LMFDB - Embedded newform 144.14.a.t.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,14,Mod(1,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	144.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-50256,0,-14532] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$154.412537691$
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} - 14629x - 625725 $ x^3 - 14629*x - 625725
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-85.5039$ of defining polynomial
Character	$\chi$	$=$	144.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+28326.9 q^{5} -120030. q^{7} +2.03411e6 q^{11} -6.77159e6 q^{13} -4.68096e7 q^{17} +2.37852e8 q^{19} -3.19781e8 q^{23} -4.18292e8 q^{25} -6.56256e8 q^{29} +2.79615e7 q^{31} -3.40007e9 q^{35} +1.45690e10 q^{37} -2.93648e10 q^{41} -8.73722e9 q^{43} -5.43201e10 q^{47} -8.24818e10 q^{49} -6.14513e10 q^{53} +5.76198e10 q^{55} +2.17020e11 q^{59} +3.22228e10 q^{61} -1.91818e11 q^{65} -3.18737e11 q^{67} +1.22188e12 q^{71} +7.02020e11 q^{73} -2.44153e11 q^{77} +5.01365e11 q^{79} +3.38646e12 q^{83} -1.32597e12 q^{85} -6.98143e12 q^{89} +8.12793e11 q^{91} +6.73761e12 q^{95} -5.81870e12 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 50256 q^{5} - 14532 q^{7} + 32448 q^{11} + 12923574 q^{13} + 9909216 q^{17} - 356030136 q^{19} + 253392000 q^{23} + 1405874241 q^{25} + 2200998480 q^{29} + 2967736956 q^{31} + 13208644032 q^{35} - 20166914670 q^{37}+ \cdots - 1700880850134 q^{97}+O(q^{100}) $ 3 * q - 50256 * q^5 - 14532 * q^7 + 32448 * q^11 + 12923574 * q^13 + 9909216 * q^17 - 356030136 * q^19 + 253392000 * q^23 + 1405874241 * q^25 + 2200998480 * q^29 + 2967736956 * q^31 + 13208644032 * q^35 - 20166914670 * q^37 + 31161545760 * q^41 + 9538966728 * q^43 + 111693895296 * q^47 + 94963151355 * q^49 - 33679447728 * q^53 - 191082451968 * q^55 + 299191569024 * q^59 - 72974149830 * q^61 - 1350236874144 * q^65 + 651715795824 * q^67 + 1331692113408 * q^71 - 1221222435774 * q^73 - 6117252995328 * q^77 - 661902595332 * q^79 + 6970892668992 * q^83 + 2740394122752 * q^85 - 15287064432192 * q^89 - 4923933166536 * q^91 + 32496957393024 * q^95 - 1700880850134 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	28326.9	0.810762	0.405381	−	0.914148i	$-0.367139\pi$
$5$	28326.9	0.810762	0.405381	+	0.914148i	$0.367139\pi$
$6$	0	0
$7$	−120030.	−0.385613	−0.192807	−	0.981237i	$-0.561759\pi$
$7$	−120030.	−0.385613	−0.192807	+	0.981237i	$0.561759\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.03411e6	0.346195	0.173098	−	0.984905i	$-0.444622\pi$
$11$	2.03411e6	0.346195	0.173098	+	0.984905i	$0.444622\pi$
$12$	0	0
$13$	−6.77159e6	−0.389098	−0.194549	−	0.980893i	$-0.562324\pi$
$13$	−6.77159e6	−0.389098	−0.194549	+	0.980893i	$0.562324\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.68096e7	−0.470345	−0.235173	−	0.971954i	$-0.575566\pi$
$17$	−4.68096e7	−0.470345	−0.235173	+	0.971954i	$0.575566\pi$
$18$	0	0
$19$	2.37852e8	1.15987	0.579935	−	0.814663i	$-0.303078\pi$
$19$	2.37852e8	1.15987	0.579935	+	0.814663i	$0.303078\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.19781e8	−0.450424	−0.225212	−	0.974310i	$-0.572307\pi$
$23$	−3.19781e8	−0.450424	−0.225212	+	0.974310i	$0.572307\pi$
$24$	0	0
$25$	−4.18292e8	−0.342665
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.56256e8	−0.204874	−0.102437	−	0.994740i	$-0.532664\pi$
$29$	−6.56256e8	−0.204874	−0.102437	+	0.994740i	$0.532664\pi$
$30$	0	0
$31$	2.79615e7	0.00565861	0.00282931	−	0.999996i	$-0.499099\pi$
$31$	2.79615e7	0.00565861	0.00282931	+	0.999996i	$0.499099\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.40007e9	−0.312641
$36$	0	0
$37$	1.45690e10	0.933507	0.466753	−	0.884387i	$-0.345424\pi$
$37$	1.45690e10	0.933507	0.466753	+	0.884387i	$0.345424\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.93648e10	−0.965455	−0.482727	−	0.875771i	$-0.660354\pi$
$41$	−2.93648e10	−0.965455	−0.482727	+	0.875771i	$0.660354\pi$
$42$	0	0
$43$	−8.73722e9	−0.210779	−0.105390	−	0.994431i	$-0.533609\pi$
$43$	−8.73722e9	−0.210779	−0.105390	+	0.994431i	$0.533609\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.43201e10	−0.735063	−0.367532	−	0.930011i	$-0.619797\pi$
$47$	−5.43201e10	−0.735063	−0.367532	+	0.930011i	$0.619797\pi$
$48$	0	0
$49$	−8.24818e10	−0.851302
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.14513e10	−0.380836	−0.190418	−	0.981703i	$-0.560984\pi$
$53$	−6.14513e10	−0.380836	−0.190418	+	0.981703i	$0.560984\pi$
$54$	0	0
$55$	5.76198e10	0.280682
