	0.150827	+	0.988560i	$0.451806\pi$
$522$	0	0
$523$	2.13593e23	1.66849	0.834244	−	0.551395i	$-0.185904\pi$
$523$	2.13593e23	1.66849	0.834244	+	0.551395i	$0.185904\pi$
$524$	0	0
$525$	−3.95592e23	−2.99153
$526$	0	0
$527$	−1.58248e22	−0.115864
$528$	0	0
$529$	−8.01486e22	−0.568228
$530$	0	0
$531$	1.12547e23	0.772738
$532$	0	0
$533$	1.33505e22	0.0887802
$534$	0	0
$535$	−3.16099e23	−2.03618
$536$	0	0
$537$	4.67070e23	2.91474
$538$	0	0
$539$	−2.18470e23	−1.32095
$540$	0	0
$541$	−1.52458e23	−0.893253	−0.446626	−	0.894721i	$-0.647375\pi$
$541$	−1.52458e23	−0.893253	−0.446626	+	0.894721i	$0.647375\pi$
$542$	0	0
$543$	−1.99845e23	−1.13474
$544$	0	0
$545$	8.65470e22	0.476302
$546$	0	0
$547$	−2.00996e23	−1.07225	−0.536124	−	0.844139i	$-0.680112\pi$
$547$	−2.00996e23	−1.07225	−0.536124	+	0.844139i	$0.680112\pi$
$548$	0	0
$549$	4.52973e23	2.34265
$550$	0	0
$551$	5.57037e22	0.279316
$552$	0	0
$553$	−5.21364e22	−0.253500
$554$	0	0
$555$	6.91856e23	3.26231
$556$	0	0
$557$	4.02153e23	1.83917	0.919586	−	0.392888i	$-0.128524\pi$
$557$	4.02153e23	1.83917	0.919586	+	0.392888i	$0.128524\pi$
$558$	0	0
$559$	5.12558e22	0.227375
$560$	0	0
$561$	4.85157e22	0.208784
$562$	0	0
$563$	−1.83177e23	−0.764802	−0.382401	−	0.923996i	$-0.624903\pi$
$563$	−1.83177e23	−0.764802	−0.382401	+	0.923996i	$0.624903\pi$
$564$	0	0
$565$	−2.19468e23	−0.889119
$566$	0	0
$567$	8.55061e22	0.336156
$568$	0	0
$569$	−2.60456e23	−0.993758	−0.496879	−	0.867820i	$-0.665521\pi$
$569$	−2.60456e23	−0.993758	−0.496879	+	0.867820i	$0.665521\pi$
$570$	0	0
$571$	4.07462e23	1.50897	0.754484	−	0.656319i	$-0.227887\pi$
$571$	4.07462e23	1.50897	0.754484	+	0.656319i	$0.227887\pi$
$572$	0	0
$573$	−5.25734e23	−1.88996
$574$	0	0
$575$	−1.86512e23	−0.650924
$576$	0	0
$577$	−4.27739e23	−1.44939	−0.724693	−	0.689071i	$-0.758019\pi$
$577$	−4.27739e23	−1.44939	−0.724693	+	0.689071i	$0.758019\pi$
$578$	0	0
$579$	−3.48859e23	−1.14784
$580$	0	0
$581$	1.74880e23	0.558783
$582$	0	0
$583$	2.12089e23	0.658166
$584$	0	0
$585$	1.55786e23	0.469573
$586$	0	0
$587$	2.98640e23	0.874427	0.437214	−	0.899358i	$-0.355965\pi$
$587$	2.98640e23	0.874427	0.437214	+	0.899358i	$0.355965\pi$
$588$	0	0
$589$	7.59988e22	0.216186
$590$	0	0
$591$	5.16324e23	1.42702
$592$	0	0
$593$	−4.06757e23	−1.09237	−0.546186	−	0.837664i	$-0.683921\pi$
$593$	−4.06757e23	−1.09237	−0.546186	+	0.837664i	$0.683921\pi$
$594$	0	0
$595$	−2.64221e23	−0.689562
$596$	0	0
$597$	−1.66386e23	−0.422022
