LMFDB - Embedded newform 8.18.a.a.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2146}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2146 $ x^2 - 2146
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-46.3249$ 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)

$f(q)$	$=$	$q-12335.2 q^{3} -738361. q^{5} -1.17649e7 q^{7} +2.30166e7 q^{9} +O(q^{10})$	$q-12335.2 q^{3} -738361. q^{5} -1.17649e7 q^{7} +2.30166e7 q^{9} +1.10233e9 q^{11} -2.57536e9 q^{13} +9.10782e9 q^{15} +4.96650e10 q^{17} +7.58630e10 q^{19} +1.45122e11 q^{21} +4.80343e10 q^{23} -2.17762e11 q^{25} +1.30905e12 q^{27} -4.46967e12 q^{29} -2.89743e12 q^{31} -1.35974e13 q^{33} +8.68672e12 q^{35} +1.98048e13 q^{37} +3.17675e13 q^{39} -5.72233e13 q^{41} +1.17723e14 q^{43} -1.69946e13 q^{45} +1.99303e14 q^{47} -9.42184e13 q^{49} -6.12627e14 q^{51} +3.19512e14 q^{53} -8.13917e14 q^{55} -9.35784e14 q^{57} +3.92786e14 q^{59} +2.54361e15 q^{61} -2.70787e14 q^{63} +1.90154e15 q^{65} +2.36089e14 q^{67} -5.92511e14 q^{69} -7.35004e14 q^{71} +7.70307e15 q^{73} +2.68614e15 q^{75} -1.29688e16 q^{77} -8.20252e15 q^{79} -1.91198e16 q^{81} -2.50479e15 q^{83} -3.66707e16 q^{85} +5.51342e16 q^{87} +1.99635e16 q^{89} +3.02987e16 q^{91} +3.57403e16 q^{93} -5.60143e16 q^{95} +5.80049e16 q^{97} +2.53719e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 952 q^{3} - 53620 q^{5} - 333168 q^{7} + 23453338 q^{9}+O(q^{10}) $ 2 * q - 952 * q^3 - 53620 * q^5 - 333168 * q^7 + 23453338 * q^9	$ 2 q - 952 q^{3} - 53620 q^{5} - 333168 q^{7} + 23453338 q^{9} + 430974680 q^{11} + 2667521948 q^{13} + 16902353840 q^{15} + 60673503268 q^{17} + 178629960040 q^{19} + 275250928704 q^{21} + 528756594608 q^{23} - 511831510850 q^{25} - 156001401136 q^{27} - 7240660091460 q^{29} - 1878351140288 q^{31} - 21239581915552 q^{33} + 16514472678240 q^{35} - 20332464566580 q^{37} + 91448192162288 q^{39} - 1763041905324 q^{41} + 193394525968664 q^{43} - 16695526837220 q^{45} + 100763837765472 q^{47} - 196165218276494 q^{49} - 487316066501360 q^{51} - 317818146060052 q^{53} - 12\!\cdots\!20 q^{55}+ \cdots + 25\!\cdots\!64 q^{99}+O(q^{100}) $ 2 * q - 952 * q^3 - 53620 * q^5 - 333168 * q^7 + 23453338 * q^9 + 430974680 * q^11 + 2667521948 * q^13 + 16902353840 * q^15 + 60673503268 * q^17 + 178629960040 * q^19 + 275250928704 * q^21 + 528756594608 * q^23 - 511831510850 * q^25 - 156001401136 * q^27 - 7240660091460 * q^29 - 1878351140288 * q^31 - 21239581915552 * q^33 + 16514472678240 * q^35 - 20332464566580 * q^37 + 91448192162288 * q^39 - 1763041905324 * q^41 + 193394525968664 * q^43 - 16695526837220 * q^45 + 100763837765472 * q^47 - 196165218276494 * q^49 - 487316066501360 * q^51 - 317818146060052 * q^53 - 1273620709243120 * q^55 + 234031155307168 * q^57 - 1262050788321736 * q^59 + 2765859692723708 * q^61 - 265794859238064 * q^63 + 5491559693146280 * q^65 + 1963006544550088 * q^67 + 4879639671010496 * q^69 + 483639107104528 * q^71 - 2176892348591212 * q^73 - 661303864041800 * q^75 - 20643473818671936 * q^77 - 5295839905627744 * q^79 - 35853176106728462 * q^81 + 9972518018887144 * q^83 - 29132785359362600 * q^85 + 23591579817096816 * q^87 - 711099036813900 * q^89 + 90233771615940576 * q^91 + 47340611694588160 * q^93 + 14354471748500080 * q^95 + 114870546609971908 * q^97 + 25078682365155064 * 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
$3$	−12335.2	−1.08546	−0.542731	−	0.839906i	$-0.682610\pi$
$3$	−12335.2	−1.08546	−0.542731	+	0.839906i	$0.682610\pi$
$4$	0	0
$5$	−738361.	−0.845325	−0.422663	−	0.906287i	$-0.638904\pi$
$5$	−738361.	−0.845325	−0.422663	+	0.906287i	$0.638904\pi$
$6$	0	0
$7$	−1.17649e7	−0.771354	−0.385677	−	0.922634i	$-0.626032\pi$
$7$	−1.17649e7	−0.771354	−0.385677	+	0.922634i	$0.626032\pi$
$8$	0	0
$9$	2.30166e7	0.178230
$10$	0	0
$11$	1.10233e9	1.55051	0.775253	−	0.631651i	$-0.217622\pi$
$11$	1.10233e9	1.55051	0.775253	+	0.631651i	$0.217622\pi$
$12$	0	0
$13$	−2.57536e9	−0.875627	−0.437814	−	0.899066i	$-0.644247\pi$
$13$	−2.57536e9	−0.875627	−0.437814	+	0.899066i	$0.644247\pi$
$14$	0	0
$15$	9.10782e9	0.917569
$16$	0	0
$17$	4.96650e10	1.72677	0.863386	−	0.504544i	$-0.168339\pi$
$17$	4.96650e10	1.72677	0.863386	+	0.504544i	$0.168339\pi$
$18$	0	0
$19$	7.58630e10	1.02477	0.512383	−	0.858757i	$-0.328763\pi$
$19$	7.58630e10	1.02477	0.512383	+	0.858757i	$0.328763\pi$
$20$	0	0
$21$	1.45122e11	0.837276
$22$	0	0
$23$	4.80343e10	0.127898	0.0639491	−	0.997953i	$-0.479630\pi$
$23$	4.80343e10	0.127898	0.0639491	+	0.997953i	$0.479630\pi$
$24$	0	0
$25$	−2.17762e11	−0.285426
$26$	0	0
$27$	1.30905e12	0.892001
$28$	0	0
$29$	−4.46967e12	−1.65918	−0.829589	−	0.558375i	$-0.811425\pi$
$29$	−4.46967e12	−1.65918	−0.829589	+	0.558375i	$0.811425\pi$
$30$	0	0
$31$	−2.89743e12	−0.610152	−0.305076	−	0.952328i	$-0.598682\pi$
$31$	−2.89743e12	−0.610152	−0.305076	+	0.952328i	$0.598682\pi$
$32$	0	0
$33$	−1.35974e13	−1.68302
$34$	0	0
$35$	8.68672e12	0.652045
$36$	0	0
$37$	1.98048e13	0.926950	0.463475	−	0.886110i	$-0.346602\pi$
$37$	1.98048e13	0.926950	0.463475	+	0.886110i	$0.346602\pi$
$38$	0	0
$39$	3.17675e13	0.950461
$40$	0	0
$41$	−5.72233e13	−1.11921	−0.559604	−	0.828760i	$-0.689047\pi$
$41$	−5.72233e13	−1.11921	−0.559604	+	0.828760i	$0.689047\pi$
$42$	0	0