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.17020e11	0.669826	0.334913	−	0.942249i	$-0.391293\pi$
$59$	2.17020e11	0.669826	0.334913	+	0.942249i	$0.391293\pi$
$60$	0	0
$61$	3.22228e10	0.0800791	0.0400396	−	0.999198i	$-0.487252\pi$
$61$	3.22228e10	0.0800791	0.0400396	+	0.999198i	$0.487252\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.91818e11	−0.315466
$66$	0	0
$67$	−3.18737e11	−0.430473	−0.215237	−	0.976562i	$-0.569052\pi$
$67$	−3.18737e11	−0.430473	−0.215237	+	0.976562i	$0.569052\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.22188e12	1.13201	0.566006	−	0.824401i	$-0.308488\pi$
$71$	1.22188e12	1.13201	0.566006	+	0.824401i	$0.308488\pi$
$72$	0	0
$73$	7.02020e11	0.542939	0.271469	−	0.962447i	$-0.412490\pi$
$73$	7.02020e11	0.542939	0.271469	+	0.962447i	$0.412490\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.44153e11	−0.133498
$78$	0	0
$79$	5.01365e11	0.232048	0.116024	−	0.993246i	$-0.462985\pi$
$79$	5.01365e11	0.232048	0.116024	+	0.993246i	$0.462985\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.38646e12	1.13694	0.568471	−	0.822703i	$-0.307535\pi$
$83$	3.38646e12	1.13694	0.568471	+	0.822703i	$0.307535\pi$
$84$	0	0
$85$	−1.32597e12	−0.381338
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.98143e12	−1.48905	−0.744525	−	0.667595i	$-0.767324\pi$
$89$	−6.98143e12	−1.48905	−0.744525	+	0.667595i	$0.767324\pi$
$90$	0	0
$91$	8.12793e11	0.150041
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.73761e12	0.940378
$96$	0	0
$97$	−5.81870e12	−0.709266	−0.354633	−	0.935006i	$-0.615394\pi$
$97$	−5.81870e12	−0.709266	−0.354633	+	0.935006i	$0.615394\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.57262e13	−1.47413	−0.737063	−	0.675824i	$-0.763788\pi$
$101$	−1.57262e13	−1.47413	−0.737063	+	0.675824i	$0.763788\pi$
$102$	0	0
$103$	−7.12837e12	−0.588231	−0.294116	−	0.955770i	$-0.595025\pi$
$103$	−7.12837e12	−0.588231	−0.294116	+	0.955770i	$0.595025\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.72575e13	1.11169	0.555845	−	0.831286i	$-0.312395\pi$
$107$	1.72575e13	1.11169	0.555845	+	0.831286i	$0.312395\pi$
$108$	0	0
$109$	1.45032e13	0.828310	0.414155	−	0.910206i	$-0.364077\pi$
$109$	1.45032e13	0.828310	0.414155	+	0.910206i	$0.364077\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.81407e13	−1.27153	−0.635763	−	0.771884i	$-0.719314\pi$
$113$	−2.81407e13	−1.27153	−0.635763	+	0.771884i	$0.719314\pi$
$114$	0	0
$115$	−9.05838e12	−0.365186
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.61855e12	0.181371
$120$	0	0
$121$	−3.03851e13	−0.880149
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.64276e13	−1.08858
$126$	0	0
$127$	−1.74578e12	−0.0369203	−0.0184601	−	0.999830i	$-0.505876\pi$
$127$	−1.74578e12	−0.0369203	−0.0184601	+	0.999830i	$0.505876\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.01654e14	1.75736	0.878679	−	0.477413i	$-0.158426\pi$
$131$	1.01654e14	1.75736	0.878679	+	0.477413i	$0.158426\pi$
$132$	0	0
$133$	−2.85494e13	−0.447261
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.15495e13	0.407670	0.203835	−	0.979005i	$-0.434659\pi$
$137$	3.15495e13	0.407670	0.203835	+	0.979005i	$0.434659\pi$
$138$	0	0
$139$	4.50845e12	0.0530189	0.0265095	−	0.999649i	$-0.491561\pi$
$139$	4.50845e12	0.0530189	0.0265095	+	0.999649i	$0.491561\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.37741e13	−0.134704
$144$	0	0
$145$	−1.85897e13	−0.166104
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.55776e13	0.565825	0.282913	−	0.959146i	$-0.408699\pi$
$149$	7.55776e13	0.565825	0.282913	+	0.959146i	$0.408699\pi$
$150$	0	0
$151$	−1.50063e14	−1.03020	−0.515101	−	0.857129i	$-0.672246\pi$
$151$	−1.50063e14	−1.03020	−0.515101	+	0.857129i	$0.672246\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.92062e11	0.00458779
$156$	0	0
$157$	−1.50997e14	−0.804673	−0.402336	−	0.915492i	$-0.631802\pi$
$157$	−1.50997e14	−0.804673	−0.402336	+	0.915492i	$0.631802\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.83832e13	0.173689
$162$	0	0
$163$	−3.78745e14	−1.58171	−0.790857	−	0.612001i	$-0.790365\pi$
$163$	−3.78745e14	−1.58171	−0.790857	+	0.612001i	$0.790365\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.53136e14	−1.61648	−0.808241	−	0.588851i	$-0.799580\pi$
$167$	−4.53136e14	−1.61648	−0.808241	+	0.588851i	$0.799580\pi$
$168$	0	0
$169$	−2.57021e14	−0.848603
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.63733e14	0.464340	0.232170	−	0.972675i	$-0.425417\pi$
$173$	1.63733e14	0.464340	0.232170	+	0.972675i	$0.425417\pi$
$174$	0	0
$175$	5.02076e13	0.132136