$598$	0	0
$599$	−8.44514e22	−0.208199	−0.104100	−	0.994567i	$-0.533196\pi$
$599$	−8.44514e22	−0.208199	−0.104100	+	0.994567i	$0.533196\pi$
$600$	0	0
$601$	1.76567e23	0.423134	0.211567	−	0.977364i	$-0.432143\pi$
$601$	1.76567e23	0.423134	0.211567	+	0.977364i	$0.432143\pi$
$602$	0	0
$603$	−7.88143e23	−1.83615
$604$	0	0
$605$	4.61673e23	1.04572
$606$	0	0
$607$	−6.46330e23	−1.42348	−0.711738	−	0.702445i	$-0.752092\pi$
$607$	−6.46330e23	−1.42348	−0.711738	+	0.702445i	$0.752092\pi$
$608$	0	0
$609$	−8.19051e23	−1.75414
$610$	0	0
$611$	−1.43806e23	−0.299521
$612$	0	0
$613$	2.13581e23	0.432662	0.216331	−	0.976320i	$-0.430591\pi$
$613$	2.13581e23	0.432662	0.216331	+	0.976320i	$0.430591\pi$
$614$	0	0
$615$	4.67095e23	0.920379
$616$	0	0
$617$	9.85119e23	1.88827	0.944137	−	0.329553i	$-0.106898\pi$
$617$	9.85119e23	1.88827	0.944137	+	0.329553i	$0.106898\pi$
$618$	0	0
$619$	2.62630e23	0.489750	0.244875	−	0.969555i	$-0.421253\pi$
$619$	2.62630e23	0.489750	0.244875	+	0.969555i	$0.421253\pi$
$620$	0	0
$621$	−3.08963e23	−0.560567
$622$	0	0
$623$	1.62790e24	2.87396
$624$	0	0
$625$	−5.87491e23	−1.00930
$626$	0	0
$627$	−2.32998e23	−0.389563
$628$	0	0
$629$	2.29960e23	0.374216
$630$	0	0
$631$	−6.27160e23	−0.993411	−0.496706	−	0.867919i	$-0.665457\pi$
$631$	−6.27160e23	−0.993411	−0.496706	+	0.867919i	$0.665457\pi$
$632$	0	0
$633$	1.20444e24	1.85718
$634$	0	0
$635$	−3.98886e23	−0.598787
$636$	0	0
$637$	−3.85002e23	−0.562702
$638$	0	0
$639$	1.03139e24	1.46780
$640$	0	0
$641$	2.72378e23	0.377467	0.188734	−	0.982028i	$-0.439562\pi$
$641$	2.72378e23	0.377467	0.188734	+	0.982028i	$0.439562\pi$
$642$	0	0
$643$	4.23545e23	0.571619	0.285809	−	0.958287i	$-0.407738\pi$
$643$	4.23545e23	0.571619	0.285809	+	0.958287i	$0.407738\pi$
$644$	0	0
$645$	1.79329e24	2.35718
$646$	0	0
$647$	−1.24483e24	−1.59376	−0.796880	−	0.604137i	$-0.793518\pi$
$647$	−1.24483e24	−1.59376	−0.796880	+	0.604137i	$0.793518\pi$
$648$	0	0
$649$	−2.05258e23	−0.255988
$650$	0	0
$651$	−1.11746e24	−1.35767
$652$	0	0
$653$	−1.39998e24	−1.65715	−0.828574	−	0.559879i	$-0.810847\pi$
$653$	−1.39998e24	−1.65715	−0.828574	+	0.559879i	$0.810847\pi$
$654$	0	0
$655$	−1.25169e24	−1.44360
$656$	0	0
$657$	−1.07452e24	−1.20756
$658$	0	0
$659$	−3.82737e22	−0.0419154	−0.0209577	−	0.999780i	$-0.506672\pi$
$659$	−3.82737e22	−0.0419154	−0.0209577	+	0.999780i	$0.506672\pi$
$660$	0	0
$661$	2.88048e23	0.307435	0.153717	−	0.988115i	$-0.450875\pi$