$43$	1.17723e14	1.53596	0.767978	−	0.640476i	$-0.221263\pi$
$43$	1.17723e14	1.53596	0.767978	+	0.640476i	$0.221263\pi$
$44$	0	0
$45$	−1.69946e13	−0.150662
$46$	0	0
$47$	1.99303e14	1.22090	0.610452	−	0.792053i	$-0.290988\pi$
$47$	1.99303e14	1.22090	0.610452	+	0.792053i	$0.290988\pi$
$48$	0	0
$49$	−9.42184e13	−0.405013
$50$	0	0
$51$	−6.12627e14	−1.87435
$52$	0	0
$53$	3.19512e14	0.704924	0.352462	−	0.935826i	$-0.385345\pi$
$53$	3.19512e14	0.704924	0.352462	+	0.935826i	$0.385345\pi$
$54$	0	0
$55$	−8.13917e14	−1.31068
$56$	0	0
$57$	−9.35784e14	−1.11234
$58$	0	0
$59$	3.92786e14	0.348268	0.174134	−	0.984722i	$-0.444287\pi$
$59$	3.92786e14	0.348268	0.174134	+	0.984722i	$0.444287\pi$
$60$	0	0
$61$	2.54361e15	1.69882	0.849409	−	0.527736i	$-0.176959\pi$
$61$	2.54361e15	1.69882	0.849409	+	0.527736i	$0.176959\pi$
$62$	0	0
$63$	−2.70787e14	−0.137478
$64$	0	0
$65$	1.90154e15	0.740189
$66$	0	0
$67$	2.36089e14	0.0710296	0.0355148	−	0.999369i	$-0.488693\pi$
$67$	2.36089e14	0.0710296	0.0355148	+	0.999369i	$0.488693\pi$
$68$	0	0
$69$	−5.92511e14	−0.138829
$70$	0	0
$71$	−7.35004e14	−0.135081	−0.0675404	−	0.997717i	$-0.521515\pi$
$71$	−7.35004e14	−0.135081	−0.0675404	+	0.997717i	$0.521515\pi$
$72$	0	0
$73$	7.70307e15	1.11794	0.558971	−	0.829187i	$-0.311196\pi$
$73$	7.70307e15	1.11794	0.558971	+	0.829187i	$0.311196\pi$
$74$	0	0
$75$	2.68614e15	0.309819
$76$	0	0
$77$	−1.29688e16	−1.19599
$78$	0	0
$79$	−8.20252e15	−0.608299	−0.304149	−	0.952624i	$-0.598372\pi$
$79$	−8.20252e15	−0.608299	−0.304149	+	0.952624i	$0.598372\pi$
$80$	0	0
$81$	−1.91198e16	−1.14646
$82$	0	0
$83$	−2.50479e15	−0.122070	−0.0610348	−	0.998136i	$-0.519440\pi$
$83$	−2.50479e15	−0.122070	−0.0610348	+	0.998136i	$0.519440\pi$
$84$	0	0
$85$	−3.66707e16	−1.45968
$86$	0	0
$87$	5.51342e16	1.80098
$88$	0	0
$89$	1.99635e16	0.537554	0.268777	−	0.963202i	$-0.413381\pi$
$89$	1.99635e16	0.537554	0.268777	+	0.963202i	$0.413381\pi$
$90$	0	0
$91$	3.02987e16	0.675418
$92$	0	0
$93$	3.57403e16	0.662297
$94$	0	0
$95$	−5.60143e16	−0.866260
$96$	0	0
$97$	5.80049e16	0.751458	0.375729	−	0.926730i	$-0.377392\pi$
$97$	5.80049e16	0.751458	0.375729	+	0.926730i	$0.377392\pi$
$98$	0	0
$99$	2.53719e16	0.276346
$100$	0	0
$101$	6.67603e16	0.613461	0.306730	−	0.951796i	$-0.400765\pi$
$101$	6.67603e16	0.613461	0.306730	+	0.951796i	$0.400765\pi$
$102$	0	0
$103$	−1.20710e17	−0.938913	−0.469457	−	0.882956i	$-0.655550\pi$
$103$	−1.20710e17	−0.938913	−0.469457	+	0.882956i	$0.655550\pi$
$104$	0	0
$105$	−1.07152e17	−0.707770
$106$	0	0
$107$	2.14828e16	0.120873	0.0604364	−	0.998172i	$-0.480751\pi$
$107$	2.14828e16	0.120873	0.0604364	+	0.998172i	$0.480751\pi$
$108$	0	0
$109$	1.10858e17	0.532897	0.266449	−	0.963849i	$-0.414150\pi$
$109$	1.10858e17	0.532897	0.266449	+	0.963849i	$0.414150\pi$
$110$	0	0
$111$	−2.44296e17	−1.00617
$112$	0	0
$113$	−2.42988e17	−0.859839	−0.429920	−	0.902867i	$-0.641458\pi$
$113$	−2.42988e17	−0.859839	−0.429920	+	0.902867i	$0.641458\pi$
$114$	0	0
$115$	−3.54666e16	−0.108116
$116$	0	0
$117$	−5.92760e16	−0.156063
$118$	0	0
$119$	−5.84303e17	−1.33195
$120$	0	0
$121$	7.09682e17	1.40407
$122$	0	0
$123$	7.05861e17	1.21486
$124$	0	0
$125$	7.24112e17	1.08660
$126$	0	0
$127$	1.21334e18	1.59093	0.795467	−	0.605997i	$-0.207226\pi$
$127$	1.21334e18	1.59093	0.795467	+	0.605997i	$0.207226\pi$
$128$	0	0
$129$	−1.45213e18	−1.66722
$130$	0	0
$131$	8.28209e17	0.834322	0.417161	−	0.908833i	$-0.363025\pi$
$131$	8.28209e17	0.834322	0.417161	+	0.908833i	$0.363025\pi$
$132$	0	0
$133$	−8.92518e17	−0.790457
$134$	0	0
$135$	−9.66554e17	−0.754031
$136$	0	0
$137$	6.81954e16	0.0469494	0.0234747	−	0.999724i	$-0.492527\pi$
$137$	6.81954e16	0.0469494	0.0234747	+	0.999724i	$0.492527\pi$
$138$	0	0
$139$	1.83200e18	1.11507	0.557533	−	0.830155i	$-0.311748\pi$
$139$	1.83200e18	1.11507	0.557533	+	0.830155i	$0.311748\pi$
$140$	0	0
$141$	−2.45844e18	−1.32525
$142$	0	0
$143$	−2.83889e18	−1.35766
$144$	0	0
$145$	3.30023e18	1.40254
$146$	0	0
$147$	1.16220e18	0.439627
$148$	0	0
$149$	−3.72386e18	−1.25577	−0.627886	−	0.778306i	$-0.716079\pi$
$149$	−3.72386e18	−1.25577	−0.627886	+	0.778306i	$0.716079\pi$
$150$	0	0
$151$	−5.49253e18	−1.65375	−0.826873	−	0.562389i	$-0.809882\pi$
$151$	−5.49253e18	−1.65375	−0.826873	+	0.562389i	$0.809882\pi$
$152$	0	0
$153$	1.14312e18	0.307762
$154$	0	0
$155$	2.13935e18	0.515777
$156$	0	0
$157$	8.30662e18	1.79588	0.897940	−	0.440118i	$-0.145064\pi$
$157$	8.30662e18	1.79588	0.897940	+	0.440118i	$0.145064\pi$
$158$	0	0
$159$	−3.94124e18	−0.765169
$160$	0	0
$161$	−5.65117e17	−0.0986548
$162$	0	0
$163$	9.27088e18	1.45722	0.728612	−	0.684927i	$-0.240166\pi$
$163$	9.27088e18	1.45722	0.728612	+	0.684927i	$0.240166\pi$
$164$	0	0
$165$	1.00398e19	1.42270
$166$	0	0
$167$	−2.81210e18	−0.359700	−0.179850	−	0.983694i	$-0.557561\pi$
$167$	−2.81210e18	−0.359700	−0.179850	+	0.983694i	$0.557561\pi$
$168$	0	0
$169$	−2.01795e18	−0.233277
$170$	0	0
$171$	1.74611e18	0.182644
$172$	0	0
$173$	−1.12251e19	−1.06365	−0.531825	−	0.846854i	$-0.678494\pi$
$173$	−1.12251e19	−1.06365	−0.531825	+	0.846854i	$0.678494\pi$
$174$	0	0
$175$	2.56195e18	0.220164
$176$	0	0
$177$	−4.84509e18	−0.378032
$178$	0	0
$179$	−1.87799e19	−1.33181	−0.665906	−	0.746036i	$-0.731955\pi$
$179$	−1.87799e19	−1.33181	−0.665906	+	0.746036i	$0.731955\pi$