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.38157e14	−0.541151	−0.270576	−	0.962699i	$-0.587214\pi$
$179$	−2.38157e14	−0.541151	−0.270576	+	0.962699i	$0.587214\pi$
$180$	0	0
$181$	−2.77880e14	−0.587418	−0.293709	−	0.955895i	$-0.594890\pi$
$181$	−2.77880e14	−0.587418	−0.293709	+	0.955895i	$0.594890\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.12693e14	0.756852
$186$	0	0
$187$	−9.52157e13	−0.162831
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.06267e15	−1.58374	−0.791869	−	0.610691i	$-0.790892\pi$
$191$	−1.06267e15	−1.58374	−0.791869	+	0.610691i	$0.790892\pi$
$192$	0	0
$193$	−8.30510e14	−1.15670	−0.578352	−	0.815787i	$-0.696304\pi$
$193$	−8.30510e14	−1.15670	−0.578352	+	0.815787i	$0.696304\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.87432e14	0.959804	0.479902	−	0.877322i	$-0.340672\pi$
$197$	7.87432e14	0.959804	0.479902	+	0.877322i	$0.340672\pi$
$198$	0	0
$199$	−1.46950e15	−1.67735	−0.838676	−	0.544631i	$-0.816670\pi$
$199$	−1.46950e15	−1.67735	−0.838676	+	0.544631i	$0.816670\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.87704e13	0.0790021
$204$	0	0
$205$	−8.31812e14	−0.782754
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.83817e14	0.401541
$210$	0	0
$211$	−1.42542e15	−1.11201	−0.556004	−	0.831180i	$-0.687666\pi$
$211$	−1.42542e15	−1.11201	−0.556004	+	0.831180i	$0.687666\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.47498e14	−0.170892
$216$	0	0
$217$	−3.35622e12	−0.00218204
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.16975e14	0.183010
$222$	0	0
$223$	−1.21296e15	−0.660486	−0.330243	−	0.943896i	$-0.607131\pi$
$223$	−1.21296e15	−0.660486	−0.330243	+	0.943896i	$0.607131\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.72976e15	0.839109	0.419554	−	0.907730i	$-0.362186\pi$
$227$	1.72976e15	0.839109	0.419554	+	0.907730i	$0.362186\pi$
$228$	0	0
$229$	−4.07334e15	−1.86647	−0.933233	−	0.359272i	$-0.883025\pi$
$229$	−4.07334e15	−1.86647	−0.933233	+	0.359272i	$0.883025\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.68044e14	−0.150690	−0.0753452	−	0.997158i	$-0.524006\pi$
$233$	−3.68044e14	−0.150690	−0.0753452	+	0.997158i	$0.524006\pi$
$234$	0	0
$235$	−1.53872e15	−0.595961
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.45486e15	0.504935	0.252467	−	0.967605i	$-0.418758\pi$
$239$	1.45486e15	0.504935	0.252467	+	0.967605i	$0.418758\pi$
$240$	0	0
$241$	−2.29665e15	−0.755064	−0.377532	−	0.925997i	$-0.623227\pi$
$241$	−2.29665e15	−0.755064	−0.377532	+	0.925997i	$0.623227\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.33645e15	−0.690204
$246$	0	0
$247$	−1.61064e15	−0.451303
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.70634e14	−0.169280	−0.0846401	−	0.996412i	$-0.526974\pi$
$251$	−6.70634e14	−0.169280	−0.0846401	+	0.996412i	$0.526974\pi$
$252$	0	0
$253$	−6.50468e14	−0.155935
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.94640e15	−1.07084	−0.535420	−	0.844586i	$-0.679847\pi$
$257$	−4.94640e15	−1.07084	−0.535420	+	0.844586i	$0.679847\pi$
$258$	0	0
$259$	−1.74871e15	−0.359973
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.14470e15	1.51762	0.758810	−	0.651313i	$-0.225781\pi$
$263$	8.14470e15	1.51762	0.758810	+	0.651313i	$0.225781\pi$
$264$	0	0
$265$	−1.74072e15	−0.308767
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.23457e15	−1.48603	−0.743014	−	0.669276i	$-0.766604\pi$
$269$	−9.23457e15	−1.48603	−0.743014	+	0.669276i	$0.766604\pi$
$270$	0	0
$271$	−2.35829e15	−0.361657	−0.180828	−	0.983515i	$-0.557878\pi$
$271$	−2.35829e15	−0.361657	−0.180828	+	0.983515i	$0.557878\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.50851e14	−0.118629
$276$	0	0
$277$	3.89172e15	0.517634	0.258817	−	0.965926i	$-0.416667\pi$
$277$	3.89172e15	0.517634	0.258817	+	0.965926i	$0.416667\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.96405e15	−0.237991	−0.118995	−	0.992895i	$-0.537967\pi$
$281$	−1.96405e15	−0.237991	−0.118995	+	0.992895i	$0.537967\pi$
$282$	0	0
$283$	−6.96020e15	−0.805397	−0.402698	−	0.915333i	$-0.631928\pi$
$283$	−6.96020e15	−0.805397	−0.402698	+	0.915333i	$0.631928\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.52465e15	0.372292
$288$	0	0
$289$	−7.71344e15	−0.778775
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.65088e14	−0.0706434	−0.0353217	−	0.999376i	$-0.511246\pi$
$293$	−7.65088e14	−0.0706434	−0.0353217	+	0.999376i	$0.511246\pi$
$294$	0	0
$295$	6.14750e15	0.543069
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.16542e15	0.175259
$300$	0	0
$301$	1.04873e15	0.0812794