$661$	2.88048e23	0.307435	0.153717	+	0.988115i	$0.450875\pi$
$662$	0	0
$663$	8.54975e22	0.0889382
$664$	0	0
$665$	1.26893e24	1.28663
$666$	0	0
$667$	−3.86163e23	−0.381682
$668$	0	0
$669$	−1.45572e24	−1.40268
$670$	0	0
$671$	−8.26108e23	−0.776060
$672$	0	0
$673$	1.54736e24	1.41731	0.708654	−	0.705556i	$-0.249303\pi$
$673$	1.54736e24	1.41731	0.708654	+	0.705556i	$0.249303\pi$
$674$	0	0
$675$	9.46205e23	0.845090
$676$	0	0
$677$	−1.88145e24	−1.63866	−0.819331	−	0.573321i	$-0.805655\pi$
$677$	−1.88145e24	−1.63866	−0.819331	+	0.573321i	$0.805655\pi$
$678$	0	0
$679$	4.17237e24	3.54397
$680$	0	0
$681$	−5.31453e23	−0.440265
$682$	0	0
$683$	−2.10444e24	−1.70044	−0.850218	−	0.526431i	$-0.823530\pi$
$683$	−2.10444e24	−1.70044	−0.850218	+	0.526431i	$0.823530\pi$
$684$	0	0
$685$	−1.30365e24	−1.02753
$686$	0	0
$687$	1.42641e24	1.09676
$688$	0	0
$689$	3.73757e23	0.280367
$690$	0	0
$691$	−8.36432e23	−0.612163	−0.306082	−	0.952005i	$-0.599018\pi$
$691$	−8.36432e23	−0.612163	−0.306082	+	0.952005i	$0.599018\pi$
$692$	0	0
$693$	2.07487e24	1.48169
$694$	0	0
$695$	2.42485e23	0.168972
$696$	0	0
$697$	1.55254e23	0.105576
$698$	0	0
$699$	1.65768e24	1.10013
$700$	0	0
$701$	−1.01169e24	−0.655307	−0.327654	−	0.944798i	$-0.606258\pi$
$701$	−1.01169e24	−0.655307	−0.327654	+	0.944798i	$0.606258\pi$
$702$	0	0
$703$	−1.10439e24	−0.698236
$704$	0	0
$705$	−5.03135e24	−3.10511
$706$	0	0
$707$	2.74506e24	1.65382
$708$	0	0
$709$	2.49014e24	1.46464	0.732320	−	0.680961i	$-0.238438\pi$
$709$	2.49014e24	1.46464	0.732320	+	0.680961i	$0.238438\pi$
$710$	0	0
$711$	3.57476e23	0.205284
$712$	0	0
$713$	−5.26858e23	−0.295415
$714$	0	0
$715$	−2.84114e23	−0.155558
$716$	0	0
$717$	4.73299e24	2.53059
$718$	0	0
$719$	1.77673e24	0.927739	0.463870	−	0.885903i	$-0.346461\pi$
$719$	1.77673e24	0.927739	0.463870	+	0.885903i	$0.346461\pi$
$720$	0	0
$721$	1.76714e24	0.901200
$722$	0	0
$723$	6.61352e22	0.0329426
$724$	0	0
$725$	1.18263e24	0.575409
$726$	0	0
$727$	−1.45639e24	−0.692208	−0.346104	−	0.938196i	$-0.612496\pi$
$727$	−1.45639e24	−0.692208	−0.346104	+	0.938196i	$0.612496\pi$
$728$	0	0
$729$	−3.51202e24	−1.63070
$730$	0	0
$731$	5.96058e23	0.270390
$732$	0	0
$733$	−3.43336e23	−0.152172	−0.0760862	−	0.997101i	$-0.524242\pi$
$733$	−3.43336e23	−0.152172	−0.0760862	+	0.997101i	$0.524242\pi$
$734$	0	0
$735$	−1.34701e25	−5.83349
$736$	0	0
$737$	1.43737e24	0.608269
$738$	0	0
$739$	−1.23819e23	−0.0512048	−0.0256024	−	0.999672i	$-0.508150\pi$