$180$	0	0
$181$	8.79232e18	0.567330	0.283665	−	0.958923i	$-0.408450\pi$
$181$	8.79232e18	0.567330	0.283665	+	0.958923i	$0.408450\pi$
$182$	0	0
$183$	−3.13759e19	−1.84400
$184$	0	0
$185$	−1.46231e19	−0.783574
$186$	0	0
$187$	5.47472e19	2.67737
$188$	0	0
$189$	−1.54008e19	−0.688049
$190$	0	0
$191$	2.67676e19	1.09352	0.546759	−	0.837290i	$-0.315861\pi$
$191$	2.67676e19	1.09352	0.546759	+	0.837290i	$0.315861\pi$
$192$	0	0
$193$	−9.79818e18	−0.366360	−0.183180	−	0.983079i	$-0.558639\pi$
$193$	−9.79818e18	−0.366360	−0.183180	+	0.983079i	$0.558639\pi$
$194$	0	0
$195$	−2.34559e19	−0.803448
$196$	0	0
$197$	−1.74519e19	−0.548125	−0.274062	−	0.961712i	$-0.588367\pi$
$197$	−1.74519e19	−0.548125	−0.274062	+	0.961712i	$0.588367\pi$
$198$	0	0
$199$	1.72318e19	0.496683	0.248341	−	0.968673i	$-0.420115\pi$
$199$	1.72318e19	0.496683	0.248341	+	0.968673i	$0.420115\pi$
$200$	0	0
$201$	−2.91220e18	−0.0771000
$202$	0	0
$203$	5.25851e19	1.27981
$204$	0	0
$205$	4.22515e19	0.946094
$206$	0	0
$207$	1.10559e18	0.0227953
$208$	0	0
$209$	8.36260e19	1.58890
$210$	0	0
$211$	−3.99012e19	−0.699174	−0.349587	−	0.936904i	$-0.613678\pi$
$211$	−3.99012e19	−0.699174	−0.349587	+	0.936904i	$0.613678\pi$
$212$	0	0
$213$	9.06640e18	0.146625
$214$	0	0
$215$	−8.69220e19	−1.29838
$216$	0	0
$217$	3.40878e19	0.470643
$218$	0	0
$219$	−9.50187e19	−1.21348
$220$	0	0
$221$	−1.27905e20	−1.51201
$222$	0	0
$223$	3.74752e19	0.410348	0.205174	−	0.978725i	$-0.434224\pi$
$223$	3.74752e19	0.410348	0.205174	+	0.978725i	$0.434224\pi$
$224$	0	0
$225$	−5.01215e18	−0.0508713
$226$	0	0
$227$	−1.31213e20	−1.23525	−0.617627	−	0.786471i	$-0.711906\pi$
$227$	−1.31213e20	−1.23525	−0.617627	+	0.786471i	$0.711906\pi$
$228$	0	0
$229$	2.72999e19	0.238539	0.119270	−	0.992862i	$-0.461945\pi$
$229$	2.72999e19	0.238539	0.119270	+	0.992862i	$0.461945\pi$
$230$	0	0
$231$	1.59972e20	1.29820
$232$	0	0
$233$	−3.83249e19	−0.289038	−0.144519	−	0.989502i	$-0.546164\pi$
$233$	−3.83249e19	−0.289038	−0.144519	+	0.989502i	$0.546164\pi$
$234$	0	0
$235$	−1.47157e20	−1.03206
$236$	0	0
$237$	1.01180e20	0.660286
$238$	0	0
$239$	2.81385e20	1.70970	0.854849	−	0.518878i	$-0.173650\pi$
$239$	2.81385e20	1.70970	0.854849	+	0.518878i	$0.173650\pi$
$240$	0	0
$241$	−4.17307e19	−0.236217	−0.118108	−	0.993001i	$-0.537683\pi$
$241$	−4.17307e19	−0.236217	−0.118108	+	0.993001i	$0.537683\pi$
$242$	0	0
$243$	6.67947e19	0.352443
$244$	0	0
$245$	6.95672e19	0.342368
$246$	0	0
$247$	−1.95374e20	−0.897312
$248$	0	0
$249$	3.08970e19	0.132502
$250$	0	0
$251$	1.65251e20	0.662089	0.331045	−	0.943615i	$-0.392599\pi$
$251$	1.65251e20	0.662089	0.331045	+	0.943615i	$0.392599\pi$
$252$	0	0
$253$	5.29496e19	0.198307
$254$	0	0
$255$	4.52340e20	1.58443
$256$	0	0
$257$	−3.37578e20	−1.10648	−0.553239	−	0.833023i	$-0.686608\pi$
$257$	−3.37578e20	−1.10648	−0.553239	+	0.833023i	$0.686608\pi$
$258$	0	0
$259$	−2.33001e20	−0.715007
$260$	0	0
$261$	−1.02877e20	−0.295715
$262$	0	0
$263$	3.92267e20	1.05672	0.528358	−	0.849022i	$-0.322808\pi$
$263$	3.92267e20	1.05672	0.528358	+	0.849022i	$0.322808\pi$
$264$	0	0
$265$	−2.35915e20	−0.595890
$266$	0	0
$267$	−2.46254e20	−0.583495
$268$	0	0
$269$	1.82892e20	0.406725	0.203362	−	0.979104i	$-0.434813\pi$
$269$	1.82892e20	0.406725	0.203362	+	0.979104i	$0.434813\pi$
$270$	0	0
$271$	−6.38843e20	−1.33400	−0.666998	−	0.745059i	$-0.732421\pi$
$271$	−6.38843e20	−1.33400	−0.666998	+	0.745059i	$0.732421\pi$
$272$	0	0
$273$	−3.73741e20	−0.733142
$274$	0	0
$275$	−2.40046e20	−0.442554
$276$	0	0
$277$	−7.07007e20	−1.22559	−0.612796	−	0.790241i	$-0.709955\pi$
$277$	−7.07007e20	−1.22559	−0.612796	+	0.790241i	$0.709955\pi$
$278$	0	0
$279$	−6.66889e19	−0.108747
$280$	0	0
$281$	7.43938e20	1.14165	0.570825	−	0.821072i	$-0.306623\pi$
$281$	7.43938e20	1.14165	0.570825	+	0.821072i	$0.306623\pi$
$282$	0	0
$283$	−4.12571e19	−0.0596093	−0.0298047	−	0.999556i	$-0.509489\pi$
$283$	−4.12571e19	−0.0596093	−0.0298047	+	0.999556i	$0.509489\pi$
$284$	0	0
$285$	6.90947e20	0.940293
$286$	0	0
$287$	6.73225e20	0.863305
$288$	0	0
$289$	1.63938e21	1.98174
$290$	0	0
$291$	−7.15501e20	−0.815680
$292$	0	0
$293$	1.92311e20	0.206837	0.103419	−	0.994638i	$-0.467022\pi$
$293$	1.92311e20	0.206837	0.103419	+	0.994638i	$0.467022\pi$
$294$	0	0
$295$	−2.90018e20	−0.294400
$296$	0	0
$297$	1.44301e21	1.38305
$298$	0	0
$299$	−1.23705e20	−0.111991
$300$	0	0
$301$	−1.38499e21	−1.18477
$302$	0	0
$303$	−8.23501e20	−0.665889
$304$	0	0
$305$	−1.87810e21	−1.43605
$306$	0	0
$307$	1.97369e21	1.42758	0.713792	−	0.700357i	$-0.246976\pi$
$307$	1.97369e21	1.42758	0.713792	+	0.700357i	$0.246976\pi$
$308$	0	0
$309$	1.48898e21	1.01916
$310$	0	0
$311$	1.95249e21	1.26510	0.632550	−	0.774520i	$-0.282008\pi$
$311$	1.95249e21	1.26510	0.632550	+	0.774520i	$0.282008\pi$
$312$	0	0
$313$	−1.94373e21	−1.19264	−0.596321	−	0.802746i	$-0.703371\pi$
$313$	−1.94373e21	−1.19264	−0.596321	+	0.802746i	$0.703371\pi$
$314$	0	0
$315$	1.99939e20	0.116214
$316$	0	0
$317$	−1.74953e21	−0.963646	−0.481823	−	0.876269i	$-0.660025\pi$
$317$	−1.74953e21	−0.963646	−0.481823	+	0.876269i	$0.660025\pi$
$318$	0	0
$319$	−4.92705e21	−2.57256
$320$	0	0
$321$	−2.64994e20	−0.131203
$322$	0	0
$323$	3.76774e21	1.76954
$324$	0	0
$325$	5.60816e20	0.249926
$326$	0	0