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.12771e14	0.0649251
$306$	0	0
$307$	1.41175e16	0.962409	0.481205	−	0.876608i	$-0.340199\pi$
$307$	1.41175e16	0.962409	0.481205	+	0.876608i	$0.340199\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.82245e15	0.552900	0.276450	−	0.961028i	$-0.410842\pi$
$311$	8.82245e15	0.552900	0.276450	+	0.961028i	$0.410842\pi$
$312$	0	0
$313$	4.19155e15	0.251963	0.125981	−	0.992033i	$-0.459792\pi$
$313$	4.19155e15	0.251963	0.125981	+	0.992033i	$0.459792\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.67999e16	1.48336	0.741681	−	0.670753i	$-0.234029\pi$
$317$	2.67999e16	1.48336	0.741681	+	0.670753i	$0.234029\pi$
$318$	0	0
$319$	−1.33489e15	−0.0709263
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.11338e16	−0.545539
$324$	0	0
$325$	2.83250e15	0.133330
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.52004e15	0.283450
$330$	0	0
$331$	9.45658e14	0.0395232	0.0197616	−	0.999805i	$-0.493709\pi$
$331$	9.45658e14	0.0395232	0.0197616	+	0.999805i	$0.493709\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.02882e15	−0.349011
$336$	0	0
$337$	3.74701e16	1.39345	0.696724	−	0.717339i	$-0.254640\pi$
$337$	3.74701e16	1.39345	0.696724	+	0.717339i	$0.254640\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.68767e13	0.00195898
$342$	0	0
$343$	2.15299e16	0.713887
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.14972e16	−1.58359	−0.791793	−	0.610789i	$-0.790852\pi$
$347$	−5.14972e16	−1.58359	−0.791793	+	0.610789i	$0.790852\pi$
$348$	0	0
$349$	3.46424e16	1.02622	0.513112	−	0.858322i	$-0.328493\pi$
$349$	3.46424e16	1.02622	0.513112	+	0.858322i	$0.328493\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.35849e16	0.923866	0.461933	−	0.886915i	$-0.347156\pi$
$353$	3.35849e16	0.923866	0.461933	+	0.886915i	$0.347156\pi$
$354$	0	0
$355$	3.46121e16	0.917792
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.59240e16	−1.13221	−0.566103	−	0.824334i	$-0.691550\pi$
$359$	−4.59240e16	−1.13221	−0.566103	+	0.824334i	$0.691550\pi$
$360$	0	0
$361$	1.45208e16	0.345298
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.98860e16	0.440194
$366$	0	0
$367$	4.79298e16	1.02394	0.511972	−	0.859002i	$-0.328915\pi$
$367$	4.79298e16	1.02394	0.511972	+	0.859002i	$0.328915\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.37599e15	0.146855
$372$	0	0
$373$	1.48431e16	0.285377	0.142688	−	0.989768i	$-0.454425\pi$
$373$	1.48431e16	0.285377	0.142688	+	0.989768i	$0.454425\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.44389e15	0.0797159
$378$	0	0
$379$	2.50551e16	0.434251	0.217125	−	0.976144i	$-0.430332\pi$
$379$	2.50551e16	0.434251	0.217125	+	0.976144i	$0.430332\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.71850e16	−0.440086	−0.220043	−	0.975490i	$-0.570620\pi$
$383$	−2.71850e16	−0.440086	−0.220043	+	0.975490i	$0.570620\pi$
$384$	0	0
$385$	−6.91610e15	−0.108235
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.91123e15	−0.0864979	−0.0432490	−	0.999064i	$-0.513771\pi$
$389$	−5.91123e15	−0.0864979	−0.0432490	+	0.999064i	$0.513771\pi$
$390$	0	0
$391$	1.49688e16	0.211855
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.42021e16	0.188136
$396$	0	0
$397$	4.24747e16	0.544493	0.272247	−	0.962228i	$-0.412233\pi$
$397$	4.24747e16	0.544493	0.272247	+	0.962228i	$0.412233\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.35418e16	−0.162644	−0.0813222	−	0.996688i	$-0.525914\pi$
$401$	−1.35418e16	−0.162644	−0.0813222	+	0.996688i	$0.525914\pi$
$402$	0	0
$403$	−1.89344e14	−0.00220175
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.96348e16	0.323176
$408$	0	0
$409$	1.00108e17	1.05746	0.528732	−	0.848789i	$-0.322668\pi$
$409$	1.00108e17	1.05746	0.528732	+	0.848789i	$0.322668\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.60489e16	−0.258294
$414$	0	0
$415$	9.59278e16	0.921789
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.31352e17	1.18590	0.592948	−	0.805241i	$-0.297964\pi$
$419$	1.31352e17	1.18590	0.592948	+	0.805241i	$0.297964\pi$
$420$	0	0
$421$	5.18481e16	0.453837	0.226918	−	0.973914i	$-0.427135\pi$
$421$	5.18481e16	0.453837	0.226918	+	0.973914i	$0.427135\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.95801e16	0.161171
$426$	0	0
$427$	−3.86770e15	−0.0308796
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.74822e14	0.00131369	0.000656846	−	1.00000i	$-0.499791\pi$
$431$	1.74822e14	0.00131369	0.000656846		1.00000i	$0.499791\pi$
$432$	0	0
$433$	9.52553e16	0.694573	0.347286	−	0.937759i	$-0.387103\pi$