$739$	−1.23819e23	−0.0512048	−0.0256024	+	0.999672i	$0.508150\pi$
$740$	0	0
$741$	−4.10604e23	−0.165947
$742$	0	0
$743$	−8.96803e23	−0.354236	−0.177118	−	0.984190i	$-0.556677\pi$
$743$	−8.96803e23	−0.354236	−0.177118	+	0.984190i	$0.556677\pi$
$744$	0	0
$745$	3.09162e24	1.19360
$746$	0	0
$747$	−1.19907e24	−0.452502
$748$	0	0
$749$	−7.41920e24	−2.73692
$750$	0	0
$751$	3.25270e24	1.17302	0.586508	−	0.809943i	$-0.300502\pi$
$751$	3.25270e24	1.17302	0.586508	+	0.809943i	$0.300502\pi$
$752$	0	0
$753$	1.21404e24	0.428030
$754$	0	0
$755$	7.95720e24	2.74291
$756$	0	0
$757$	2.52863e24	0.852258	0.426129	−	0.904662i	$-0.359877\pi$
$757$	2.52863e24	0.852258	0.426129	+	0.904662i	$0.359877\pi$
$758$	0	0
$759$	1.61524e24	0.532332
$760$	0	0
$761$	−5.15668e24	−1.66188	−0.830942	−	0.556359i	$-0.812198\pi$
$761$	−5.15668e24	−1.66188	−0.830942	+	0.556359i	$0.812198\pi$
$762$	0	0
$763$	2.03135e24	0.640217
$764$	0	0
$765$	1.81165e24	0.558407
$766$	0	0
$767$	−3.61719e23	−0.109046
$768$	0	0
$769$	1.80944e24	0.533543	0.266771	−	0.963760i	$-0.414043\pi$
$769$	1.80944e24	0.533543	0.266771	+	0.963760i	$0.414043\pi$
$770$	0	0
$771$	−8.27144e24	−2.38572
$772$	0	0
$773$	−2.10672e24	−0.594402	−0.297201	−	0.954815i	$-0.596053\pi$
$773$	−2.10672e24	−0.594402	−0.297201	+	0.954815i	$0.596053\pi$
$774$	0	0
$775$	1.61351e24	0.445357
$776$	0	0
$777$	1.62386e25	4.38501
$778$	0	0
$779$	−7.45610e23	−0.196990
$780$	0	0
$781$	−1.88100e24	−0.486245
$782$	0	0
$783$	1.95907e24	0.495535
$784$	0	0
$785$	−2.98991e24	−0.740059
$786$	0	0
$787$	−1.37159e24	−0.332229	−0.166115	−	0.986106i	$-0.553122\pi$
$787$	−1.37159e24	−0.332229	−0.166115	+	0.986106i	$0.553122\pi$
$788$	0	0
$789$	−9.18606e23	−0.217758
$790$	0	0
$791$	−5.15116e24	−1.19510
$792$	0	0
$793$	−1.45582e24	−0.330587
$794$	0	0
$795$	1.30767e25	2.90654
$796$	0	0
$797$	4.84784e24	1.05476	0.527378	−	0.849631i	$-0.323175\pi$
$797$	4.84784e24	1.05476	0.527378	+	0.849631i	$0.323175\pi$
$798$	0	0
$799$	−1.67233e24	−0.356184
$800$	0	0
$801$	−1.11618e25	−2.32733
$802$	0	0
$803$	1.95966e24	0.400035
$804$	0	0
$805$	−8.79678e24	−1.75816
$806$	0	0
$807$	2.62234e24	0.513174
$808$	0	0
$809$	3.62003e24	0.693665	0.346832	−	0.937927i	$-0.387257\pi$
$809$	3.62003e24	0.693665	0.346832	+	0.937927i	$0.387257\pi$
$810$	0	0
$811$	−5.05588e24	−0.948680	−0.474340	−	0.880342i	$-0.657313\pi$
$811$	−5.05588e24	−0.948680	−0.474340	+	0.880342i	$0.657313\pi$
$812$	0	0
$813$	3.26182e23	0.0599363
$814$	0	0