$327$	−1.36746e21	−0.578440
$328$	0	0
$329$	−2.34477e21	−0.941749
$330$	0	0
$331$	−1.64785e21	−0.628610	−0.314305	−	0.949322i	$-0.601771\pi$
$331$	−1.64785e21	−0.628610	−0.314305	+	0.949322i	$0.601771\pi$
$332$	0	0
$333$	4.55840e20	0.165210
$334$	0	0
$335$	−1.74319e20	−0.0600431
$336$	0	0
$337$	3.40530e21	1.11507	0.557533	−	0.830154i	$-0.311748\pi$
$337$	3.40530e21	1.11507	0.557533	+	0.830154i	$0.311748\pi$
$338$	0	0
$339$	2.99730e21	0.933324
$340$	0	0
$341$	−3.19392e21	−0.946044
$342$	0	0
$343$	3.84533e21	1.08376
$344$	0	0
$345$	4.37487e20	0.117355
$346$	0	0
$347$	5.65630e21	1.44455	0.722273	−	0.691608i	$-0.243097\pi$
$347$	5.65630e21	1.44455	0.722273	+	0.691608i	$0.243097\pi$
$348$	0	0
$349$	6.06990e21	1.47627	0.738135	−	0.674653i	$-0.235707\pi$
$349$	6.06990e21	1.47627	0.738135	+	0.674653i	$0.235707\pi$
$350$	0	0
$351$	−3.37128e21	−0.781060
$352$	0	0
$353$	−3.68173e21	−0.812769	−0.406385	−	0.913702i	$-0.633211\pi$
$353$	−3.68173e21	−0.812769	−0.406385	+	0.913702i	$0.633211\pi$
$354$	0	0
$355$	5.42698e20	0.114187
$356$	0	0
$357$	7.20748e21	1.44578
$358$	0	0
$359$	−5.93217e21	−1.13478	−0.567389	−	0.823450i	$-0.692046\pi$
$359$	−5.93217e21	−1.13478	−0.567389	+	0.823450i	$0.692046\pi$
$360$	0	0
$361$	2.74808e20	0.0501439
$362$	0	0
$363$	−8.75406e21	−1.52406
$364$	0	0
$365$	−5.68764e21	−0.945024
$366$	0	0
$367$	7.23976e20	0.114832	0.0574160	−	0.998350i	$-0.481714\pi$
$367$	7.23976e20	0.114832	0.0574160	+	0.998350i	$0.481714\pi$
$368$	0	0
$369$	−1.31709e21	−0.199476
$370$	0	0
$371$	−3.75902e21	−0.543746
$372$	0	0
$373$	4.37552e21	0.604651	0.302325	−	0.953205i	$-0.402237\pi$
$373$	4.37552e21	0.604651	0.302325	+	0.953205i	$0.402237\pi$
$374$	0	0
$375$	−8.93206e21	−1.17947
$376$	0	0
$377$	1.15110e22	1.45282
$378$	0	0
$379$	2.01932e21	0.243653	0.121827	−	0.992551i	$-0.461125\pi$
$379$	2.01932e21	0.243653	0.121827	+	0.992551i	$0.461125\pi$
$380$	0	0
$381$	−1.49668e22	−1.72690
$382$	0	0
$383$	4.16170e21	0.459284	0.229642	−	0.973275i	$-0.426245\pi$
$383$	4.16170e21	0.459284	0.229642	+	0.973275i	$0.426245\pi$
$384$	0	0
$385$	9.57562e21	1.01100
$386$	0	0
$387$	2.70958e21	0.273753
$388$	0	0
$389$	−1.10978e22	−1.07316	−0.536581	−	0.843849i	$-0.680284\pi$
$389$	−1.10978e22	−1.07316	−0.536581	+	0.843849i	$0.680284\pi$
$390$	0	0
$391$	2.38562e21	0.220851
$392$	0	0
$393$	−1.02161e22	−0.905626
$394$	0	0
$395$	6.05642e21	0.514210
$396$	0	0
$397$	3.83409e21	0.311848	0.155924	−	0.987769i	$-0.450164\pi$
$397$	3.83409e21	0.311848	0.155924	+	0.987769i	$0.450164\pi$
$398$	0	0
$399$	1.10094e22	0.858011
$400$	0	0
$401$	−1.05377e22	−0.787080	−0.393540	−	0.919308i	$-0.628750\pi$
$401$	−1.05377e22	−0.787080	−0.393540	+	0.919308i	$0.628750\pi$
$402$	0	0
$403$	7.46191e21	0.534266
$404$	0	0
$405$	1.41173e22	0.969135
$406$	0	0
$407$	2.18314e22	1.43724
$408$	0	0
$409$	2.17591e22	1.37402	0.687009	−	0.726649i	$-0.258923\pi$
$409$	2.17591e22	1.37402	0.687009	+	0.726649i	$0.258923\pi$
$410$	0	0
$411$	−8.41203e20	−0.0509618
$412$	0	0
$413$	−4.62108e21	−0.268638
$414$	0	0
$415$	1.84944e21	0.103188
$416$	0	0
$417$	−2.25981e22	−1.21036
$418$	0	0
$419$	−1.97426e22	−1.01528	−0.507638	−	0.861570i	$-0.669481\pi$
$419$	−1.97426e22	−1.01528	−0.507638	+	0.861570i	$0.669481\pi$
$420$	0	0
$421$	−1.77119e22	−0.874716	−0.437358	−	0.899288i	$-0.644086\pi$
$421$	−1.77119e22	−0.874716	−0.437358	+	0.899288i	$0.644086\pi$
$422$	0	0
$423$	4.58728e21	0.217601
$424$	0	0
$425$	−1.08152e22	−0.492865
$426$	0	0
$427$	−2.99252e22	−1.31039
$428$	0	0
$429$	3.50183e22	1.47369
$430$	0	0
$431$	−8.15558e21	−0.329912	−0.164956	−	0.986301i	$-0.552748\pi$
$431$	−8.15558e21	−0.329912	−0.164956	+	0.986301i	$0.552748\pi$
$432$	0	0
$433$	−1.37369e22	−0.534245	−0.267123	−	0.963663i	$-0.586073\pi$
$433$	−1.37369e22	−0.534245	−0.267123	+	0.963663i	$0.586073\pi$
$434$	0	0
$435$	−4.07090e22	−1.52241
$436$	0	0
$437$	3.64402e21	0.131066
$438$	0	0
$439$	2.08615e22	0.721769	0.360884	−	0.932611i	$-0.382475\pi$
$439$	2.08615e22	0.721769	0.360884	+	0.932611i	$0.382475\pi$
$440$	0	0
$441$	−2.16859e21	−0.0721854
$442$	0	0
$443$	1.14158e22	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$443$	1.14158e22	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$444$	0	0
$445$	−1.47403e22	−0.454408
$446$	0	0
$447$	4.59346e22	1.36309
$448$	0	0
$449$	5.18457e21	0.148122	0.0740609	−	0.997254i	$-0.476404\pi$
$449$	5.18457e21	0.148122	0.0740609	+	0.997254i	$0.476404\pi$
$450$	0	0
$451$	−6.30790e22	−1.73534
$452$	0	0
$453$	6.77514e22	1.79508
$454$	0	0
$455$	−2.23714e22	−0.570948
$456$	0	0
$457$	4.30871e22	1.05940	0.529699	−	0.848185i	$-0.322305\pi$
$457$	4.30871e22	1.05940	0.529699	+	0.848185i	$0.322305\pi$
$458$	0	0
$459$	6.50142e22	1.54028
$460$	0	0
$461$	−3.34197e22	−0.763036	−0.381518	−	0.924361i	$-0.624599\pi$
$461$	−3.34197e22	−0.763036	−0.381518	+	0.924361i	$0.624599\pi$
$462$	0	0
$463$	3.60638e21	0.0793656	0.0396828	−	0.999212i	$-0.487365\pi$
$463$	3.60638e21	0.0793656	0.0396828	+	0.999212i	$0.487365\pi$
$464$	0	0
$465$	−2.63892e22	−0.559857
$466$	0	0
$467$	1.89207e22	0.387028	0.193514	−	0.981097i	$-0.438011\pi$
$467$	1.89207e22	0.387028	0.193514	+	0.981097i	$0.438011\pi$
$468$	0	0
$469$	−2.77755e21	−0.0547889
$470$	0	0
$471$	−1.02464e23	−1.94936
$472$	0	0