$433$	9.52553e16	0.694573	0.347286	+	0.937759i	$0.387103\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.60606e16	−0.522433
$438$	0	0
$439$	−1.00081e16	−0.0667314	−0.0333657	−	0.999443i	$-0.510623\pi$
$439$	−1.00081e16	−0.0667314	−0.0333657	+	0.999443i	$0.510623\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.91776e17	1.83411	0.917054	−	0.398764i	$-0.130561\pi$
$443$	2.91776e17	1.83411	0.917054	+	0.398764i	$0.130561\pi$
$444$	0	0
$445$	−1.97762e17	−1.20726
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.41123e17	−1.38879	−0.694395	−	0.719594i	$-0.744328\pi$
$449$	−2.41123e17	−1.38879	−0.694395	+	0.719594i	$0.744328\pi$
$450$	0	0
$451$	−5.97311e16	−0.334236
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.30239e16	0.121648
$456$	0	0
$457$	−1.28788e17	−0.661333	−0.330666	−	0.943748i	$-0.607273\pi$
$457$	−1.28788e17	−0.661333	−0.330666	+	0.943748i	$0.607273\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.04488e17	−0.507002	−0.253501	−	0.967335i	$-0.581582\pi$
$461$	−1.04488e17	−0.507002	−0.253501	+	0.967335i	$0.581582\pi$
$462$	0	0
$463$	1.61325e17	0.761071	0.380536	−	0.924766i	$-0.375740\pi$
$463$	1.61325e17	0.761071	0.380536	+	0.924766i	$0.375740\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.49623e16	−0.200580	−0.100290	−	0.994958i	$-0.531977\pi$
$467$	−4.49623e16	−0.200580	−0.100290	+	0.994958i	$0.531977\pi$
$468$	0	0
$469$	3.82580e16	0.165996
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.77724e16	−0.0729708
$474$	0	0
$475$	−9.94918e16	−0.397447
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.13574e16	0.232106	0.116053	−	0.993243i	$-0.462976\pi$
$479$	6.13574e16	0.232106	0.116053	+	0.993243i	$0.462976\pi$
$480$	0	0
$481$	−9.86550e16	−0.363225
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.64825e17	−0.575046
$486$	0	0
$487$	−4.36624e17	−1.48310	−0.741548	−	0.670900i	$-0.765908\pi$
$487$	−4.36624e17	−1.48310	−0.741548	+	0.670900i	$0.765908\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.04800e17	−0.659629	−0.329814	−	0.944046i	$-0.606986\pi$
$491$	−2.04800e17	−0.659629	−0.329814	+	0.944046i	$0.606986\pi$
$492$	0	0
$493$	3.07191e16	0.0963614
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.46663e17	−0.436519
$498$	0	0
$499$	−2.61990e17	−0.759680	−0.379840	−	0.925052i	$-0.624021\pi$
$499$	−2.61990e17	−0.759680	−0.379840	+	0.925052i	$0.624021\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.72135e16	−0.0749191	−0.0374595	−	0.999298i	$-0.511927\pi$
$503$	−2.72135e16	−0.0749191	−0.0374595	+	0.999298i	$0.511927\pi$
$504$	0	0
$505$	−4.45474e17	−1.19517
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.79832e17	0.968113	0.484056	−	0.875037i	$-0.339163\pi$
$509$	3.79832e17	0.968113	0.484056	+	0.875037i	$0.339163\pi$
$510$	0	0
$511$	−8.42634e16	−0.209365
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.01924e17	−0.476915
$516$	0	0
$517$	−1.10493e17	−0.254475
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.33395e17	0.949377	0.474688	−	0.880154i	$-0.342561\pi$
$521$	4.33395e17	0.949377	0.474688	+	0.880154i	$0.342561\pi$
$522$	0	0
$523$	−5.45127e17	−1.16476	−0.582380	−	0.812917i	$-0.697878\pi$
$523$	−5.45127e17	−1.16476	−0.582380	+	0.812917i	$0.697878\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.30887e15	−0.00266150
$528$	0	0
$529$	−4.01777e17	−0.797118
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.98846e17	0.375656
$534$	0	0
$535$	4.88851e17	0.901316
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.67777e17	−0.294717
$540$	0	0
$541$	−2.81001e17	−0.481865	−0.240933	−	0.970542i	$-0.577453\pi$
$541$	−2.81001e17	−0.481865	−0.240933	+	0.970542i	$0.577453\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.10831e17	0.671562
$546$	0	0
$547$	−3.61702e17	−0.577343	−0.288671	−	0.957428i	$-0.593214\pi$
$547$	−3.61702e17	−0.577343	−0.288671	+	0.957428i	$0.593214\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.56092e17	−0.237627
$552$	0	0
$553$	−6.01788e16	−0.0894809
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.22966e17	1.16768	0.583839	−	0.811870i	$-0.301550\pi$
$557$	8.22966e17	1.16768	0.583839	+	0.811870i	$0.301550\pi$
$558$	0	0
$559$	5.91648e16	0.0820138
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.50885e17	−1.25841	−0.629207	−	0.777238i	$-0.716620\pi$
$563$	−9.50885e17	−1.25841	−0.629207	+	0.777238i	$0.716620\pi$
$564$	0	0
$565$	−7.97138e17	−1.03090
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.41130e17	0.915513	0.457756	−	0.889078i	$-0.348653\pi$
