LMFDB - Embedded newform 64.14.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,14,Mod(1,64)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	64.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:	$68.6277945292$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2195x - 37995 $ x^3 - x^2 - 2195*x - 37995
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$54.3044$ of defining polynomial
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1553.74 q^{3} -4052.28 q^{5} -23239.8 q^{7} +819790. q^{9} +176072. q^{11} -2.27385e7 q^{13} -6.29619e6 q^{15} +1.40715e8 q^{17} -7.16239e7 q^{19} -3.61086e7 q^{21} -7.31400e8 q^{23} -1.20428e9 q^{25} -1.20342e9 q^{27} +2.69757e9 q^{29} -8.20983e9 q^{31} +2.73570e8 q^{33} +9.41740e7 q^{35} -1.53462e10 q^{37} -3.53298e10 q^{39} +2.07764e10 q^{41} +5.27322e10 q^{43} -3.32202e9 q^{45} -5.64800e10 q^{47} -9.63489e10 q^{49} +2.18635e11 q^{51} +9.36107e10 q^{53} -7.13491e8 q^{55} -1.11285e11 q^{57} +4.12378e11 q^{59} -4.02716e11 q^{61} -1.90517e10 q^{63} +9.21428e10 q^{65} -5.18649e11 q^{67} -1.13641e12 q^{69} +3.19747e11 q^{71} -1.07513e12 q^{73} -1.87114e12 q^{75} -4.09186e9 q^{77} -2.17970e11 q^{79} -3.17682e12 q^{81} -4.92681e12 q^{83} -5.70216e11 q^{85} +4.19132e12 q^{87} -4.56960e12 q^{89} +5.28438e11 q^{91} -1.27560e13 q^{93} +2.90240e11 q^{95} -3.61421e12 q^{97} +1.44342e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 520 q^{3} - 11594 q^{5} - 96304 q^{7} - 196793 q^{9} + 1923816 q^{11} + 11211006 q^{13} + 10944368 q^{15} - 16892538 q^{17} + 74665048 q^{19} + 115146880 q^{21} - 486642576 q^{23} + 442696461 q^{25}+ \cdots + 2082116453576 q^{99}+O(q^{100}) $ 3 * q - 520 * q^3 - 11594 * q^5 - 96304 * q^7 - 196793 * q^9 + 1923816 * q^11 + 11211006 * q^13 + 10944368 * q^15 - 16892538 * q^17 + 74665048 * q^19 + 115146880 * q^21 - 486642576 * q^23 + 442696461 * q^25 + 3111524144 * q^27 - 1747468946 * q^29 - 6471356864 * q^31 - 2901436352 * q^33 + 32905782432 * q^35 + 1189828278 * q^37 - 73495249232 * q^39 - 12181275970 * q^41 + 151151885096 * q^43 - 19024866434 * q^45 - 273552663072 * q^47 - 26716022709 * q^49 + 361685604336 * q^51 - 151381970362 * q^53 - 594621149104 * q^55 - 253183022656 * q^57 + 534333367560 * q^59 - 598414222482 * q^61 - 138475736176 * q^63 - 1313304118564 * q^65 - 703511405320 * q^67 - 1246094112384 * q^69 + 1867178467152 * q^71 - 2138161799730 * q^73 - 3649896495992 * q^75 - 4774654840448 * q^77 + 6521880375328 * q^79 - 6029061199109 * q^81 - 5725896739752 * q^83 - 8753386407828 * q^85 + 9313876797488 * q^87 - 11930629520834 * q^89 - 10949018828768 * q^91 - 15654956180992 * q^93 + 3955366679088 * q^95 - 23076828823050 * q^97 + 2082116453576 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1553.74	1.23053	0.615263	−	0.788322i	$-0.289050\pi$
$3$	1553.74	1.23053	0.615263	+	0.788322i	$0.289050\pi$
$4$	0	0
$5$	−4052.28	−0.115983	−0.0579915	−	0.998317i	$-0.518470\pi$
$5$	−4052.28	−0.115983	−0.0579915	+	0.998317i	$0.518470\pi$
$6$	0	0
$7$	−23239.8	−0.0746611	−0.0373306	−	0.999303i	$-0.511885\pi$
$7$	−23239.8	−0.0746611	−0.0373306	+	0.999303i	$0.511885\pi$
$8$	0	0
$9$	819790.	0.514193
$10$	0	0
$11$	176072.	0.0299666	0.0149833	−	0.999888i	$-0.495230\pi$
$11$	176072.	0.0299666	0.0149833	+	0.999888i	$0.495230\pi$
$12$	0	0
$13$	−2.27385e7	−1.30656	−0.653282	−	0.757115i	$-0.726608\pi$
$13$	−2.27385e7	−1.30656	−0.653282	+	0.757115i	$0.726608\pi$
$14$	0	0
$15$	−6.29619e6	−0.142720
$16$	0	0
$17$	1.40715e8	1.41391	0.706955	−	0.707258i	$-0.250068\pi$
$17$	1.40715e8	1.41391	0.706955	+	0.707258i	$0.250068\pi$
$18$	0	0
$19$	−7.16239e7	−0.349268	−0.174634	−	0.984633i	$-0.555874\pi$
$19$	−7.16239e7	−0.349268	−0.174634	+	0.984633i	$0.555874\pi$
$20$	0	0
$21$	−3.61086e7	−0.0918724
$22$	0	0
$23$	−7.31400e8	−1.03021	−0.515103	−	0.857128i	$-0.672246\pi$
$23$	−7.31400e8	−1.03021	−0.515103	+	0.857128i	$0.672246\pi$
$24$	0	0
$25$	−1.20428e9	−0.986548
$26$	0	0
$27$	−1.20342e9	−0.597797
$28$	0	0
$29$	2.69757e9	0.842142	0.421071	−	0.907028i	$-0.361654\pi$
$29$	2.69757e9	0.842142	0.421071	+	0.907028i	$0.361654\pi$
$30$	0	0
$31$	−8.20983e9	−1.66143	−0.830717	−	0.556694i	$-0.812069\pi$
$31$	−8.20983e9	−1.66143	−0.830717	+	0.556694i	$0.812069\pi$
$32$	0	0
$33$	2.73570e8	0.0368746
$34$	0	0
$35$	9.41740e7	0.00865942
$36$	0	0
$37$	−1.53462e10	−0.983310	−0.491655	−	0.870790i	$-0.663608\pi$
$37$	−1.53462e10	−0.983310	−0.491655	+	0.870790i	$0.663608\pi$
$38$	0	0
$39$	−3.53298e10	−1.60776
$40$	0	0
$41$	2.07764e10	0.683085	0.341543	−	0.939866i	$-0.389051\pi$
$41$	2.07764e10	0.683085	0.341543	+	0.939866i	$0.389051\pi$
$42$	0	0
$43$	5.27322e10	1.27213	0.636064	−	0.771637i	$-0.280562\pi$
$43$	5.27322e10	1.27213	0.636064	+	0.771637i	$0.280562\pi$
$44$	0	0
$45$	−3.32202e9	−0.0596377
$46$	0	0
$47$	−5.64800e10	−0.764290	−0.382145	−	0.924102i	$-0.624815\pi$
$47$	−5.64800e10	−0.764290	−0.382145	+	0.924102i	$0.624815\pi$
$48$	0	0
$49$	−9.63489e10	−0.994426
$50$	0	0
$51$	2.18635e11	1.73985
$52$	0	0
$53$	9.36107e10	0.580140	0.290070	−	0.957005i	$-0.406321\pi$
$53$	9.36107e10	0.580140	0.290070	+	0.957005i	$0.406321\pi$