$473$	1.29769e23	2.38151
$474$	0	0
$475$	−1.65201e22	−0.292494
$476$	0	0
$477$	7.35409e21	0.125638
$478$	0	0
$479$	9.87835e22	1.62867	0.814334	−	0.580396i	$-0.197102\pi$
$479$	9.87835e22	1.62867	0.814334	+	0.580396i	$0.197102\pi$
$480$	0	0
$481$	−5.10045e22	−0.811663
$482$	0	0
$483$	6.97082e21	0.107086
$484$	0	0
$485$	−4.28285e22	−0.635226
$486$	0	0
$487$	4.66690e22	0.668394	0.334197	−	0.942503i	$-0.391535\pi$
$487$	4.66690e22	0.668394	0.334197	+	0.942503i	$0.391535\pi$
$488$	0	0
$489$	−1.14358e23	−1.58176
$490$	0	0
$491$	6.30360e21	0.0842162	0.0421081	−	0.999113i	$-0.486593\pi$
$491$	6.30360e21	0.0842162	0.0421081	+	0.999113i	$0.486593\pi$
$492$	0	0
$493$	−2.21986e23	−2.86502
$494$	0	0
$495$	−1.87336e22	−0.233602
$496$	0	0
$497$	8.64722e21	0.104195
$498$	0	0
$499$	5.57436e22	0.649143	0.324571	−	0.945861i	$-0.394780\pi$
$499$	5.57436e22	0.649143	0.324571	+	0.945861i	$0.394780\pi$
$500$	0	0
$501$	3.46878e22	0.390441
$502$	0	0
$503$	−9.22523e22	−1.00380	−0.501902	−	0.864924i	$-0.667366\pi$
$503$	−9.22523e22	−1.00380	−0.501902	+	0.864924i	$0.667366\pi$
$504$	0	0
$505$	−4.92932e22	−0.518574
$506$	0	0
$507$	2.48917e22	0.253214
$508$	0	0
$509$	5.10776e22	0.502493	0.251246	−	0.967923i	$-0.419160\pi$
$509$	5.10776e22	0.502493	0.251246	+	0.967923i	$0.419160\pi$
$510$	0	0
$511$	−9.06255e22	−0.862329
$512$	0	0
$513$	9.93087e22	0.914092
$514$	0	0
$515$	8.91273e22	0.793687
$516$	0	0
$517$	2.19697e23	1.89302
$518$	0	0
$519$	1.38464e23	1.15455
$520$	0	0
$521$	6.17836e22	0.498600	0.249300	−	0.968426i	$-0.419799\pi$
$521$	6.17836e22	0.498600	0.249300	+	0.968426i	$0.419799\pi$
$522$	0	0
$523$	−1.51385e21	−0.0118255	−0.00591274	−	0.999983i	$-0.501882\pi$
$523$	−1.51385e21	−0.0118255	−0.00591274	+	0.999983i	$0.501882\pi$
$524$	0	0
$525$	−3.16021e22	−0.238980
$526$	0	0
$527$	−1.43901e23	−1.05359
$528$	0	0
$529$	−1.38743e23	−0.983642
$530$	0	0
$531$	9.04061e21	0.0620718
$532$	0	0
$533$	1.47371e23	0.980008
$534$	0	0
$535$	−1.58621e22	−0.102177
$536$	0	0
$537$	2.31654e23	1.44563
$538$	0	0
$539$	−1.03860e23	−0.627975
$540$	0	0
$541$	−1.82889e23	−1.07154	−0.535772	−	0.844363i	$-0.679979\pi$
$541$	−1.82889e23	−1.07154	−0.535772	+	0.844363i	$0.679979\pi$
$542$	0	0
$543$	−1.08455e23	−0.615816
$544$	0	0
$545$	−8.18536e22	−0.450471
$546$	0	0
$547$	3.33097e23	1.77696	0.888481	−	0.458913i	$-0.151761\pi$
$547$	3.33097e23	1.77696	0.888481	+	0.458913i	$0.151761\pi$
$548$	0	0
$549$	5.85453e22	0.302780
$550$	0	0
$551$	−3.39083e23	−1.70027
$552$	0	0
$553$	9.65016e22	0.469214
$554$	0	0
$555$	1.80379e23	0.850541
$556$	0	0
$557$	1.61270e23	0.737539	0.368769	−	0.929521i	$-0.379779\pi$
$557$	1.61270e23	0.737539	0.368769	+	0.929521i	$0.379779\pi$
$558$	0	0
$559$	−3.03179e23	−1.34492
$560$	0	0
$561$	−6.75317e23	−2.90619
$562$	0	0
$563$	−2.08376e23	−0.870014	−0.435007	−	0.900427i	$-0.643254\pi$
$563$	−2.08376e23	−0.870014	−0.435007	+	0.900427i	$0.643254\pi$
$564$	0	0
$565$	1.79413e23	0.726844
$566$	0	0
$567$	2.24942e23	0.884329
$568$	0	0
$569$	1.08748e23	0.414924	0.207462	−	0.978243i	$-0.433480\pi$
$569$	1.08748e23	0.414924	0.207462	+	0.978243i	$0.433480\pi$
$570$	0	0
$571$	−1.01838e23	−0.377141	−0.188570	−	0.982060i	$-0.560385\pi$
$571$	−1.01838e23	−0.377141	−0.188570	+	0.982060i	$0.560385\pi$
$572$	0	0
$573$	−3.30183e23	−1.18697
$574$	0	0
$575$	−1.04601e22	−0.0365054
$576$	0	0
$577$	1.78939e23	0.606333	0.303166	−	0.952938i	$-0.401956\pi$
$577$	1.78939e23	0.606333	0.303166	+	0.952938i	$0.401956\pi$
$578$	0	0
$579$	1.20862e23	0.397670
$580$	0	0
$581$	2.94685e22	0.0941589
$582$	0	0
$583$	3.52207e23	1.09299
$584$	0	0
$585$	4.37671e22	0.131924
$586$	0	0
$587$	−5.83709e22	−0.170912	−0.0854560	−	0.996342i	$-0.527235\pi$
$587$	−5.83709e22	−0.170912	−0.0854560	+	0.996342i	$0.527235\pi$
$588$	0	0
$589$	−2.19807e23	−0.625263
$590$	0	0
$591$	2.15272e23	0.594969
$592$	0	0
$593$	2.99461e23	0.804220	0.402110	−	0.915591i	$-0.368277\pi$
$593$	2.99461e23	0.804220	0.402110	+	0.915591i	$0.368277\pi$
$594$	0	0
$595$	4.31426e23	1.12593
$596$	0	0
$597$	−2.12557e23	−0.539131
$598$	0	0
$599$	5.08779e23	1.25430	0.627149	−	0.778899i	$-0.284222\pi$
$599$	5.08779e23	1.25430	0.627149	+	0.778899i	$0.284222\pi$
$600$	0	0
$601$	7.61602e23	1.82514	0.912568	−	0.408925i	$-0.134096\pi$
$601$	7.61602e23	1.82514	0.912568	+	0.408925i	$0.134096\pi$
$602$	0	0
$603$	5.43396e21	0.0126596
$604$	0	0
$605$	−5.24002e23	−1.18689
$606$	0	0
$607$	3.29833e23	0.726424	0.363212	−	0.931707i	$-0.381680\pi$
$607$	3.29833e23	0.726424	0.363212	+	0.931707i	$0.381680\pi$
$608$	0	0
$609$	−6.48647e23	−1.38919
$610$	0	0
$611$	−5.13276e23	−1.06906
$612$	0	0
$613$	−1.14510e23	−0.231968	−0.115984	−	0.993251i	$-0.537002\pi$
$613$	−1.14510e23	−0.231968	−0.115984	+	0.993251i	$0.537002\pi$
$614$	0	0
$615$	−5.21180e23	−1.02695
$616$	0	0
$617$	1.25024e22	0.0239646	0.0119823	−	0.999928i	$-0.496186\pi$
$617$	1.25024e22	0.0239646	0.0119823	+	0.999928i	$0.496186\pi$
$618$	0	0
$619$	−9.20447e22	−0.171644	−0.0858219	−	0.996310i	$-0.527352\pi$
$619$	−9.20447e22	−0.171644	−0.0858219	+	0.996310i	$0.527352\pi$
$620$	0	0
$621$	6.28794e22	0.114085
$622$	0	0
$623$	−2.34868e23	−0.414644
$624$	0	0
$625$	−3.68517e23	−0.633107
$626$	0	0
$627$	−1.03154e24	−1.72470
$628$	0	0
$629$	9.83608e23	1.60063
$630$	0	0