$569$	7.41130e17	0.915513	0.457756	+	0.889078i	$0.348653\pi$
$570$	0	0
$571$	−1.40040e18	−1.69089	−0.845446	−	0.534061i	$-0.820665\pi$
$571$	−1.40040e18	−1.69089	−0.845446	+	0.534061i	$0.820665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.33762e17	0.154344
$576$	0	0
$577$	8.63725e16	0.0974390	0.0487195	−	0.998812i	$-0.484486\pi$
$577$	8.63725e16	0.0974390	0.0487195	+	0.998812i	$0.484486\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.06476e17	−0.438420
$582$	0	0
$583$	−1.24998e17	−0.131844
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.59248e17	−0.463340	−0.231670	−	0.972794i	$-0.574419\pi$
$587$	−4.59248e17	−0.463340	−0.231670	+	0.972794i	$0.574419\pi$
$588$	0	0
$589$	6.65072e15	0.00656325
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.28709e17	−0.593737	−0.296869	−	0.954918i	$-0.595942\pi$
$593$	−6.28709e17	−0.593737	−0.296869	+	0.954918i	$0.595942\pi$
$594$	0	0
$595$	1.59156e17	0.147049
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.14761e17	−0.366880	−0.183440	−	0.983031i	$-0.558723\pi$
$599$	−4.14761e17	−0.366880	−0.183440	+	0.983031i	$0.558723\pi$
$600$	0	0
$601$	5.50998e17	0.476943	0.238471	−	0.971150i	$-0.423354\pi$
$601$	5.50998e17	0.476943	0.238471	+	0.971150i	$0.423354\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.60715e17	−0.713591
$606$	0	0
$607$	1.05843e17	0.0858883	0.0429442	−	0.999077i	$-0.486326\pi$
$607$	1.05843e17	0.0858883	0.0429442	+	0.999077i	$0.486326\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.67834e17	0.286011
$612$	0	0
$613$	−2.03011e17	−0.154535	−0.0772673	−	0.997010i	$-0.524619\pi$
$613$	−2.03011e17	−0.154535	−0.0772673	+	0.997010i	$0.524619\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.91382e17	0.285593	0.142796	−	0.989752i	$-0.454391\pi$
$617$	3.91382e17	0.285593	0.142796	+	0.989752i	$0.454391\pi$
$618$	0	0
$619$	−9.67789e17	−0.691499	−0.345749	−	0.938327i	$-0.612375\pi$
$619$	−9.67789e17	−0.691499	−0.345749	+	0.938327i	$0.612375\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.37980e17	0.574198
$624$	0	0
$625$	−8.04537e17	−0.539916
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.81967e17	−0.439071
$630$	0	0
$631$	2.62249e18	1.65395	0.826977	−	0.562235i	$-0.190058\pi$
$631$	2.62249e18	1.65395	0.826977	+	0.562235i	$0.190058\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.94524e16	−0.0299336
$636$	0	0
$637$	5.58533e17	0.331240
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.65719e18	−1.51302	−0.756512	−	0.653979i	$-0.773098\pi$
$641$	−2.65719e18	−1.51302	−0.756512	+	0.653979i	$0.773098\pi$
$642$	0	0
$643$	−3.69318e17	−0.206077	−0.103038	−	0.994677i	$-0.532856\pi$
$643$	−3.69318e17	−0.206077	−0.103038	+	0.994677i	$0.532856\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.81215e18	0.971216	0.485608	−	0.874177i	$-0.338598\pi$
$647$	1.81215e18	0.971216	0.485608	+	0.874177i	$0.338598\pi$
$648$	0	0
$649$	4.41442e17	0.231891
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.85288e18	0.935215	0.467607	−	0.883936i	$-0.345116\pi$
$653$	1.85288e18	0.935215	0.467607	+	0.883936i	$0.345116\pi$
$654$	0	0
$655$	2.87953e18	1.42480
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.11072e18	1.00386	0.501932	−	0.864907i	$-0.332623\pi$
$659$	2.11072e18	1.00386	0.501932	+	0.864907i	$0.332623\pi$
$660$	0	0
$661$	2.13919e18	0.997564	0.498782	−	0.866728i	$-0.333781\pi$
$661$	2.13919e18	0.997564	0.498782	+	0.866728i	$0.333781\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.08715e17	−0.362622
$666$	0	0
$667$	2.09858e17	0.0922800
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.55446e16	0.0277230
$672$	0	0
$673$	2.92051e18	1.21160	0.605801	−	0.795616i	$-0.292853\pi$
$673$	2.92051e18	1.21160	0.605801	+	0.795616i	$0.292853\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.58911e18	−1.83190	−0.915951	−	0.401290i	$-0.868562\pi$
$677$	−4.58911e18	−1.83190	−0.915951	+	0.401290i	$0.868562\pi$
$678$	0	0
$679$	6.98417e17	0.273503
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.19249e18	1.20336	0.601678	−	0.798739i	$-0.294499\pi$
$683$	3.19249e18	1.20336	0.601678	+	0.798739i	$0.294499\pi$
$684$	0	0
$685$	8.93697e17	0.330523
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.16123e17	0.148182
$690$	0	0
$691$	−1.92130e18	−0.671411	−0.335706	−	0.941967i	$-0.608975\pi$
$691$	−1.92130e18	−0.671411	−0.335706	+	0.941967i	$0.608975\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.27710e17	0.0429857
$696$	0	0
$697$	1.37455e18	0.454097
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.87273e18	0.914383	0.457192	−	0.889368i	$-0.348855\pi$