$815$	9.36268e24	1.68485
$816$	0	0
$817$	−2.86258e24	−0.504510
$818$	0	0
$819$	3.65647e24	0.631174
$820$	0	0
$821$	2.03537e24	0.344133	0.172067	−	0.985085i	$-0.444956\pi$
$821$	2.03537e24	0.344133	0.172067	+	0.985085i	$0.444956\pi$
$822$	0	0
$823$	−3.90337e24	−0.646459	−0.323229	−	0.946321i	$-0.604769\pi$
$823$	−3.90337e24	−0.646459	−0.323229	+	0.946321i	$0.604769\pi$
$824$	0	0
$825$	−4.94672e24	−0.802525
$826$	0	0
$827$	−5.68416e24	−0.903377	−0.451689	−	0.892176i	$-0.649178\pi$
$827$	−5.68416e24	−0.903377	−0.451689	+	0.892176i	$0.649178\pi$
$828$	0	0
$829$	−6.91406e24	−1.07651	−0.538257	−	0.842781i	$-0.680917\pi$
$829$	−6.91406e24	−1.07651	−0.538257	+	0.842781i	$0.680917\pi$
$830$	0	0
$831$	−2.21360e24	−0.337669
$832$	0	0
$833$	−4.47722e24	−0.669153
$834$	0	0
$835$	−7.23399e24	−1.05936
$836$	0	0
$837$	2.67283e24	0.383535
$838$	0	0
$839$	−9.93397e24	−1.39684	−0.698419	−	0.715689i	$-0.746113\pi$
$839$	−9.93397e24	−1.39684	−0.698419	+	0.715689i	$0.746113\pi$
$840$	0	0
$841$	−4.80857e24	−0.662598
$842$	0	0
$843$	2.16951e24	0.292973
$844$	0	0
$845$	1.01598e25	1.34463
$846$	0	0
$847$	1.08360e25	1.40559
$848$	0	0
$849$	1.66669e25	2.11904
$850$	0	0
$851$	7.65612e24	0.954130
$852$	0	0
$853$	4.07127e24	0.497351	0.248675	−	0.968587i	$-0.420005\pi$
$853$	4.07127e24	0.497351	0.248675	+	0.968587i	$0.420005\pi$
$854$	0	0
$855$	−8.70047e24	−1.04191
$856$	0	0
$857$	7.83692e24	0.920044	0.460022	−	0.887907i	$-0.347841\pi$
$857$	7.83692e24	0.920044	0.460022	+	0.887907i	$0.347841\pi$
$858$	0	0
$859$	−7.75004e24	−0.891994	−0.445997	−	0.895034i	$-0.647151\pi$
$859$	−7.75004e24	−0.891994	−0.445997	+	0.895034i	$0.647151\pi$
$860$	0	0
$861$	1.09632e25	1.23712
$862$	0	0
$863$	9.90022e24	1.09535	0.547675	−	0.836691i	$-0.315513\pi$
$863$	9.90022e24	1.09535	0.547675	+	0.836691i	$0.315513\pi$
$864$	0	0
$865$	1.79929e25	1.95193
$866$	0	0
$867$	−1.39755e25	−1.48664
$868$	0	0
$869$	−6.51946e23	−0.0680054
$870$	0	0
$871$	2.53303e24	0.259111
$872$	0	0
$873$	−2.86081e25	−2.86991
$874$	0	0
$875$	−2.55360e23	−0.0251237
$876$	0	0
$877$	9.81741e24	0.947328	0.473664	−	0.880706i	$-0.342931\pi$
$877$	9.81741e24	0.947328	0.473664	+	0.880706i	$0.342931\pi$
$878$	0	0
$879$	2.79542e25	2.64570
$880$	0	0
$881$	−1.13021e25	−1.04921	−0.524607	−	0.851345i	$-0.675788\pi$
$881$	−1.13021e25	−1.04921	−0.524607	+	0.851345i	$0.675788\pi$
$882$	0	0
$883$	−1.05781e25	−0.963255	−0.481627	−	0.876376i	$-0.659954\pi$