$54$	0	0
$55$	−7.13491e8	−0.00347561
$56$	0	0
$57$	−1.11285e11	−0.429784
$58$	0	0
$59$	4.12378e11	1.27279	0.636396	−	0.771363i	$-0.280424\pi$
$59$	4.12378e11	1.27279	0.636396	+	0.771363i	$0.280424\pi$
$60$	0	0
$61$	−4.02716e11	−1.00082	−0.500408	−	0.865790i	$-0.666817\pi$
$61$	−4.02716e11	−1.00082	−0.500408	+	0.865790i	$0.666817\pi$
$62$	0	0
$63$	−1.90517e10	−0.0383903
$64$	0	0
$65$	9.21428e10	0.151539
$66$	0	0
$67$	−5.18649e11	−0.700466	−0.350233	−	0.936663i	$-0.613898\pi$
$67$	−5.18649e11	−0.700466	−0.350233	+	0.936663i	$0.613898\pi$
$68$	0	0
$69$	−1.13641e12	−1.26769
$70$	0	0
$71$	3.19747e11	0.296229	0.148114	−	0.988970i	$-0.452680\pi$
$71$	3.19747e11	0.296229	0.148114	+	0.988970i	$0.452680\pi$
$72$	0	0
$73$	−1.07513e12	−0.831504	−0.415752	−	0.909478i	$-0.636481\pi$
$73$	−1.07513e12	−0.831504	−0.415752	+	0.909478i	$0.636481\pi$
$74$	0	0
$75$	−1.87114e12	−1.21397
$76$	0	0
$77$	−4.09186e9	−0.00223734
$78$	0	0
$79$	−2.17970e11	−0.100884	−0.0504419	−	0.998727i	$-0.516063\pi$
$79$	−2.17970e11	−0.100884	−0.0504419	+	0.998727i	$0.516063\pi$
$80$	0	0
$81$	−3.17682e12	−1.24980
$82$	0	0
$83$	−4.92681e12	−1.65409	−0.827044	−	0.562138i	$-0.809979\pi$
$83$	−4.92681e12	−1.65409	−0.827044	+	0.562138i	$0.809979\pi$
$84$	0	0
$85$	−5.70216e11	−0.163990
$86$	0	0
$87$	4.19132e12	1.03628
$88$	0	0
$89$	−4.56960e12	−0.974638	−0.487319	−	0.873224i	$-0.662025\pi$
$89$	−4.56960e12	−0.974638	−0.487319	+	0.873224i	$0.662025\pi$
$90$	0	0
$91$	5.28438e11	0.0975495
$92$	0	0
$93$	−1.27560e13	−2.04444
$94$	0	0
$95$	2.90240e11	0.0405092
$96$	0	0
$97$	−3.61421e12	−0.440552	−0.220276	−	0.975438i	$-0.570696\pi$
$97$	−3.61421e12	−0.440552	−0.220276	+	0.975438i	$0.570696\pi$
$98$	0	0
$99$	1.44342e11	0.0154086
$100$	0	0
$101$	−1.04974e13	−0.983996	−0.491998	−	0.870596i	$-0.663733\pi$
$101$	−1.04974e13	−0.983996	−0.491998	+	0.870596i	$0.663733\pi$
$102$	0	0
$103$	1.09103e13	0.900319	0.450159	−	0.892948i	$-0.351367\pi$
$103$	1.09103e13	0.900319	0.450159	+	0.892948i	$0.351367\pi$
$104$	0	0
$105$	1.46322e11	0.0106556
$106$	0	0
$107$	2.95074e13	1.90080	0.950399	−	0.311033i	$-0.100675\pi$
$107$	2.95074e13	1.90080	0.950399	+	0.311033i	$0.100675\pi$
$108$	0	0
$109$	4.32360e12	0.246930	0.123465	−	0.992349i	$-0.460599\pi$
$109$	4.32360e12	0.246930	0.123465	+	0.992349i	$0.460599\pi$
$110$	0	0
$111$	−2.38441e13	−1.20999
$112$	0	0
$113$	−3.50360e13	−1.58309	−0.791543	−	0.611113i	$-0.790722\pi$
$113$	−3.50360e13	−1.58309	−0.791543	+	0.611113i	$0.790722\pi$
$114$	0	0
$115$	2.96383e12	0.119486
$116$	0	0
$117$	−1.86408e13	−0.671826
$118$	0	0
$119$	−3.27018e12	−0.105564
$120$	0	0
$121$	−3.44917e13	−0.999102
$122$	0	0
$123$	3.22811e13	0.840554
$124$	0	0
$125$	9.82671e12	0.230406
$126$	0	0
$127$	−8.01799e13	−1.69567	−0.847835	−	0.530261i	$-0.822094\pi$
$127$	−8.01799e13	−1.69567	−0.847835	+	0.530261i	$0.822094\pi$
$128$	0	0
$129$	8.19322e13	1.56539
$130$	0	0
$131$	−6.25517e13	−1.08137	−0.540686	−	0.841224i	$-0.681835\pi$
$131$	−6.25517e13	−1.08137	−0.540686	+	0.841224i	$0.681835\pi$
$132$	0	0
$133$	1.66452e12	0.0260768
$134$	0	0
$135$	4.87661e12	0.0693343
$136$	0	0
$137$	1.80014e13	0.232607	0.116303	−	0.993214i	$-0.462896\pi$
$137$	1.80014e13	0.232607	0.116303	+	0.993214i	$0.462896\pi$
$138$	0	0
$139$	1.14479e13	0.134626	0.0673129	−	0.997732i	$-0.478557\pi$
$139$	1.14479e13	0.134626	0.0673129	+	0.997732i	$0.478557\pi$
$140$	0	0
$141$	−8.77553e13	−0.940479
$142$	0	0
$143$	−4.00361e12	−0.0391532
$144$	0	0
$145$	−1.09313e13	−0.0976741
$146$	0	0
$147$	−1.49701e14	−1.22367
$148$	0	0
$149$	2.03204e14	1.52133	0.760663	−	0.649147i	$-0.224874\pi$
$149$	2.03204e14	1.52133	0.760663	+	0.649147i	$0.224874\pi$
$150$	0	0
$151$	2.28427e14	1.56818	0.784091	−	0.620646i	$-0.213129\pi$
$151$	2.28427e14	1.56818	0.784091	+	0.620646i	$0.213129\pi$
$152$	0	0
$153$	1.15357e14	0.727024
$154$	0	0
$155$	3.32685e13	0.192698
$156$	0	0
$157$	3.02124e14	1.61004	0.805022	−	0.593244i	$-0.202153\pi$
$157$	3.02124e14	1.61004	0.805022	+	0.593244i	$0.202153\pi$
$158$	0	0
$159$	1.45447e14	0.713877
$160$	0	0
$161$	1.69976e13	0.0769163
$162$	0	0
$163$	7.44730e12	0.0311013	0.0155507	−	0.999879i	$-0.495050\pi$
$163$	7.44730e12	0.0311013	0.0155507	+	0.999879i	$0.495050\pi$
$164$	0	0
$165$	−1.10858e12	−0.00427683
$166$	0	0
$167$	2.80750e14	1.00153	0.500764	−	0.865584i	$-0.333053\pi$
$167$	2.80750e14	1.00153	0.500764	+	0.865584i	$0.333053\pi$
$168$	0	0
$169$	2.14166e14	0.707108
$170$	0	0
$171$	−5.87165e13	−0.179592
$172$	0	0
$173$	6.36564e13	0.180527	0.0902635	−	0.995918i	$-0.471229\pi$
$173$	6.36564e13	0.180527	0.0902635	+	0.995918i	$0.471229\pi$
$174$	0	0
$175$	2.79872e13	0.0736568
$176$	0	0
$177$	6.40729e14	1.56620
$178$	0	0
$179$	1.73421e14	0.394054	0.197027	−	0.980398i	$-0.436871\pi$
$179$	1.73421e14	0.394054	0.197027	+	0.980398i	$0.436871\pi$
$180$	0	0
$181$	6.07821e13	0.128489	0.0642443	−	0.997934i	$-0.479536\pi$
$181$	6.07821e13	0.128489	0.0642443	+	0.997934i	$0.479536\pi$
$182$	0	0
$183$	−6.25716e14	−1.23153
$184$	0	0
$185$	6.21872e13	0.114047
$186$	0	0
$187$	2.47759e13	0.0423700
$188$	0	0
$189$	2.79673e13	0.0446322
$190$	0	0