$631$	6.82055e23	1.08036	0.540181	−	0.841549i	$-0.318356\pi$
$631$	6.82055e23	1.08036	0.540181	+	0.841549i	$0.318356\pi$
$632$	0	0
$633$	4.92189e23	0.758927
$634$	0	0
$635$	−8.95885e23	−1.34486
$636$	0	0
$637$	2.42646e23	0.354641
$638$	0	0
$639$	−1.69173e22	−0.0240754
$640$	0	0
$641$	−9.47755e23	−1.31342	−0.656709	−	0.754144i	$-0.728052\pi$
$641$	−9.47755e23	−1.31342	−0.656709	+	0.754144i	$0.728052\pi$
$642$	0	0
$643$	2.06099e23	0.278153	0.139076	−	0.990282i	$-0.455587\pi$
$643$	2.06099e23	0.278153	0.139076	+	0.990282i	$0.455587\pi$
$644$	0	0
$645$	1.07220e24	1.40935
$646$	0	0
$647$	−7.16276e23	−0.917052	−0.458526	−	0.888681i	$-0.651622\pi$
$647$	−7.16276e23	−0.917052	−0.458526	+	0.888681i	$0.651622\pi$
$648$	0	0
$649$	4.32980e23	0.539992
$650$	0	0
$651$	−4.20480e23	−0.510866
$652$	0	0
$653$	−8.48252e23	−1.00407	−0.502034	−	0.864848i	$-0.667415\pi$
$653$	−8.48252e23	−1.00407	−0.502034	+	0.864848i	$0.667415\pi$
$654$	0	0
$655$	−6.11517e23	−0.705274
$656$	0	0
$657$	1.77298e23	0.199250
$658$	0	0
$659$	1.22073e24	1.33688	0.668442	−	0.743764i	$-0.266961\pi$
$659$	1.22073e24	1.33688	0.668442	+	0.743764i	$0.266961\pi$
$660$	0	0
$661$	2.43310e23	0.259686	0.129843	−	0.991535i	$-0.458553\pi$
$661$	2.43310e23	0.259686	0.129843	+	0.991535i	$0.458553\pi$
$662$	0	0
$663$	1.57774e24	1.64123
$664$	0	0
$665$	6.59001e23	0.668193
$666$	0	0
$667$	−2.14697e23	−0.212206
$668$	0	0
$669$	−4.62264e23	−0.445418
$670$	0	0
$671$	2.80389e24	2.63403
$672$	0	0
$673$	−1.72736e24	−1.58217	−0.791086	−	0.611704i	$-0.790484\pi$
$673$	−1.72736e24	−1.58217	−0.791086	+	0.611704i	$0.790484\pi$
$674$	0	0
$675$	−2.85063e23	−0.254600
$676$	0	0
$677$	9.08667e23	0.791409	0.395704	−	0.918378i	$-0.370500\pi$
$677$	9.08667e23	0.791409	0.395704	+	0.918378i	$0.370500\pi$
$678$	0	0
$679$	−6.82420e23	−0.579640
$680$	0	0
$681$	1.61853e24	1.34082
$682$	0	0
$683$	1.57751e22	0.0127467	0.00637333	−	0.999980i	$-0.497971\pi$
$683$	1.57751e22	0.0127467	0.00637333	+	0.999980i	$0.497971\pi$
$684$	0	0
$685$	−5.03528e22	−0.0396875
$686$	0	0
$687$	−3.36750e23	−0.258925
$688$	0	0
$689$	−8.22858e23	−0.617251
$690$	0	0
$691$	−1.54734e24	−1.13246	−0.566228	−	0.824248i	$-0.691598\pi$
$691$	−1.54734e24	−1.13246	−0.566228	+	0.824248i	$0.691598\pi$
$692$	0	0
$693$	−2.98497e23	−0.213161
$694$	0	0
$695$	−1.35268e24	−0.942593
$696$	0	0
$697$	−2.84200e24	−1.93262
$698$	0	0
$699$	4.72745e23	0.313741
$700$	0	0
$701$	−1.78409e24	−1.15561	−0.577807	−	0.816174i	$-0.696091\pi$
$701$	−1.78409e24	−1.15561	−0.577807	+	0.816174i	$0.696091\pi$
$702$	0	0
$703$	1.50245e24	0.949907
$704$	0	0
$705$	1.81521e24	1.12026
$706$	0	0
$707$	−7.85427e23	−0.473195
$708$	0	0
$709$	−1.62194e23	−0.0953985	−0.0476993	−	0.998862i	$-0.515189\pi$
$709$	−1.62194e23	−0.0953985	−0.0476993	+	0.998862i	$0.515189\pi$
$710$	0	0
$711$	−1.88794e23	−0.108417
$712$	0	0
$713$	−1.39176e23	−0.0780373
$714$	0	0
$715$	2.09613e24	1.14767
$716$	0	0
$717$	−3.47094e24	−1.85581
$718$	0	0
$719$	−1.20674e22	−0.00630110	−0.00315055	−	0.999995i	$-0.501003\pi$
$719$	−1.20674e22	−0.00630110	−0.00315055	+	0.999995i	$0.501003\pi$
$720$	0	0
$721$	1.42013e24	0.724234
$722$	0	0
$723$	5.14755e23	0.256404
$724$	0	0
$725$	9.73327e23	0.473571
$726$	0	0
$727$	−3.36296e24	−1.59838	−0.799188	−	0.601081i	$-0.794737\pi$
$727$	−3.36296e24	−1.59838	−0.799188	+	0.601081i	$0.794737\pi$
$728$	0	0
$729$	1.64521e24	0.763900
$730$	0	0
$731$	5.84671e24	2.65225
$732$	0	0
$733$	1.77549e24	0.786927	0.393463	−	0.919340i	$-0.371277\pi$
$733$	1.77549e24	0.786927	0.393463	+	0.919340i	$0.371277\pi$
$734$	0	0
$735$	−8.58125e23	−0.371628
$736$	0	0
$737$	2.60247e23	0.110132
$738$	0	0
$739$	−4.41138e24	−1.82430	−0.912152	−	0.409853i	$-0.865580\pi$
$739$	−4.41138e24	−1.82430	−0.912152	+	0.409853i	$0.865580\pi$
$740$	0	0
$741$	2.40998e24	0.973999
$742$	0	0
$743$	−3.23478e23	−0.127773	−0.0638867	−	0.997957i	$-0.520350\pi$
$743$	−3.23478e23	−0.127773	−0.0638867	+	0.997957i	$0.520350\pi$
$744$	0	0
$745$	2.74956e24	1.06154
$746$	0	0
$747$	−5.76518e22	−0.0217564
$748$	0	0
$749$	−2.52742e23	−0.0932358
$750$	0	0
$751$	−3.72468e24	−1.34323	−0.671614	−	0.740902i	$-0.734398\pi$
$751$	−3.72468e24	−1.34323	−0.671614	+	0.740902i	$0.734398\pi$
$752$	0	0
$753$	−2.03840e24	−0.718674
$754$	0	0
$755$	4.05547e24	1.39795
$756$	0	0
$757$	3.95820e24	1.33408	0.667042	−	0.745020i	$-0.267560\pi$
$757$	3.95820e24	1.33408	0.667042	+	0.745020i	$0.267560\pi$
$758$	0	0
$759$	−6.53143e23	−0.215255
$760$	0	0
$761$	1.77221e24	0.571144	0.285572	−	0.958357i	$-0.407816\pi$
$761$	1.77221e24	0.571144	0.285572	+	0.958357i	$0.407816\pi$
$762$	0	0
$763$	−1.30423e24	−0.411052
$764$	0	0
$765$	−8.44036e23	−0.260159
$766$	0	0
$767$	−1.01156e24	−0.304953
$768$	0	0
$769$	−3.15665e24	−0.930793	−0.465396	−	0.885102i	$-0.654088\pi$
$769$	−3.15665e24	−0.930793	−0.465396	+	0.885102i	$0.654088\pi$
$770$	0	0
$771$	4.16409e24	1.20104
$772$	0	0
$773$	−2.46595e24	−0.695759	−0.347880	−	0.937539i	$-0.613098\pi$
$773$	−2.46595e24	−0.695759	−0.347880	+	0.937539i	$0.613098\pi$
$774$	0	0
$775$	6.30950e23	0.174153
$776$	0	0
$777$	2.87411e24	0.776113
$778$	0	0
$779$	−4.34113e24	−1.14692
$780$	0	0
$781$	−8.10216e23	−0.209444
$782$	0	0
$783$	−5.85104e24	−1.47999
$784$	0	0
$785$	−6.13328e24	−1.51810
$786$	0	0