$883$	−1.05781e25	−0.963255	−0.481627	+	0.876376i	$0.659954\pi$
$884$	0	0
$885$	−1.26555e25	−1.13048
$886$	0	0
$887$	−1.16661e25	−1.02230	−0.511148	−	0.859493i	$-0.670780\pi$
$887$	−1.16661e25	−1.02230	−0.511148	+	0.859493i	$0.670780\pi$
$888$	0	0
$889$	−9.36231e24	−0.804855
$890$	0	0
$891$	1.06922e24	0.0901792
$892$	0	0
$893$	8.03141e24	0.664591
$894$	0	0
$895$	−3.18081e25	−2.58251
$896$	0	0
$897$	2.84649e24	0.226764
$898$	0	0
$899$	3.34069e24	0.261143
$900$	0	0
$901$	4.34645e24	0.333406
$902$	0	0
$903$	4.20905e25	3.16839
$904$	0	0
$905$	1.36097e25	1.00540
$906$	0	0
$907$	7.57602e24	0.549261	0.274631	−	0.961550i	$-0.411444\pi$
$907$	7.57602e24	0.549261	0.274631	+	0.961550i	$0.411444\pi$
$908$	0	0
$909$	−1.88217e25	−1.33926
$910$	0	0
$911$	1.05833e25	0.739118	0.369559	−	0.929207i	$-0.379509\pi$
$911$	1.05833e25	0.739118	0.369559	+	0.929207i	$0.379509\pi$
$912$	0	0
$913$	2.18681e24	0.149903
$914$	0	0
$915$	−5.09350e25	−3.42718
$916$	0	0
$917$	−2.93786e25	−1.94040
$918$	0	0
$919$	1.44343e25	0.935867	0.467933	−	0.883764i	$-0.344999\pi$
$919$	1.44343e25	0.935867	0.467933	+	0.883764i	$0.344999\pi$
$920$	0	0
$921$	−1.34550e24	−0.0856398
$922$	0	0
$923$	−3.31482e24	−0.207131
$924$	0	0
$925$	−2.34470e25	−1.43841
$926$	0	0
$927$	−1.21165e25	−0.729792
$928$	0	0
$929$	2.30190e25	1.36130	0.680648	−	0.732610i	$-0.261698\pi$
$929$	2.30190e25	1.36130	0.680648	+	0.732610i	$0.261698\pi$
$930$	0	0
$931$	2.15019e25	1.24855
$932$	0	0
$933$	5.76136e23	0.0328496
$934$	0	0
$935$	−3.30398e24	−0.184986
$936$	0	0
$937$	−9.05092e24	−0.497630	−0.248815	−	0.968551i	$-0.580041\pi$
$937$	−9.05092e24	−0.497630	−0.248815	+	0.968551i	$0.580041\pi$
$938$	0	0
$939$	3.30100e25	1.78233
$940$	0	0
$941$	1.82487e25	0.967654	0.483827	−	0.875164i	$-0.339246\pi$
$941$	1.82487e25	0.967654	0.483827	+	0.875164i	$0.339246\pi$
$942$	0	0
$943$	5.16890e24	0.269184
$944$	0	0
$945$	4.46275e25	2.28261
$946$	0	0
$947$	−1.28506e25	−0.645580	−0.322790	−	0.946471i	$-0.604621\pi$
$947$	−1.28506e25	−0.645580	−0.322790	+	0.946471i	$0.604621\pi$
$948$	0	0
$949$	3.45344e24	0.170408
$950$	0	0
$951$	−6.70525e24	−0.324998
$952$	0	0
$953$	−2.06250e25	−0.981984	−0.490992	−	0.871164i	$-0.663366\pi$
$953$	−2.06250e25	−0.981984	−0.490992	+	0.871164i	$0.663366\pi$
$954$	0	0
$955$	3.58032e25	1.67453
$956$	0	0
$957$	−1.02419e25	−0.470576
$958$	0	0
$959$	−3.05982e25	−1.38114
$960$	0	0
$961$	−1.79923e25	−0.797880
$962$	0	0
$963$	5.08701e25	2.21635
$964$	0	0
$965$	2.37578e25	1.01701