$191$	−3.14388e14	−0.468543	−0.234271	−	0.972171i	$-0.575270\pi$
$191$	−3.14388e14	−0.468543	−0.234271	+	0.972171i	$0.575270\pi$
$192$	0	0
$193$	−3.24465e14	−0.451903	−0.225951	−	0.974139i	$-0.572549\pi$
$193$	−3.24465e14	−0.451903	−0.225951	+	0.974139i	$0.572549\pi$
$194$	0	0
$195$	1.43166e14	0.186473
$196$	0	0
$197$	4.61063e14	0.561992	0.280996	−	0.959709i	$-0.409335\pi$
$197$	4.61063e14	0.561992	0.280996	+	0.959709i	$0.409335\pi$
$198$	0	0
$199$	−1.29671e15	−1.48013	−0.740063	−	0.672538i	$-0.765204\pi$
$199$	−1.29671e15	−1.48013	−0.740063	+	0.672538i	$0.765204\pi$
$200$	0	0
$201$	−8.05846e14	−0.861941
$202$	0	0
$203$	−6.26908e13	−0.0628753
$204$	0	0
$205$	−8.41917e13	−0.0792262
$206$	0	0
$207$	−5.99594e14	−0.529725
$208$	0	0
$209$	−1.26109e13	−0.0104664
$210$	0	0
$211$	1.56891e15	1.22394	0.611972	−	0.790880i	$-0.290377\pi$
$211$	1.56891e15	1.22394	0.611972	+	0.790880i	$0.290377\pi$
$212$	0	0
$213$	4.96804e14	0.364517
$214$	0	0
$215$	−2.13685e14	−0.147545
$216$	0	0
$217$	1.90795e14	0.124045
$218$	0	0
$219$	−1.67048e15	−1.02319
$220$	0	0
$221$	−3.19965e15	−1.84736
$222$	0	0
$223$	2.75415e15	1.49971	0.749853	−	0.661604i	$-0.230124\pi$
$223$	2.75415e15	1.49971	0.749853	+	0.661604i	$0.230124\pi$
$224$	0	0
$225$	−9.87259e14	−0.507276
$226$	0	0
$227$	1.58251e15	0.767679	0.383839	−	0.923400i	$-0.374602\pi$
$227$	1.58251e15	0.767679	0.383839	+	0.923400i	$0.374602\pi$
$228$	0	0
$229$	−2.47800e15	−1.13546	−0.567729	−	0.823215i	$-0.692178\pi$
$229$	−2.47800e15	−1.13546	−0.567729	+	0.823215i	$0.692178\pi$
$230$	0	0
$231$	−6.35770e12	−0.00275310
$232$	0	0
$233$	−1.56530e15	−0.640890	−0.320445	−	0.947267i	$-0.603832\pi$
$233$	−1.56530e15	−0.640890	−0.320445	+	0.947267i	$0.603832\pi$
$234$	0	0
$235$	2.28872e14	0.0886446
$236$	0	0
$237$	−3.38670e14	−0.124140
$238$	0	0
$239$	4.54070e15	1.57593	0.787964	−	0.615721i	$-0.211135\pi$
$239$	4.54070e15	1.57593	0.787964	+	0.615721i	$0.211135\pi$
$240$	0	0
$241$	4.59058e14	0.150923	0.0754617	−	0.997149i	$-0.475957\pi$
$241$	4.59058e14	0.150923	0.0754617	+	0.997149i	$0.475957\pi$
$242$	0	0
$243$	−3.01731e15	−0.940112
$244$	0	0
$245$	3.90433e14	0.115336
$246$	0	0
$247$	1.62862e15	0.456341
$248$	0	0
$249$	−7.65499e15	−2.03540
$250$	0	0
$251$	−2.50049e15	−0.631169	−0.315584	−	0.948898i	$-0.602201\pi$
$251$	−2.50049e15	−0.631169	−0.315584	+	0.948898i	$0.602201\pi$
$252$	0	0
$253$	−1.28779e14	−0.0308717
$254$	0	0
$255$	−8.85968e14	−0.201793
$256$	0	0
$257$	1.52977e15	0.331178	0.165589	−	0.986195i	$-0.447048\pi$
$257$	1.52977e15	0.331178	0.165589	+	0.986195i	$0.447048\pi$
$258$	0	0
$259$	3.56643e14	0.0734151
$260$	0	0
$261$	2.21144e15	0.433024
$262$	0	0
$263$	7.02008e14	0.130807	0.0654033	−	0.997859i	$-0.479167\pi$
$263$	7.02008e14	0.130807	0.0654033	+	0.997859i	$0.479167\pi$
$264$	0	0
$265$	−3.79337e14	−0.0672863
$266$	0	0
$267$	−7.09998e15	−1.19932
$268$	0	0
$269$	−6.45337e15	−1.03848	−0.519239	−	0.854629i	$-0.673784\pi$
$269$	−6.45337e15	−1.03848	−0.519239	+	0.854629i	$0.673784\pi$
$270$	0	0
$271$	−7.59564e15	−1.16483	−0.582417	−	0.812890i	$-0.697893\pi$
$271$	−7.59564e15	−1.16483	−0.582417	+	0.812890i	$0.697893\pi$
$272$	0	0
$273$	8.21057e14	0.120037
$274$	0	0
$275$	−2.12040e14	−0.0295634
$276$	0	0
$277$	−2.40405e15	−0.319760	−0.159880	−	0.987136i	$-0.551111\pi$
$277$	−2.40405e15	−0.319760	−0.159880	+	0.987136i	$0.551111\pi$
$278$	0	0
$279$	−6.73034e15	−0.854299
$280$	0	0
$281$	7.15842e15	0.867413	0.433707	−	0.901054i	$-0.357205\pi$
$281$	7.15842e15	0.867413	0.433707	+	0.901054i	$0.357205\pi$
$282$	0	0
$283$	1.36939e16	1.58459	0.792294	−	0.610139i	$-0.208887\pi$
$283$	1.36939e16	1.58459	0.792294	+	0.610139i	$0.208887\pi$
$284$	0	0
$285$	4.50958e14	0.0498476
$286$	0	0
$287$	−4.82838e14	−0.0509999
$288$	0	0
$289$	9.89609e15	0.999143
$290$	0	0
$291$	−5.61556e15	−0.542111
$292$	0	0
$293$	1.96166e16	1.81127	0.905636	−	0.424057i	$-0.139394\pi$
$293$	1.96166e16	1.81127	0.905636	+	0.424057i	$0.139394\pi$
$294$	0	0
$295$	−1.67107e15	−0.147622
$296$	0	0
$297$	−2.11889e14	−0.0179139
$298$	0	0
$299$	1.66310e16	1.34603
$300$	0	0
$301$	−1.22548e15	−0.0949785
$302$	0	0
$303$	−1.63103e16	−1.21083
$304$	0	0
$305$	1.63192e15	0.116078
$306$	0	0
$307$	1.54129e16	1.05072	0.525359	−	0.850881i	$-0.323931\pi$
$307$	1.54129e16	1.05072	0.525359	+	0.850881i	$0.323931\pi$
$308$	0	0
$309$	1.69519e16	1.10787
$310$	0	0
$311$	2.56624e16	1.60826	0.804128	−	0.594457i	$-0.202633\pi$
$311$	2.56624e16	1.60826	0.804128	+	0.594457i	$0.202633\pi$
$312$	0	0
$313$	−2.44135e16	−1.46754	−0.733772	−	0.679396i	$-0.762242\pi$
$313$	−2.44135e16	−1.46754	−0.733772	+	0.679396i	$0.762242\pi$
$314$	0	0
$315$	7.72030e13	0.00445262
$316$	0	0
$317$	−4.41228e15	−0.244218	−0.122109	−	0.992517i	$-0.538966\pi$
$317$	−4.41228e15	−0.244218	−0.122109	+	0.992517i	$0.538966\pi$
$318$	0	0
$319$	4.74965e14	0.0252361
$320$	0	0
$321$	4.58468e16	2.33898
$322$	0	0
$323$	−1.00785e16	−0.493834
$324$	0	0
$325$	2.73836e16	1.28899
$326$	0	0
$327$	6.71775e15	0.303853
$328$	0	0
$329$	1.31258e15	0.0570628
$330$	0	0
$331$	1.97246e16	0.824377	0.412188	−	0.911099i	$-0.364765\pi$