$701$	2.87273e18	0.914383	0.457192	+	0.889368i	$0.348855\pi$
$702$	0	0
$703$	3.46526e18	1.08275
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.88761e18	0.568443
$708$	0	0
$709$	−8.72658e17	−0.258014	−0.129007	−	0.991644i	$-0.541179\pi$
$709$	−8.72658e17	−0.258014	−0.129007	+	0.991644i	$0.541179\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.94156e15	−0.00254877
$714$	0	0
$715$	−3.90178e17	−0.109213
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.26088e18	−0.610296	−0.305148	−	0.952305i	$-0.598706\pi$
$719$	−2.26088e18	−0.610296	−0.305148	+	0.952305i	$0.598706\pi$
$720$	0	0
$721$	8.55617e17	0.226830
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.74507e17	0.0702031
$726$	0	0
$727$	7.74296e18	1.94506	0.972530	−	0.232776i	$-0.0747808\pi$
$727$	7.74296e18	1.94506	0.972530	+	0.232776i	$0.0747808\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.08986e17	0.0991391
$732$	0	0
$733$	1.79013e18	0.426293	0.213146	−	0.977020i	$-0.431629\pi$
$733$	1.79013e18	0.426293	0.213146	+	0.977020i	$0.431629\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.48345e17	−0.149028
$738$	0	0
$739$	−4.22852e18	−0.954991	−0.477495	−	0.878634i	$-0.658455\pi$
$739$	−4.22852e18	−0.954991	−0.477495	+	0.878634i	$0.658455\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.04422e18	−1.53605	−0.768025	−	0.640420i	$-0.778760\pi$
$743$	−7.04422e18	−1.53605	−0.768025	+	0.640420i	$0.778760\pi$
$744$	0	0
$745$	2.14088e18	0.458749
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.07142e18	−0.428683
$750$	0	0
$751$	8.87613e18	1.80536	0.902680	−	0.430312i	$-0.141596\pi$
$751$	8.87613e18	1.80536	0.902680	+	0.430312i	$0.141596\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.25080e18	−0.835249
$756$	0	0
$757$	4.08129e18	0.788268	0.394134	−	0.919053i	$-0.371045\pi$
$757$	4.08129e18	0.788268	0.394134	+	0.919053i	$0.371045\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.13871e18	1.51899	0.759496	−	0.650512i	$-0.225446\pi$
$761$	8.13871e18	1.51899	0.759496	+	0.650512i	$0.225446\pi$
$762$	0	0
$763$	−1.74082e18	−0.319408
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.46957e18	−0.260628
$768$	0	0
$769$	−3.31598e18	−0.578216	−0.289108	−	0.957296i	$-0.593359\pi$
$769$	−3.31598e18	−0.578216	−0.289108	+	0.957296i	$0.593359\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.30481e17	−0.0219979	−0.0109990	−	0.999940i	$-0.503501\pi$
$773$	−1.30481e17	−0.0219979	−0.0109990	+	0.999940i	$0.503501\pi$
$774$	0	0
$775$	−1.16961e16	−0.00193901
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.98449e18	−1.11980
$780$	0	0
$781$	2.48544e18	0.391897
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.27726e18	−0.652398
$786$	0	0
$787$	−8.54302e18	−1.28167	−0.640834	−	0.767679i	$-0.721411\pi$
$787$	−8.54302e18	−1.28167	−0.640834	+	0.767679i	$0.721411\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.37773e18	0.490317
$792$	0	0
$793$	−2.18199e17	−0.0311586
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.33651e19	1.84711	0.923554	−	0.383468i	$-0.125270\pi$
$797$	1.33651e19	1.84711	0.923554	+	0.383468i	$0.125270\pi$
$798$	0	0
$799$	2.54270e18	0.345734
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.42798e18	0.187963
$804$	0	0
$805$	1.08728e18	0.140821
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.28151e18	0.913180	0.456590	−	0.889677i	$-0.349071\pi$
$809$	7.28151e18	0.913180	0.456590	+	0.889677i	$0.349071\pi$
$810$	0	0
$811$	−5.91711e18	−0.730254	−0.365127	−	0.930958i	$-0.618974\pi$
$811$	−5.91711e18	−0.730254	−0.365127	+	0.930958i	$0.618974\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.07287e19	−1.28239
$816$	0	0
$817$	−2.07817e18	−0.244477
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.29881e18	0.148018	0.0740092	−	0.997258i	$-0.476421\pi$
$821$	1.29881e18	0.148018	0.0740092	+	0.997258i	$0.476421\pi$
$822$	0	0
$823$	1.14772e19	1.28747	0.643735	−	0.765248i	$-0.277384\pi$
$823$	1.14772e19	1.28747	0.643735	+	0.765248i	$0.277384\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.30445e18	−0.141789	−0.0708945	−	0.997484i	$-0.522585\pi$
$827$	−1.30445e18	−0.141789	−0.0708945	+	0.997484i	$0.522585\pi$
$828$	0	0
$829$	−3.67389e18	−0.393117	−0.196558	−	0.980492i	$-0.562977\pi$
$829$	−3.67389e18	−0.393117	−0.196558	+	0.980492i	$0.562977\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.86094e18	0.400406
$834$	0	0
$835$	−1.28359e19	−1.31058
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.41441e19	1.39998	0.699991	−	0.714151i	$-0.253187\pi$