$966$	0	0
$967$	9.79148e24	0.411835	0.205918	−	0.978569i	$-0.433982\pi$
$967$	9.79148e24	0.411835	0.205918	+	0.978569i	$0.433982\pi$
$968$	0	0
$969$	−4.77494e24	−0.197340
$970$	0	0
$971$	−2.45503e25	−0.996995	−0.498498	−	0.866891i	$-0.666115\pi$
$971$	−2.45503e25	−0.996995	−0.498498	+	0.866891i	$0.666115\pi$
$972$	0	0
$973$	5.69140e24	0.227122
$974$	0	0
$975$	−8.71743e24	−0.341861
$976$	0	0
$977$	4.70253e24	0.181229	0.0906145	−	0.995886i	$-0.471117\pi$
$977$	4.70253e24	0.181229	0.0906145	+	0.995886i	$0.471117\pi$
$978$	0	0
$979$	2.03563e25	0.770985
$980$	0	0
$981$	−1.39281e25	−0.518447
$982$	0	0
$983$	−3.79842e25	−1.38962	−0.694812	−	0.719191i	$-0.744513\pi$
$983$	−3.79842e25	−1.38962	−0.694812	+	0.719191i	$0.744513\pi$
$984$	0	0
$985$	−3.51624e25	−1.26436
$986$	0	0
$987$	−1.18091e26	−4.17371
$988$	0	0
$989$	1.98447e25	0.689407
$990$	0	0
$991$	−2.78114e24	−0.0949724	−0.0474862	−	0.998872i	$-0.515121\pi$
$991$	−2.78114e24	−0.0949724	−0.0474862	+	0.998872i	$0.515121\pi$
$992$	0	0
$993$	4.00107e23	0.0134310
$994$	0	0
$995$	1.13311e25	0.373918
$996$	0	0
$997$	5.10311e25	1.65549	0.827744	−	0.561106i	$-0.189624\pi$
$997$	5.10311e25	1.65549	0.827744	+	0.561106i	$0.189624\pi$
$998$	0	0
$999$	−3.88407e25	−1.23874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.18.a.b.1.2	✓	2
3.2	odd	2		72.18.a.d.1.2		2
4.3	odd	2		16.18.a.c.1.1		2
8.3	odd	2		64.18.a.m.1.2		2
8.5	even	2		64.18.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.18.a.b.1.2	✓	2	1.1	even	1	trivial
16.18.a.c.1.1		2	4.3	odd	2
64.18.a.f.1.1		2	8.5	even	2
64.18.a.m.1.2		2	8.3	odd	2
72.18.a.d.1.2		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:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q+18096.0 q^{3} -1.23236e6 q^{5} -2.89249e7 q^{7} +1.98325e8 q^{9} +O(q^{10})\)	\(q+18096.0 q^{3} -1.23236e6 q^{5} -2.89249e7 q^{7} +1.98325e8 q^{9} -3.61694e8 q^{11} -6.37402e8 q^{13} -2.23008e10 q^{15} -7.41238e9 q^{17} +3.55981e10 q^{19} -5.23425e11 q^{21} -2.46782e11 q^{23} +7.55776e11 q^{25} +1.25197e12 q^{27} +1.56479e12 q^{29} +2.13491e12 q^{31} -6.54522e12 q^{33} +3.56459e13 q^{35} -3.10238e13 q^{37} -1.15344e13 q^{39} -2.09452e13 q^{41} -8.04137e13 q^{43} -2.44408e14 q^{45} +2.25613e14 q^{47} +6.04019e14 q^{49} -1.34134e14 q^{51} -5.86376e14 q^{53} +4.45738e14 q^{55} +6.44184e14 q^{57} +5.67491e14 q^{59} +2.28400e15 q^{61} -5.73652e15 q^{63} +7.85509e14 q^{65} -3.97400e15 q^{67} -4.46577e15 q^{69} +5.20052e15 