$331$	1.97246e16	0.824377	0.412188	+	0.911099i	$0.364765\pi$
$332$	0	0
$333$	−1.25807e16	−0.505612
$334$	0	0
$335$	2.10171e15	0.0812421
$336$	0	0
$337$	3.70654e16	1.37840	0.689199	−	0.724572i	$-0.257963\pi$
$337$	3.70654e16	1.37840	0.689199	+	0.724572i	$0.257963\pi$
$338$	0	0
$339$	−5.44369e16	−1.94803
$340$	0	0
$341$	−1.44552e15	−0.0497875
$342$	0	0
$343$	4.49081e15	0.148906
$344$	0	0
$345$	4.60503e15	0.147031
$346$	0	0
$347$	−3.29535e16	−1.01335	−0.506676	−	0.862137i	$-0.669126\pi$
$347$	−3.29535e16	−1.01335	−0.506676	+	0.862137i	$0.669126\pi$
$348$	0	0
$349$	−1.58132e16	−0.468442	−0.234221	−	0.972183i	$-0.575254\pi$
$349$	−1.58132e16	−0.468442	−0.234221	+	0.972183i	$0.575254\pi$
$350$	0	0
$351$	2.73641e16	0.781060
$352$	0	0
$353$	−4.15488e16	−1.14294	−0.571469	−	0.820624i	$-0.693626\pi$
$353$	−4.15488e16	−1.14294	−0.571469	+	0.820624i	$0.693626\pi$
$354$	0	0
$355$	−1.29570e15	−0.0343575
$356$	0	0
$357$	−5.08102e15	−0.129899
$358$	0	0
$359$	4.48817e15	0.110651	0.0553255	−	0.998468i	$-0.482380\pi$
$359$	4.48817e15	0.110651	0.0553255	+	0.998468i	$0.482380\pi$
$360$	0	0
$361$	−3.69230e16	−0.878012
$362$	0	0
$363$	−5.35912e16	−1.22942
$364$	0	0
$365$	4.35674e15	0.0964402
$366$	0	0
$367$	−6.55889e16	−1.40120	−0.700602	−	0.713553i	$-0.747085\pi$
$367$	−6.55889e16	−1.40120	−0.700602	+	0.713553i	$0.747085\pi$
$368$	0	0
$369$	1.70323e16	0.351238
$370$	0	0
$371$	−2.17549e15	−0.0433139
$372$	0	0
$373$	8.28290e16	1.59248	0.796242	−	0.604978i	$-0.206818\pi$
$373$	8.28290e16	1.59248	0.796242	+	0.604978i	$0.206818\pi$
$374$	0	0
$375$	1.52682e16	0.283520
$376$	0	0
$377$	−6.13387e16	−1.10031
$378$	0	0
$379$	5.12943e16	0.889025	0.444513	−	0.895773i	$-0.353377\pi$
$379$	5.12943e16	0.889025	0.444513	+	0.895773i	$0.353377\pi$
$380$	0	0
$381$	−1.24579e17	−2.08656
$382$	0	0
$383$	8.13579e16	1.31707	0.658533	−	0.752552i	$-0.271177\pi$
$383$	8.13579e16	1.31707	0.658533	+	0.752552i	$0.271177\pi$
$384$	0	0
$385$	1.65814e13	0.000259493 0
$386$	0	0
$387$	4.32293e16	0.654120
$388$	0	0
$389$	−1.97328e16	−0.288746	−0.144373	−	0.989523i	$-0.546116\pi$
$389$	−1.97328e16	−0.288746	−0.144373	+	0.989523i	$0.546116\pi$
$390$	0	0
$391$	−1.02919e17	−1.45662
$392$	0	0
$393$	−9.71891e16	−1.33066
$394$	0	0
$395$	8.83277e14	0.0117008
$396$	0	0
$397$	−1.28139e17	−1.64264	−0.821318	−	0.570470i	$-0.806761\pi$
$397$	−1.28139e17	−1.64264	−0.821318	+	0.570470i	$0.806761\pi$
$398$	0	0
$399$	2.58624e15	0.0320881
$400$	0	0
$401$	1.26064e17	1.51409	0.757046	−	0.653362i	$-0.226642\pi$
$401$	1.26064e17	1.51409	0.757046	+	0.653362i	$0.226642\pi$
$402$	0	0
$403$	1.86680e17	2.17077
$404$	0	0
$405$	1.28734e16	0.144955
$406$	0	0
$407$	−2.70204e15	−0.0294664
$408$	0	0
$409$	−5.84658e16	−0.617590	−0.308795	−	0.951129i	$-0.599926\pi$
$409$	−5.84658e16	−0.617590	−0.308795	+	0.951129i	$0.599926\pi$
$410$	0	0
$411$	2.79695e16	0.286228
$412$	0	0
$413$	−9.58357e15	−0.0950281
$414$	0	0
$415$	1.99648e16	0.191846
$416$	0	0
$417$	1.77870e16	0.165661
$418$	0	0
$419$	7.57809e15	0.0684177	0.0342089	−	0.999415i	$-0.489109\pi$
$419$	7.57809e15	0.0684177	0.0342089	+	0.999415i	$0.489109\pi$
$420$	0	0
$421$	−3.05150e16	−0.267104	−0.133552	−	0.991042i	$-0.542638\pi$
$421$	−3.05150e16	−0.267104	−0.133552	+	0.991042i	$0.542638\pi$
$422$	0	0
$423$	−4.63017e16	−0.392993
$424$	0	0
$425$	−1.69460e17	−1.39489
$426$	0	0
$427$	9.35902e15	0.0747221
$428$	0	0
$429$	−6.22057e15	−0.0481790
$430$	0	0
$431$	−6.41105e16	−0.481756	−0.240878	−	0.970555i	$-0.577435\pi$
$431$	−6.41105e16	−0.481756	−0.240878	+	0.970555i	$0.577435\pi$
$432$	0	0
$433$	−6.27323e16	−0.457425	−0.228712	−	0.973494i	$-0.573452\pi$
$433$	−6.27323e16	−0.457425	−0.228712	+	0.973494i	$0.573452\pi$
$434$	0	0
$435$	−1.69844e16	−0.120191
$436$	0	0
$437$	5.23857e16	0.359818
$438$	0	0
$439$	−1.66208e17	−1.10823	−0.554117	−	0.832439i	$-0.686944\pi$
$439$	−1.66208e17	−1.10823	−0.554117	+	0.832439i	$0.686944\pi$
$440$	0	0
$441$	−7.89859e16	−0.511327
$442$	0	0
$443$	−2.50550e17	−1.57496	−0.787482	−	0.616338i	$-0.788615\pi$
$443$	−2.50550e17	−1.57496	−0.787482	+	0.616338i	$0.788615\pi$
$444$	0	0
$445$	1.85173e16	0.113041
$446$	0	0
$447$	3.15727e17	1.87203
$448$	0	0
$449$	1.58027e17	0.910187	0.455093	−	0.890444i	$-0.349606\pi$
$449$	1.58027e17	0.910187	0.455093	+	0.890444i	$0.349606\pi$
$450$	0	0
$451$	3.65813e15	0.0204697
$452$	0	0
$453$	3.54916e17	1.92969
$454$	0	0
$455$	−2.14138e15	−0.0113141
$456$	0	0
$457$	2.52753e17	1.29790	0.648949	−	0.760832i	$-0.275209\pi$
$457$	2.52753e17	1.29790	0.648949	+	0.760832i	$0.275209\pi$
$458$	0	0
$459$	−1.69340e17	−0.845232
$460$	0	0
$461$	−3.34322e17	−1.62222	−0.811109	−	0.584896i	$-0.801136\pi$
$461$	−3.34322e17	−1.62222	−0.811109	+	0.584896i	$0.801136\pi$
$462$	0	0
$463$	−3.56352e17	−1.68113	−0.840567	−	0.541708i	$-0.817778\pi$
$463$	−3.56352e17	−1.68113	−0.840567	+	0.541708i	$0.817778\pi$
$464$	0	0
$465$	5.16907e16	0.237120
$466$	0	0
$467$	−3.24921e17	−1.44950	−0.724749	−	0.689013i	$-0.758044\pi$
$467$	−3.24921e17	−1.44950	−0.724749	+	0.689013i	$0.758044\pi$
$468$	0	0
$469$	1.20533e16	0.0522976
$470$	0	0
$471$	4.69423e17	1.98120
$472$	0	0