$787$	−3.37302e24	−0.817022	−0.408511	−	0.912753i	$-0.633952\pi$
$787$	−3.37302e24	−0.817022	−0.408511	+	0.912753i	$0.633952\pi$
$788$	0	0
$789$	−4.83869e24	−1.14703
$790$	0	0
$791$	2.85872e24	0.663240
$792$	0	0
$793$	−6.55070e24	−1.48753
$794$	0	0
$795$	2.91006e24	0.646817
$796$	0	0
$797$	4.09256e24	0.890428	0.445214	−	0.895424i	$-0.353128\pi$
$797$	4.09256e24	0.890428	0.445214	+	0.895424i	$0.353128\pi$
$798$	0	0
$799$	9.89838e24	2.10822
$800$	0	0
$801$	4.59492e23	0.0958080
$802$	0	0
$803$	8.49131e24	1.73338
$804$	0	0
$805$	4.17260e23	0.0833954
$806$	0	0
$807$	−2.25601e24	−0.441484
$808$	0	0
$809$	−4.90388e24	−0.939674	−0.469837	−	0.882753i	$-0.655687\pi$
$809$	−4.90388e24	−0.939674	−0.469837	+	0.882753i	$0.655687\pi$
$810$	0	0
$811$	2.78957e24	0.523431	0.261716	−	0.965145i	$-0.415712\pi$
$811$	2.78957e24	0.523431	0.261716	+	0.965145i	$0.415712\pi$
$812$	0	0
$813$	7.88024e24	1.44800
$814$	0	0
$815$	−6.84525e24	−1.23183
$816$	0	0
$817$	8.93081e24	1.57399
$818$	0	0
$819$	6.97375e23	0.120380
$820$	0	0
$821$	8.98127e23	0.151852	0.0759260	−	0.997113i	$-0.475809\pi$
$821$	8.98127e23	0.151852	0.0759260	+	0.997113i	$0.475809\pi$
$822$	0	0
$823$	3.92273e24	0.649666	0.324833	−	0.945771i	$-0.394692\pi$
$823$	3.92273e24	0.649666	0.324833	+	0.945771i	$0.394692\pi$
$824$	0	0
$825$	2.96101e24	0.480376
$826$	0	0
$827$	−7.96954e24	−1.26659	−0.633295	−	0.773910i	$-0.718298\pi$
$827$	−7.96954e24	−1.26659	−0.633295	+	0.773910i	$0.718298\pi$
$828$	0	0
$829$	−7.37502e24	−1.14829	−0.574143	−	0.818755i	$-0.694665\pi$
$829$	−7.37502e24	−1.14829	−0.574143	+	0.818755i	$0.694665\pi$
$830$	0	0
$831$	8.72106e24	1.33033
$832$	0	0
$833$	−4.67936e24	−0.699365
$834$	0	0
$835$	2.07635e24	0.304064
$836$	0	0
$837$	−3.79289e24	−0.544256
$838$	0	0
$839$	5.27056e23	0.0741106	0.0370553	−	0.999313i	$-0.488202\pi$
$839$	5.27056e23	0.0741106	0.0370553	+	0.999313i	$0.488202\pi$
$840$	0	0
$841$	1.27208e25	1.75287
$842$	0	0
$843$	−9.17661e24	−1.23922
$844$	0	0
$845$	1.48997e24	0.197195
$846$	0	0
$847$	−8.34932e24	−1.08303
$848$	0	0
$849$	5.08914e23	0.0647037
$850$	0	0
$851$	9.51310e23	0.118555
$852$	0	0
$853$	−1.48605e24	−0.181538	−0.0907688	−	0.995872i	$-0.528932\pi$
$853$	−1.48605e24	−0.181538	−0.0907688	+	0.995872i	$0.528932\pi$
$854$	0	0
$855$	−1.28926e24	−0.154393
$856$	0	0
$857$	−4.94768e24	−0.580851	−0.290426	−	0.956898i	$-0.593797\pi$
$857$	−4.94768e24	−0.580851	−0.290426	+	0.956898i	$0.593797\pi$
$858$	0	0
$859$	9.89526e24	1.13890	0.569450	−	0.822026i	$-0.307156\pi$
$859$	9.89526e24	1.13890	0.569450	+	0.822026i	$0.307156\pi$
$860$	0	0
$861$	−8.30435e24	−0.937085
$862$	0	0
$863$	2.12084e24	0.234648	0.117324	−	0.993094i	$-0.462568\pi$
$863$	2.12084e24	0.234648	0.117324	+	0.993094i	$0.462568\pi$
$864$	0	0
$865$	8.28819e24	0.899130
$866$	0	0
$867$	−2.02220e25	−2.15111
$868$	0	0
$869$	−9.04188e24	−0.943171
$870$	0	0
$871$	−6.08013e23	−0.0621954
$872$	0	0
$873$	1.33508e24	0.133932
$874$	0	0
$875$	−8.51908e24	−0.838155
$876$	0	0
$877$	8.14461e24	0.785912	0.392956	−	0.919557i	$-0.371453\pi$
$877$	8.14461e24	0.785912	0.392956	+	0.919557i	$0.371453\pi$
$878$	0	0
$879$	−2.37219e24	−0.224514
$880$	0	0
$881$	8.56282e24	0.794917	0.397458	−	0.917620i	$-0.369892\pi$
$881$	8.56282e24	0.794917	0.397458	+	0.917620i	$0.369892\pi$
$882$	0	0
$883$	−7.54039e24	−0.686638	−0.343319	−	0.939219i	$-0.611551\pi$
$883$	−7.54039e24	−0.686638	−0.343319	+	0.939219i	$0.611551\pi$
$884$	0	0
$885$	3.57743e24	0.319560
$886$	0	0
$887$	1.35804e25	1.19004	0.595020	−	0.803711i	$-0.297144\pi$
$887$	1.35804e25	1.19004	0.595020	+	0.803711i	$0.297144\pi$
$888$	0	0
$889$	−1.42748e25	−1.22717
$890$	0	0
$891$	−2.10763e25	−1.77760
$892$	0	0
$893$	1.51197e25	1.25114
$894$	0	0
$895$	1.38664e25	1.12581
$896$	0	0
$897$	1.52593e24	0.121562
$898$	0	0
$899$	1.29505e25	1.01235
$900$	0	0
$901$	1.58686e25	1.21724
$902$	0	0
$903$	1.70842e25	1.28602
$904$	0	0
$905$	−6.49191e24	−0.479578
$906$	0	0
$907$	−1.22769e24	−0.0890076	−0.0445038	−	0.999009i	$-0.514171\pi$
$907$	−1.22769e24	−0.0890076	−0.0445038	+	0.999009i	$0.514171\pi$
$908$	0	0
$909$	1.53660e24	0.109337
$910$	0	0
$911$	−5.89739e24	−0.411864	−0.205932	−	0.978566i	$-0.566023\pi$
$911$	−5.89739e24	−0.411864	−0.205932	+	0.978566i	$0.566023\pi$
$912$	0	0
$913$	−2.76110e24	−0.189270
$914$	0	0
$915$	2.31667e25	1.55878
$916$	0	0
$917$	−9.74377e24	−0.643558
$918$	0	0
$919$	2.92359e25	1.89555	0.947775	−	0.318941i	$-0.103327\pi$
$919$	2.92359e25	1.89555	0.947775	+	0.318941i	$0.103327\pi$
$920$	0	0
$921$	−2.43458e25	−1.54959
$922$	0	0
$923$	1.89290e24	0.118280
$924$	0	0
$925$	−4.31275e24	−0.264575
$926$	0	0
$927$	−2.77833e24	−0.167342
$928$	0	0
$929$	1.37120e25	0.810900	0.405450	−	0.914117i	$-0.367115\pi$
$929$	1.37120e25	0.810900	0.405450	+	0.914117i	$0.367115\pi$
$930$	0	0
$931$	−7.14769e24	−0.415043
$932$	0	0
$933$	−2.40843e25	−1.37322
$934$	0	0
$935$	−4.04232e25	−2.26325
$936$	0	0
$937$	−2.37505e25	−1.30583	−0.652915	−	0.757431i	$-0.726454\pi$
$937$	−2.37505e25	−1.30583	−0.652915	+	0.757431i	$0.726454\pi$
$938$	0	0
$939$	2.39763e25	1.29457
$940$	0	0
$941$	−8.71956e24	−0.462363	−0.231181	−	0.972911i	$-0.574259\pi$
$941$	−8.71956e24	−0.462363	−0.231181	+	0.972911i	$0.574259\pi$
$942$	0	0
$943$	−2.74868e24	−0.143145
$944$	0	0
$945$	1.13714e25	0.581625