q^{71} -5.41800e15 q^{73} +1.36765e16 q^{75} +1.04620e16 q^{77} +1.80248e15 q^{79} -2.95614e15 q^{81} -6.04601e15 q^{83} +9.13474e15 q^{85} +2.83165e16 q^{87} -5.62804e16 q^{89} +1.84368e16 q^{91} +3.86333e16 q^{93} -4.38698e16 q^{95} -1.44249e17 q^{97} -7.17330e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 11592 q^{3} - 791924 q^{5} - 18932592 q^{7} + 111486618 q^{9}+O(q^{10}) \) 2 * q + 11592 * q^3 - 791924 * q^5 - 18932592 * q^7 + 111486618 * q^9	\( 2 q + 11592 q^{3} - 791924 q^{5} - 18932592 q^{7} + 111486618 q^{9} - 470771240 q^{11} - 2503523428 q^{13} - 25165411920 q^{15} - 48444688348 q^{17} - 82483300760 q^{19} - 588414407616 q^{21} - 307097031248 q^{23} + 186821492926 q^{25} + 2656687689936 q^{27} + 1989575387580 q^{29} + 10752133575232 q^{31} - 5835781169568 q^{33} + 40046902091616 q^{35} - 51174452749620 q^{37} + 602829666288 q^{39} - 114227291044524 q^{41} - 56757554203624 q^{43} - 282654737698788 q^{45} + 201553163158368 q^{47} + 471233999080306 q^{49} + 132739385855760 q^{51} + 217275463587052 q^{53} + 397696513984784 q^{55} + 14\!\cdots\!32 q^{57}+ \cdots - 62\!\cdots\!36 q^{99}+O(q^{100}) \) 2 * q + 11592 * q^3 - 791924 * q^5 - 18932592 * q^7 + 111486618 * q^9 - 470771240 * q^11 - 2503523428 * q^13 - 25165411920 * q^15 - 48444688348 * q^17 - 82483300760 * q^19 - 588414407616 * q^21 - 307097031248 * q^23 + 186821492926 * q^25 + 2656687689936 * q^27 + 1989575387580 * q^29 + 10752133575232 * q^31 - 5835781169568 * q^33 + 40046902091616 * q^35 - 51174452749620 * q^37 + 602829666288 * q^39 - 114227291044524 * q^41 - 56757554203624 * q^43 - 282654737698788 * q^45 + 201553163158368 * q^47 + 471233999080306 * q^49 + 132739385855760 * q^51 + 217275463587052 * q^53 + 397696513984784 * q^55 + 1412184913398432 * q^57 - 466311448890056 * q^59 + 2063817263626748 * q^61 - 6604236688046256 * q^63 - 36400955340632 * q^65 - 6637225962358328 * q^67 - 4073481953319744 * q^69 + 1543684997984528 * q^71 - 3734106800358508 * q^73 + 17376986610519480 * q^75 + 9372036167046336 * q^77 + 17755378507330976 * q^79 - 878143806358542 * q^81 + 25453597558158056 * q^83 - 8937430679761448 * q^85 + 25553707384497264 * q^87 - 67216838659203660 * q^89 - 210069989389344 * q^91 - 17413067555884800 * q^93 - 95877299737499536 * q^95 - 130979024861443388 * q^97 - 62260905496457736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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