$473$	9.28464e15	0.0381213
$474$	0	0
$475$	8.62553e16	0.344570
$476$	0	0
$477$	7.67412e16	0.298304
$478$	0	0
$479$	3.51173e17	1.32843	0.664217	−	0.747540i	$-0.268765\pi$
$479$	3.51173e17	1.32843	0.664217	+	0.747540i	$0.268765\pi$
$480$	0	0
$481$	3.48951e17	1.28476
$482$	0	0
$483$	2.64098e16	0.0946475
$484$	0	0
$485$	1.46458e16	0.0510966
$486$	0	0
$487$	−2.50106e17	−0.849545	−0.424773	−	0.905300i	$-0.639646\pi$
$487$	−2.50106e17	−0.849545	−0.424773	+	0.905300i	$0.639646\pi$
$488$	0	0
$489$	1.15712e16	0.0382710
$490$	0	0
$491$	1.32539e17	0.426889	0.213445	−	0.976955i	$-0.431532\pi$
$491$	1.32539e17	0.426889	0.213445	+	0.976955i	$0.431532\pi$
$492$	0	0
$493$	3.79588e17	1.19071
$494$	0	0
$495$	−5.84913e14	−0.00178714
$496$	0	0
$497$	−7.43085e15	−0.0221168
$498$	0	0
$499$	6.59853e16	0.191335	0.0956674	−	0.995413i	$-0.469501\pi$
$499$	6.59853e16	0.191335	0.0956674	+	0.995413i	$0.469501\pi$
$500$	0	0
$501$	4.36213e17	1.23241
$502$	0	0
$503$	3.29052e17	0.905883	0.452942	−	0.891540i	$-0.350375\pi$
$503$	3.29052e17	0.905883	0.452942	+	0.891540i	$0.350375\pi$
$504$	0	0
$505$	4.25384e16	0.114127
$506$	0	0
$507$	3.32758e17	0.870115
$508$	0	0
$509$	2.09762e17	0.534639	0.267320	−	0.963608i	$-0.413862\pi$
$509$	2.09762e17	0.534639	0.267320	+	0.963608i	$0.413862\pi$
$510$	0	0
$511$	2.49859e16	0.0620810
$512$	0	0
$513$	8.61938e16	0.208792
$514$	0	0
$515$	−4.42117e16	−0.104422
$516$	0	0
$517$	−9.94452e15	−0.0229031
$518$	0	0
$519$	9.89055e16	0.222143
$520$	0	0
$521$	−1.65580e17	−0.362713	−0.181356	−	0.983417i	$-0.558049\pi$
$521$	−1.65580e17	−0.362713	−0.181356	+	0.983417i	$0.558049\pi$
$522$	0	0
$523$	5.53861e17	1.18342	0.591711	−	0.806150i	$-0.298453\pi$
$523$	5.53861e17	1.18342	0.591711	+	0.806150i	$0.298453\pi$
$524$	0	0
$525$	4.34849e16	0.0906366
$526$	0	0
$527$	−1.15525e18	−2.34912
$528$	0	0
$529$	3.09091e16	0.0613232
$530$	0	0
$531$	3.38064e17	0.654461
$532$	0	0
$533$	−4.72424e17	−0.892494
$534$	0	0
$535$	−1.19572e17	−0.220460
$536$	0	0
$537$	2.69451e17	0.484894
$538$	0	0
$539$	−1.69643e16	−0.0297995
$540$	0	0
$541$	−4.16988e17	−0.715059	−0.357529	−	0.933902i	$-0.616381\pi$
$541$	−4.16988e17	−0.715059	−0.357529	+	0.933902i	$0.616381\pi$
$542$	0	0
$543$	9.44396e16	0.158109
$544$	0	0
$545$	−1.75204e16	−0.0286396
$546$	0	0
$547$	−4.24918e17	−0.678246	−0.339123	−	0.940742i	$-0.610130\pi$
$547$	−4.24918e17	−0.678246	−0.339123	+	0.940742i	$0.610130\pi$
$548$	0	0
$549$	−3.30142e17	−0.514613
$550$	0	0
$551$	−1.93210e17	−0.294134
$552$	0	0
$553$	5.06558e15	0.00753210
$554$	0	0
$555$	9.66229e16	0.140338
$556$	0	0
$557$	4.31083e16	0.0611648	0.0305824	−	0.999532i	$-0.490264\pi$
$557$	4.31083e16	0.0611648	0.0305824	+	0.999532i	$0.490264\pi$
$558$	0	0
$559$	−1.19905e18	−1.66212
$560$	0	0
$561$	3.84953e16	0.0521374
$562$	0	0
$563$	−5.25558e17	−0.695531	−0.347765	−	0.937582i	$-0.613059\pi$
$563$	−5.25558e17	−0.695531	−0.347765	+	0.937582i	$0.613059\pi$
$564$	0	0
$565$	1.41976e17	0.183611
$566$	0	0
$567$	7.38286e16	0.0933114
$568$	0	0
$569$	−2.47151e17	−0.305304	−0.152652	−	0.988280i	$-0.548781\pi$
$569$	−2.47151e17	−0.305304	−0.152652	+	0.988280i	$0.548781\pi$
$570$	0	0
$571$	1.41469e17	0.170815	0.0854076	−	0.996346i	$-0.472781\pi$
$571$	1.41469e17	0.170815	0.0854076	+	0.996346i	$0.472781\pi$
$572$	0	0
$573$	−4.88478e17	−0.576554
$574$	0	0
$575$	8.80812e17	1.01635
$576$	0	0
$577$	−1.61476e18	−1.82166	−0.910828	−	0.412787i	$-0.864555\pi$
$577$	−1.61476e18	−1.82166	−0.910828	+	0.412787i	$0.864555\pi$
$578$	0	0
$579$	−5.04134e17	−0.556078
$580$	0	0
$581$	1.14498e17	0.123496
$582$	0	0
$583$	1.64822e16	0.0173848
$584$	0	0
$585$	7.55378e16	0.0779204
$586$	0	0
$587$	−6.46274e17	−0.652033	−0.326016	−	0.945364i	$-0.605706\pi$
$587$	−6.46274e17	−0.652033	−0.326016	+	0.945364i	$0.605706\pi$
$588$	0	0
$589$	5.88020e17	0.580287
$590$	0	0
$591$	7.16373e17	0.691546
$592$	0	0
$593$	−1.64843e18	−1.55674	−0.778368	−	0.627808i	$-0.783952\pi$
$593$	−1.64843e18	−1.55674	−0.778368	+	0.627808i	$0.783952\pi$
$594$	0	0
$595$	1.32517e16	0.0122436
$596$	0	0
$597$	−2.01476e18	−1.82133
$598$	0	0
$599$	9.40114e17	0.831584	0.415792	−	0.909460i	$-0.363504\pi$
$599$	9.40114e17	0.831584	0.415792	+	0.909460i	$0.363504\pi$
$600$	0	0
$601$	−4.84299e16	−0.0419208	−0.0209604	−	0.999780i	$-0.506672\pi$
$601$	−4.84299e16	−0.0419208	−0.0209604	+	0.999780i	$0.506672\pi$
$602$	0	0
$603$	−4.25183e17	−0.360175
$604$	0	0
$605$	1.39770e17	0.115879
$606$	0	0
$607$	1.70396e18	1.38272	0.691358	−	0.722513i	$-0.257013\pi$
$607$	1.70396e18	1.38272	0.691358	+	0.722513i	$0.257013\pi$
$608$	0	0
$609$	−9.74054e16	−0.0773696
$610$	0	0
$611$	1.28427e18	0.998594
$612$	0	0
$613$	−1.25520e18	−0.955476	−0.477738	−	0.878502i	$-0.658543\pi$
$613$	−1.25520e18	−0.955476	−0.477738	+	0.878502i	$0.658543\pi$
$614$	0	0
$615$	−1.30812e17	−0.0974899
$616$	0	0
$617$	1.76186e17	0.128563	0.0642816	−	0.997932i	$-0.479524\pi$
$617$	1.76186e17	0.128563	0.0642816	+	0.997932i	$0.479524\pi$
$618$	0	0
$619$	−9.95121e17	−0.711028	−0.355514	−	0.934671i	$-0.615694\pi$
$619$	−9.95121e17	−0.711028	−0.355514	+	0.934671i	$0.615694\pi$
$620$	0	0