$946$	0	0
$947$	−2.81777e25	−1.41557	−0.707785	−	0.706428i	$-0.750305\pi$
$947$	−2.81777e25	−1.41557	−0.707785	+	0.706428i	$0.750305\pi$
$948$	0	0
$949$	−1.98382e25	−0.978900
$950$	0	0
$951$	2.15808e25	1.04600
$952$	0	0
$953$	3.23840e25	1.54185	0.770923	−	0.636928i	$-0.219795\pi$
$953$	3.23840e25	1.54185	0.770923	+	0.636928i	$0.219795\pi$
$954$	0	0
$955$	−1.97642e25	−0.924378
$956$	0	0
$957$	6.07761e25	2.79242
$958$	0	0
$959$	−8.02309e23	−0.0362146
$960$	0	0
$961$	−1.41550e25	−0.627715
$962$	0	0
$963$	4.94461e23	0.0215431
$964$	0	0
$965$	7.23459e24	0.309693
$966$	0	0
$967$	−4.38141e25	−1.84285	−0.921423	−	0.388562i	$-0.872972\pi$
$967$	−4.38141e25	−1.84285	−0.921423	+	0.388562i	$0.872972\pi$
$968$	0	0
$969$	−4.64758e25	−1.92077
$970$	0	0
$971$	2.53693e25	1.03026	0.515129	−	0.857113i	$-0.327744\pi$
$971$	2.53693e25	1.03026	0.515129	+	0.857113i	$0.327744\pi$
$972$	0	0
$973$	−2.15533e25	−0.860110
$974$	0	0
$975$	−6.91777e24	−0.271286
$976$	0	0
$977$	3.84698e25	1.48257	0.741286	−	0.671189i	$-0.234216\pi$
$977$	3.84698e25	1.48257	0.741286	+	0.671189i	$0.234216\pi$
$978$	0	0
$979$	2.20064e25	0.833480
$980$	0	0
$981$	2.55159e24	0.0949781
$982$	0	0
$983$	−2.49403e25	−0.912423	−0.456211	−	0.889871i	$-0.650794\pi$
$983$	−2.49403e25	−0.912423	−0.456211	+	0.889871i	$0.650794\pi$
$984$	0	0
$985$	1.28858e25	0.463344
$986$	0	0
$987$	2.89232e25	1.02223
$988$	0	0
$989$	5.65473e24	0.196446
$990$	0	0
$991$	−9.79749e24	−0.334571	−0.167286	−	0.985908i	$-0.553500\pi$
$991$	−9.79749e24	−0.334571	−0.167286	+	0.985908i	$0.553500\pi$
$992$	0	0
$993$	2.03266e25	0.682332
$994$	0	0
$995$	−1.27233e25	−0.419858
$996$	0	0
$997$	1.87197e25	0.607282	0.303641	−	0.952787i	$-0.401798\pi$
$997$	1.87197e25	0.607282	0.303641	+	0.952787i	$0.401798\pi$
$998$	0	0
$999$	2.59256e25	0.826841

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.18.a.a.1.1	✓	2	1.1	even	1	trivial
16.18.a.d.1.2		2	4.3	odd	2
64.18.a.h.1.1		2	8.3	odd	2
64.18.a.k.1.2		2	8.5	even	2
72.18.a.c.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)

\(f(q)\)	\(=\)	\(q-12335.2 q^{3} -738361. q^{5} -1.17649e7 q^{7} +2.30166e7 q^{9} +O(q^{10})\)	\(q-12335.2 q^{3} -738361. q^{5} -1.17649e7 q^{7} +2.30166e7 q^{9} +1.10233e9 q^{11} -2.57536e9 q^{13} +9.10782e9 q^{15} +4.96650e10 q^{17} +7.58630e10 q^{19} +1.45122e11 q^{21} +4.80343e10 q^{23} -2.17762e11 q^{25} +1.30905e12 q^{27} -4.46967e12 q^{29} -2.89743e12 q^{31} -1.35974e13 q^{33} +8.68672e12 q^{35} +1.98048e13 q^{37} +3.17675e13 q^{39} -5.72233e13 q^{41} +1.17723e14 q^{43} -1.69946e13 q^{45} +1.99303e14 q^{47} -9.42184e13 q^{49} -6.12627e14 q^{51} +3.19512e14 q^{53} -8.13917e14 q^{55} -9.35784e14 q^{57} +3.92786e14 q^{59} +2.54361e15 q^{61} -2.70787e14 q^{63} +1.90154e15 q^{65} +2.36089e14 q^{67} -5.92511e14 q^{69} -7.35004e14 q^{71} +7.70307e15 q^{73} +2.68614e15 q^{75} -1.29688e16 q^{77} -8.20252e15 q^{79} -1.91198e16 q^{81} -2.50479e15 q^{83} -3.66707e16 q^{85} +5.51342e16 q^{87} +1.99635e16 q^{89} +3.02987e16 q^{91} +3.57403e16 q^{93} -5.60143e16 q^{95} +5.80049e16 q^{97} +2.53719e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 952 q^{3} - 53620 q^{5} - 333168 q^{7} + 23453338 q^{9}+O(q^{10}) \) 2 * q - 952 * q^3 - 53620 * q^5 - 333168 * q^7 + 23453338 * q^9	\( 2 q - 952 q^{3} - 53620 q^{5} - 333168 q^{7} + 23453338 q^{9} + 430974680 q^{11} + 2667521948 q^{13} + 16902353840 q^{15} + 60673503268 q^{17} + 178629960040 q^{19} + 275250928704 q^{21} + 528756594608 q^{23} - 511831510850 q^{25} - 156001401136 q^{27} - 7240660091460 q^{29} - 1878351140288 q^{31} - 21239581915552 q^{33} + 16514472678240 q^{35} - 20332464566580 q^{37} + 91448192162288 q^{39} - 1763041905324 q^{41} + 193394525968664 q^{43} - 16695526837220 q^{45} + 100763837765472 q^{47} - 196165218276494 q^{49} - 487316066501360 q^{51} - 317818146060052 q^{53} - 12\!\cdots\!20 q^{55}+ \cdots + 25\!\cdots\!64 q^{99}+O(q^{100}) \) 2 * q - 952 * q^3 - 53620 * q^5 - 333168 * q^7 + 23453338 * q^9 + 430974680 * q^11 + 2667521948 * q^13 + 16902353840 * q^15 + 60673503268 * q^17 + 178629960040 * q^19 + 275250928704 * q^21 + 528756594608 * q^23 - 511831510850 * q^25 - 156001401136 * q^27 - 7240660091460 * q^29 - 1878351140288 * q^31 - 21239581915552 * q^33 + 16514472678240 * q^35 - 20332464566580 * q^37 + 91448192162288 * q^39 - 1763041905324 * q^41 + 193394525968664 * q^43 - 16695526837220 * q^45 + 100763837765472 * q^47 - 196165218276494 * q^49 - 487316066501360 * q^51 - 317818146060052 * q^53 - 1273620709243120 * q^55 + 234031155307168 * q^57 - 1262050788321736 * q^59 + 2765859692723708 * q^61 - 265794859238064 * q^63 + 5491559693146280 * q^65 + 1963006544550088 * q^67 + 4879639671010496 * q^69 + 483639107104528 * q^71 - 2176892348591212 * q^73 - 661303864041800 * q^75 - 20643473818671936 * q^77 - 5295839905627744 * q^79 - 35853176106728462 * q^81 + 9972518018887144 * q^83 - 29132785359362600 * q^85 + 23591579817096816 * q^87 - 711099036813900 * q^89 + 90233771615940576 * q^91 + 47340611694588160 * q^93 + 14354471748500080 * q^95 + 114870546609971908 * q^97 + 25078682365155064 * q^99

Introduction

L-functions

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Database

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