$621$	8.80184e17	0.615854
$622$	0	0
$623$	1.06196e17	0.0727675
$624$	0	0
$625$	1.43025e18	0.959825
$626$	0	0
$627$	−1.95941e16	−0.0128791
$628$	0	0
$629$	−2.15944e18	−1.39031
$630$	0	0
$631$	1.59895e18	1.00842	0.504212	−	0.863580i	$-0.331783\pi$
$631$	1.59895e18	1.00842	0.504212	+	0.863580i	$0.331783\pi$
$632$	0	0
$633$	2.43768e18	1.50609
$634$	0	0
$635$	3.24911e17	0.196669
$636$	0	0
$637$	2.19083e18	1.29928
$638$	0	0
$639$	2.62126e17	0.152319
$640$	0	0
$641$	−3.09289e18	−1.76111	−0.880555	−	0.473943i	$-0.842830\pi$
$641$	−3.09289e18	−1.76111	−0.880555	+	0.473943i	$0.842830\pi$
$642$	0	0
$643$	4.98981e17	0.278428	0.139214	−	0.990262i	$-0.455542\pi$
$643$	4.98981e17	0.278428	0.139214	+	0.990262i	$0.455542\pi$
$644$	0	0
$645$	−3.32012e17	−0.181558
$646$	0	0
$647$	−2.65007e18	−1.42030	−0.710149	−	0.704052i	$-0.751372\pi$
$647$	−2.65007e18	−1.42030	−0.710149	+	0.704052i	$0.751372\pi$
$648$	0	0
$649$	7.26081e16	0.0381412
$650$	0	0
$651$	2.96446e17	0.152640
$652$	0	0
$653$	−2.22020e18	−1.12062	−0.560308	−	0.828284i	$-0.689317\pi$
$653$	−2.22020e18	−1.12062	−0.560308	+	0.828284i	$0.689317\pi$
$654$	0	0
$655$	2.53477e17	0.125421
$656$	0	0
$657$	−8.81385e17	−0.427554
$658$	0	0
$659$	7.37791e17	0.350896	0.175448	−	0.984489i	$-0.443863\pi$
$659$	7.37791e17	0.350896	0.175448	+	0.984489i	$0.443863\pi$
$660$	0	0
$661$	3.40095e18	1.58596	0.792978	−	0.609251i	$-0.208530\pi$
$661$	3.40095e18	1.58596	0.792978	+	0.609251i	$0.208530\pi$
$662$	0	0
$663$	−4.97143e18	−2.27323
$664$	0	0
$665$	−6.74511e15	−0.00302446
$666$	0	0
$667$	−1.97300e18	−0.867579
$668$	0	0
$669$	4.27924e18	1.84543
$670$	0	0
$671$	−7.09068e16	−0.0299910
$672$	0	0
$673$	2.57576e18	1.06858	0.534291	−	0.845301i	$-0.320579\pi$
$673$	2.57576e18	1.06858	0.534291	+	0.845301i	$0.320579\pi$
$674$	0	0
$675$	1.44926e18	0.589756
$676$	0	0
$677$	8.19888e17	0.327287	0.163643	−	0.986520i	$-0.447675\pi$
$677$	8.19888e17	0.327287	0.163643	+	0.986520i	$0.447675\pi$
$678$	0	0
$679$	8.39935e16	0.0328921
$680$	0	0
$681$	2.45882e18	0.944649
$682$	0	0
$683$	1.48474e18	0.559647	0.279824	−	0.960051i	$-0.409724\pi$
$683$	1.48474e18	0.559647	0.279824	+	0.960051i	$0.409724\pi$
$684$	0	0
$685$	−7.29466e16	−0.0269784
$686$	0	0
$687$	−3.85018e18	−1.39721
$688$	0	0
$689$	−2.12857e18	−0.757989
$690$	0	0
$691$	1.65745e18	0.579207	0.289603	−	0.957147i	$-0.406477\pi$
$691$	1.65745e18	0.579207	0.289603	+	0.957147i	$0.406477\pi$
$692$	0	0
$693$	−3.35447e15	−0.00115042
$694$	0	0
$695$	−4.63899e16	−0.0156143
$696$	0	0
$697$	2.92355e18	0.965821
$698$	0	0
$699$	−2.43207e18	−0.788632
$700$	0	0
$701$	−1.71107e17	−0.0544628	−0.0272314	−	0.999629i	$-0.508669\pi$
$701$	−1.71107e17	−0.0544628	−0.0272314	+	0.999629i	$0.508669\pi$
$702$	0	0
$703$	1.09916e18	0.343439
$704$	0	0
$705$	3.55609e17	0.109079
$706$	0	0
$707$	2.43957e17	0.0734663
$708$	0	0
$709$	−4.40222e18	−1.30158	−0.650791	−	0.759257i	$-0.725563\pi$
$709$	−4.40222e18	−1.30158	−0.650791	+	0.759257i	$0.725563\pi$
$710$	0	0
$711$	−1.78690e17	−0.0518738
$712$	0	0
$713$	6.00467e18	1.71162
$714$	0	0
$715$	1.62237e16	0.00454111
$716$	0	0
$717$	7.05508e18	1.93922
$718$	0	0
$719$	1.54992e18	0.418380	0.209190	−	0.977875i	$-0.432917\pi$
$719$	1.54992e18	0.418380	0.209190	+	0.977875i	$0.432917\pi$
$720$	0	0
$721$	−2.53554e17	−0.0672188
$722$	0	0
$723$	7.13257e17	0.185715
$724$	0	0
$725$	−3.24863e18	−0.830813
$726$	0	0
$727$	4.24477e18	1.06630	0.533152	−	0.846020i	$-0.321007\pi$
$727$	4.24477e18	1.06630	0.533152	+	0.846020i	$0.321007\pi$
$728$	0	0
$729$	3.76753e17	0.0929669
$730$	0	0
$731$	7.42020e18	1.79867
$732$	0	0
$733$	3.17450e18	0.755962	0.377981	−	0.925813i	$-0.376619\pi$
$733$	3.17450e18	0.755962	0.377981	+	0.925813i	$0.376619\pi$
$734$	0	0
$735$	6.06631e17	0.141924
$736$	0	0
$737$	−9.13193e16	−0.0209906
$738$	0	0
$739$	−7.43070e18	−1.67819	−0.839094	−	0.543986i	$-0.816914\pi$
$739$	−7.43070e18	−1.67819	−0.839094	+	0.543986i	$0.816914\pi$
$740$	0	0
$741$	2.53046e18	0.561540
$742$	0	0
$743$	3.74540e18	0.816715	0.408358	−	0.912822i	$-0.366102\pi$
$743$	3.74540e18	0.816715	0.408358	+	0.912822i	$0.366102\pi$
$744$	0	0
$745$	−8.23440e17	−0.176448
$746$	0	0
$747$	−4.03895e18	−0.850521
$748$	0	0
$749$	−6.85745e17	−0.141916
$750$	0	0
$751$	9.46402e18	1.92493	0.962467	−	0.271398i	$-0.0874858\pi$
$751$	9.46402e18	1.92493	0.962467	+	0.271398i	$0.0874858\pi$
$752$	0	0
$753$	−3.88511e18	−0.776669
$754$	0	0
$755$	−9.25648e17	−0.181882
$756$	0	0
$757$	6.04674e17	0.116788	0.0583940	−	0.998294i	$-0.481402\pi$
$757$	6.04674e17	0.116788	0.0583940	+	0.998294i	$0.481402\pi$
$758$	0	0
$759$	−2.00089e17	−0.0379884
$760$	0	0
$761$	−8.78731e18	−1.64005	−0.820023	−	0.572331i	$-0.806039\pi$
$761$	−8.78731e18	−1.64005	−0.820023	+	0.572331i	$0.806039\pi$
$762$	0	0
$763$	−1.00479e17	−0.0184360
$764$	0	0
$765$	−4.67457e17	−0.0843223
$766$	0	0
$767$	−9.37687e18	−1.66298
$768$	0	0
$769$	2.70998e18	0.472546	0.236273	−	0.971687i	$-0.424074\pi$
$769$	2.70998e18	0.472546	0.236273	+	0.971687i	$0.424074\pi$
$770$	0	0
$771$	2.37687e18	0.407523
$772$	0	0
$773$	−4.50254e18	−0.759086	−0.379543	−	0.925174i	$-0.623919\pi$
$773$	−4.50254e18	−0.759086	−0.379543	+	0.925174i	$0.623919\pi$
$774$	0	0
$775$	9.88696e18	1.63909
$776$	0	0
$777$	5.54131e17	0.0903391
$778$	0	0
$779$	−1.48808e18	−0.238580
$780$	0	0
$781$	5.62984e16	0.00887696
$782$	0	0
$783$	−3.24632e18	−0.503430
$784$	0	0
$785$	−1.22429e18	−0.186738
$786$	0	0
$787$	−1.10845e19	−1.66295	−0.831475	−	0.555562i	$-0.812503\pi$
$787$	−1.10845e19	−1.66295	−0.831475	+	0.555562i	$0.812503\pi$
$788$	0	0
$789$	1.09074e18	0.160961
$790$	0	0
$791$	8.14229e17	0.118195
$792$	0	0
$793$	9.15716e18	1.30763
$794$	0	0
$795$	−5.89391e17	−0.0827975
$796$	0	0
$797$	−1.23845e19	−1.71158	−0.855792	−	0.517320i	$-0.826930\pi$
$797$	−1.23845e19	−1.71158	−0.855792	+	0.517320i	$0.826930\pi$
$798$	0	0
$799$	−7.94757e18	−1.08064
$800$	0	0
$801$	−3.74611e18	−0.501152
$802$	0	0
$803$	−1.89301e17	−0.0249173
$804$	0	0
$805$	−6.88788e16	−0.00892098
$806$	0	0
$807$	−1.00269e19	−1.27787
$808$	0	0
$809$	−1.05373e19	−1.32149	−0.660747	−	0.750609i	$-0.729760\pi$
$809$	−1.05373e19	−1.32149	−0.660747	+	0.750609i	$0.729760\pi$
$810$	0	0
$811$	8.99512e18	1.11012	0.555062	−	0.831809i	$-0.312695\pi$
$811$	8.99512e18	1.11012	0.555062	+	0.831809i	$0.312695\pi$
$812$	0	0
$813$	−1.18017e19	−1.43336
$814$	0	0
$815$	−3.01785e16	−0.00360723
$816$	0	0
$817$	−3.77688e18	−0.444314
$818$	0	0
$819$	4.33209e17	0.0501593
$820$	0	0
$821$	−1.26415e19	−1.44068	−0.720339	−	0.693622i	$-0.756014\pi$
$821$	−1.26415e19	−1.44068	−0.720339	+	0.693622i	$0.756014\pi$
$822$	0	0
$823$	−7.27293e18	−0.815850	−0.407925	−	0.913015i	$-0.633748\pi$
$823$	−7.27293e18	−0.815850	−0.407925	+	0.913015i	$0.633748\pi$
$824$	0	0
$825$	−3.29455e17	−0.0363786
$826$	0	0
$827$	2.84826e18	0.309595	0.154798	−	0.987946i	$-0.450527\pi$
$827$	2.84826e18	0.309595	0.154798	+	0.987946i	$0.450527\pi$
$828$	0	0
$829$	9.55683e18	1.02261	0.511304	−	0.859400i	$-0.329162\pi$
$829$	9.55683e18	1.02261	0.511304	+	0.859400i	$0.329162\pi$
$830$	0	0
$831$	−3.73527e18	−0.393473
$832$	0	0
$833$	−1.35577e19	−1.40603
$834$	0	0
$835$	−1.13768e18	−0.116160
$836$	0	0
$837$	9.87991e18	0.993201
$838$	0	0
$839$	4.47785e18	0.443218	0.221609	−	0.975136i	$-0.428869\pi$
$839$	4.47785e18	0.443218	0.221609	+	0.975136i	$0.428869\pi$
$840$	0	0
$841$	−2.98376e18	−0.290797
$842$	0	0
$843$	1.11223e19	1.06737
$844$	0	0
$845$	−8.67858e17	−0.0820125
$846$	0	0
$847$	8.01580e17	0.0745941
$848$	0	0
$849$	2.12768e19	1.94988
$850$	0	0
$851$	1.12242e19	1.01301
$852$	0	0
$853$	1.01502e19	0.902208	0.451104	−	0.892471i	$-0.351030\pi$
$853$	1.01502e19	0.902208	0.451104	+	0.892471i	$0.351030\pi$
$854$	0	0
$855$	2.37936e17	0.0208296
$856$	0	0
$857$	−1.88859e19	−1.62841	−0.814203	−	0.580580i	$-0.802826\pi$
$857$	−1.88859e19	−1.62841	−0.814203	+	0.580580i	$0.802826\pi$
$858$	0	0
$859$	−6.92193e18	−0.587857	−0.293929	−	0.955827i	$-0.594963\pi$
$859$	−6.92193e18	−0.587857	−0.293929	+	0.955827i	$0.594963\pi$
$860$	0	0
$861$	−7.50206e17	−0.0627567
$862$	0	0
$863$	−1.51401e19	−1.24755	−0.623776	−	0.781603i	$-0.714402\pi$
$863$	−1.51401e19	−1.24755	−0.623776	+	0.781603i	$0.714402\pi$
$864$	0	0
$865$	−2.57953e17	−0.0209381
$866$	0	0
$867$	1.53760e19	1.22947
$868$	0	0
$869$	−3.83784e16	−0.00302314
$870$	0	0
$871$	1.17933e19	0.915203
$872$	0	0
$873$	−2.96290e18	−0.226529
$874$	0	0
$875$	−2.28371e17	−0.0172024
$876$	0	0
$877$	−1.72930e19	−1.28343	−0.641715	−	0.766943i	$-0.721777\pi$
$877$	−1.72930e19	−1.28343	−0.641715	+	0.766943i	$0.721777\pi$
$878$	0	0
$879$	3.04791e19	2.22882
$880$	0	0
$881$	9.78427e18	0.704993	0.352497	−	0.935813i	$-0.385333\pi$
$881$	9.78427e18	0.704993	0.352497	+	0.935813i	$0.385333\pi$
$882$	0	0
$883$	5.79779e18	0.411640	0.205820	−	0.978590i	$-0.434014\pi$
$883$	5.79779e18	0.411640	0.205820	+	0.978590i	$0.434014\pi$
$884$	0	0
$885$	−2.59641e18	−0.181653
$886$	0	0
$887$	−1.09014e19	−0.751586	−0.375793	−	0.926704i	$-0.622630\pi$
$887$	−1.09014e19	−0.751586	−0.375793	+	0.926704i	$0.622630\pi$
$888$	0	0
$889$	1.86336e18	0.126601
$890$	0	0
$891$	−5.59348e17	−0.0374522
$892$	0	0
$893$	4.04531e18	0.266942
$894$	0	0
$895$	−7.02748e17	−0.0457036
$896$	0	0
$897$	2.58402e19	1.65632
$898$	0	0
$899$	−2.21466e19	−1.39916
$900$	0	0
$901$	1.31724e19	0.820266
$902$	0	0
$903$	−1.90408e18	−0.116873
$904$	0	0
$905$	−2.46306e17	−0.0149025
$906$	0	0
$907$	−2.19849e19	−1.31122	−0.655612	−	0.755098i	$-0.727589\pi$
$907$	−2.19849e19	−1.31122	−0.655612	+	0.755098i	$0.727589\pi$
$908$	0	0
$909$	−8.60568e18	−0.505964
$910$	0	0
$911$	2.14120e19	1.24104	0.620522	−	0.784189i	$-0.286921\pi$
$911$	2.14120e19	1.24104	0.620522	+	0.784189i	$0.286921\pi$
$912$	0	0
$913$	−8.67472e17	−0.0495673
$914$	0	0
$915$	2.53558e18	0.142837
$916$	0	0
$917$	1.45369e18	0.0807365
$918$	0	0
$919$	−4.39133e18	−0.240461	−0.120231	−	0.992746i	$-0.538363\pi$
$919$	−4.39133e18	−0.240461	−0.120231	+	0.992746i	$0.538363\pi$
$920$	0	0
$921$	2.39477e19	1.29293
$922$	0	0
$923$	−7.27058e18	−0.387042
$924$	0	0
$925$	1.84812e19	0.970083
$926$	0	0
$927$	8.94419e18	0.462938
$928$	0	0
$929$	−5.87144e18	−0.299669	−0.149835	−	0.988711i	$-0.547874\pi$
$929$	−5.87144e18	−0.299669	−0.149835	+	0.988711i	$0.547874\pi$
$930$	0	0
$931$	6.90088e18	0.347321
$932$	0	0
$933$	3.98728e19	1.97900
$934$	0	0
$935$	−1.00399e17	−0.00491420
$936$	0	0
$937$	2.52648e18	0.121958	0.0609788	−	0.998139i	$-0.480578\pi$
$937$	2.52648e18	0.121958	0.0609788	+	0.998139i	$0.480578\pi$
$938$	0	0
$939$	−3.79322e19	−1.80585
$940$	0	0
$941$	−1.11075e19	−0.521537	−0.260769	−	0.965401i	$-0.583976\pi$
$941$	−1.11075e19	−0.521537	−0.260769	+	0.965401i	$0.583976\pi$
$942$	0	0
$943$	−1.51958e19	−0.703718
$944$	0	0
$945$	−1.13331e17	−0.00517658
$946$	0	0
$947$	−2.32105e19	−1.04570	−0.522852	−	0.852423i	$-0.675132\pi$
$947$	−2.32105e19	−1.04570	−0.522852	+	0.852423i	$0.675132\pi$
$948$	0	0
$949$	2.44470e19	1.08641
$950$	0	0
$951$	−6.85554e18	−0.300517
$952$	0	0
$953$	−2.94491e19	−1.27341	−0.636704	−	0.771108i	$-0.719703\pi$
$953$	−2.94491e19	−1.27341	−0.636704	+	0.771108i	$0.719703\pi$
$954$	0	0
$955$	1.27399e18	0.0543430
$956$	0	0
$957$	7.37973e17	0.0310537
$958$	0	0
$959$	−4.18348e17	−0.0173667
$960$	0	0
$961$	4.29838e19	1.76037
$962$	0	0
$963$	2.41899e19	0.977378
$964$	0	0
$965$	1.31482e18	0.0524130
$966$	0	0
$967$	−1.64375e19	−0.646493	−0.323247	−	0.946315i	$-0.604774\pi$
$967$	−1.64375e19	−0.646493	−0.323247	+	0.946315i	$0.604774\pi$
$968$	0	0
$969$	−1.56594e19	−0.607676
$970$	0	0
$971$	1.43874e19	0.550880	0.275440	−	0.961318i	$-0.411176\pi$
$971$	1.43874e19	0.550880	0.275440	+	0.961318i	$0.411176\pi$
$972$	0	0
$973$	−2.66046e17	−0.0100513
$974$	0	0
$975$	4.25470e19	1.58613
$976$	0	0
$977$	1.13141e19	0.416203	0.208102	−	0.978107i	$-0.433272\pi$
$977$	1.13141e19	0.416203	0.208102	+	0.978107i	$0.433272\pi$
$978$	0	0
$979$	−8.04577e17	−0.0292065
$980$	0	0
$981$	3.54444e18	0.126970
$982$	0	0
$983$	−1.38897e19	−0.491016	−0.245508	−	0.969395i	$-0.578955\pi$
$983$	−1.38897e19	−0.491016	−0.245508	+	0.969395i	$0.578955\pi$
$984$	0	0
$985$	−1.86836e18	−0.0651815
$986$	0	0
$987$	2.03941e18	0.0702172
$988$	0	0
$989$	−3.85683e19	−1.31055
$990$	0	0
$991$	−4.80221e19	−1.61050	−0.805252	−	0.592932i	$-0.797970\pi$
$991$	−4.80221e19	−1.61050	−0.805252	+	0.592932i	$0.797970\pi$
$992$	0	0
$993$	3.06469e19	1.01442
$994$	0	0
$995$	5.25464e18	0.171669
$996$	0	0
$997$	−8.83733e17	−0.0284972	−0.0142486	−	0.999898i	$-0.504536\pi$
$997$	−8.83733e17	−0.0284972	−0.0142486	+	0.999898i	$0.504536\pi$
$998$	0	0
$999$	1.84680e19	0.587820

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.14.a.m.1.3		3
4.3	odd	2		64.14.a.n.1.1		3
8.3	odd	2		32.14.a.c.1.3	✓	3
8.5	even	2		32.14.a.d.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.14.a.c.1.3	✓	3	8.3	odd	2
32.14.a.d.1.1	yes	3	8.5	even	2
64.14.a.m.1.3		3	1.1	even	1	trivial
64.14.a.n.1.1		3	4.3	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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q+1553.74 q^{3} -4052.28 q^{5} -23239.8 q^{7} +819790. q^{9} +176072. q^{11} -2.27385e7 q^{13} -6.29619e6 q^{15} +1.40715e8 q^{17} -7.16239e7 q^{19} -3.61086e7 q^{21} -7.31400e8 q^{23} -1.20428e9 q^{25} -1.20342e9 q^{27} +2.69757e9 q^{29} -8.20983e9 q^{31} +2.73570e8 q^{33} +9.41740e7 q^{35} -1.53462e10 q^{37} -3.53298e10 q^{39} +2.07764e10 q^{41} +5.27322e10 q^{43} -3.32202e9 q^{45} -5.64800e10 q^{47} -9.63489e10 q^{49} +2.18635e11 q^{51} +9.36107e10 q^{53} -7.13491e8 q^{55} -1.11285e11 q^{57} +4.12378e11 q^{59} -4.02716e11 q^{61} -1.90517e10 q^{63} +9.21428e10 q^{65} -5.18649e11 q^{67} -1.13641e12 q^{69} +3.19747e11 q^{71} -1.07513e12 q^{73} -1.87114e12 q^{75} -4.09186e9 q^{77} -2.17970e11 q^{79} -3.17682e12 q^{81} -4.92681e12 q^{83} -5.70216e11 q^{85} +4.19132e12 q^{87} -4.56960e12 q^{89} +5.28438e11 q^{91} -1.27560e13 q^{93} +2.90240e11 q^{95} -3.61421e12 q^{97} +1.44342e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 520 q^{3} - 11594 q^{5} - 96304 q^{7} - 196793 q^{9} + 1923816 q^{11} + 11211006 q^{13} + 10944368 q^{15} - 16892538 q^{17} + 74665048 q^{19} + 115146880 q^{21} - 486642576 q^{23} + 442696461 q^{25}+ \cdots + 2082116453576 q^{99}+O(q^{100}) \) 3 * q - 520 * q^3 - 11594 * q^5 - 96304 * q^7 - 196793 * q^9 + 1923816 * q^11 + 11211006 * q^13 + 10944368 * q^15 - 16892538 * q^17 + 74665048 * q^19 + 115146880 * q^21 - 486642576 * q^23 + 442696461 * q^25 + 3111524144 * q^27 - 1747468946 * q^29 - 6471356864 * q^31 - 2901436352 * q^33 + 32905782432 * q^35 + 1189828278 * q^37 - 73495249232 * q^39 - 12181275970 * q^41 + 151151885096 * q^43 - 19024866434 * q^45 - 273552663072 * q^47 - 26716022709 * q^49 + 361685604336 * q^51 - 151381970362 * q^53 - 594621149104 * q^55 - 253183022656 * q^57 + 534333367560 * q^59 - 598414222482 * q^61 - 138475736176 * q^63 - 1313304118564 * q^65 - 703511405320 * q^67 - 1246094112384 * q^69 + 1867178467152 * q^71 - 2138161799730 * q^73 - 3649896495992 * q^75 - 4774654840448 * q^77 + 6521880375328 * q^79 - 6029061199109 * q^81 - 5725896739752 * q^83 - 8753386407828 * q^85 + 9313876797488 * q^87 - 11930629520834 * q^89 - 10949018828768 * q^91 - 15654956180992 * q^93 + 3955366679088 * q^95 - 23076828823050 * q^97 + 2082116453576 * q^99

Introduction

L-functions

Modular forms

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