$839$	1.41441e19	1.39998	0.699991	+	0.714151i	$0.253187\pi$
$840$	0	0
$841$	−9.82996e18	−0.958027
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.28059e18	−0.688015
$846$	0	0
$847$	3.64712e18	0.339397
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.65887e18	−0.420474
$852$	0	0
$853$	−1.51294e19	−1.34479	−0.672394	−	0.740194i	$-0.734734\pi$
$853$	−1.51294e19	−1.34479	−0.672394	+	0.740194i	$0.734734\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.46368e18	−0.212427	−0.106213	−	0.994343i	$-0.533873\pi$
$857$	−2.46368e18	−0.212427	−0.106213	+	0.994343i	$0.533873\pi$
$858$	0	0
$859$	−1.50879e19	−1.28136	−0.640682	−	0.767806i	$-0.721348\pi$
$859$	−1.50879e19	−1.28136	−0.640682	+	0.767806i	$0.721348\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.22921e19	1.01288	0.506438	−	0.862276i	$-0.330962\pi$
$863$	1.22921e19	1.01288	0.506438	+	0.862276i	$0.330962\pi$
$864$	0	0
$865$	4.63804e18	0.376469
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.01983e18	0.0803339
$870$	0	0
$871$	2.15835e18	0.167496
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.57270e18	0.419772
$876$	0	0
$877$	−1.43823e19	−1.06741	−0.533706	−	0.845670i	$-0.679201\pi$
$877$	−1.43823e19	−1.06741	−0.533706	+	0.845670i	$0.679201\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.57727e19	−1.13648	−0.568239	−	0.822863i	$-0.692375\pi$
$881$	−1.57727e19	−1.13648	−0.568239	+	0.822863i	$0.692375\pi$
$882$	0	0
$883$	−1.54911e19	−1.09986	−0.549929	−	0.835211i	$-0.685345\pi$
$883$	−1.54911e19	−1.09986	−0.549929	+	0.835211i	$0.685345\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.71721e18	−0.532055	−0.266028	−	0.963965i	$-0.585711\pi$
$887$	−7.71721e18	−0.532055	−0.266028	+	0.963965i	$0.585711\pi$
$888$	0	0
$889$	2.09546e17	0.0142370
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.29202e19	−0.852578
$894$	0	0
$895$	−6.74624e18	−0.438745
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.83499e16	−0.00115930
$900$	0	0
$901$	2.87651e18	0.179124
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.87148e18	−0.476256
$906$	0	0
$907$	−2.40141e19	−1.43225	−0.716126	−	0.697971i	$-0.754086\pi$
$907$	−2.40141e19	−1.43225	−0.716126	+	0.697971i	$0.754086\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.69361e19	1.56123	0.780613	−	0.625015i	$-0.214907\pi$
$911$	2.69361e19	1.56123	0.780613	+	0.625015i	$0.214907\pi$
$912$	0	0
$913$	6.88842e18	0.393604
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.22015e19	−0.677661
$918$	0	0
$919$	2.85938e19	1.56574	0.782872	−	0.622183i	$-0.213754\pi$
$919$	2.85938e19	1.56574	0.782872	+	0.622183i	$0.213754\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.27409e18	−0.440463
$924$	0	0
$925$	−6.09409e18	−0.319880
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.43292e19	−0.731340	−0.365670	−	0.930744i	$-0.619160\pi$
$929$	−1.43292e19	−0.731340	−0.365670	+	0.930744i	$0.619160\pi$
$930$	0	0
$931$	−1.96185e19	−0.987400
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.69716e18	−0.132017
$936$	0	0
$937$	−2.72707e19	−1.31640	−0.658201	−	0.752843i	$-0.728682\pi$
$937$	−2.72707e19	−1.31640	−0.658201	+	0.752843i	$0.728682\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.05988e19	0.497649	0.248824	−	0.968549i	$-0.419956\pi$
$941$	1.05988e19	0.497649	0.248824	+	0.968549i	$0.419956\pi$
$942$	0	0
$943$	9.39030e18	0.434864
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.01781e19	1.35962	0.679810	−	0.733389i	$-0.262062\pi$
$947$	3.01781e19	1.35962	0.679810	+	0.733389i	$0.262062\pi$
$948$	0	0
$949$	−4.75379e18	−0.211256
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.26002e19	−0.544845	−0.272422	−	0.962178i	$-0.587825\pi$
$953$	−1.26002e19	−0.544845	−0.272422	+	0.962178i	$0.587825\pi$
$954$	0	0
$955$	−3.01022e19	−1.28403
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.78688e18	−0.157203
$960$	0	0
$961$	−2.44168e19	−0.999968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.35257e19	−0.937812
$966$	0	0
$967$	1.79166e19	0.704668	0.352334	−	0.935874i	$-0.385388\pi$
$967$	1.79166e19	0.704668	0.352334	+	0.935874i	$0.385388\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.93584e17	−0.0227278	−0.0113639	−	0.999935i	$-0.503617\pi$
$971$	−5.93584e17	−0.0227278	−0.0113639	+	0.999935i	$0.503617\pi$
$972$	0	0
$973$	−5.41148e17	−0.0204448
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.28901e19	1.57776	0.788882	−	0.614545i	$-0.210660\pi$
$977$	4.28901e19	1.57776	0.788882	+	0.614545i	$0.210660\pi$
$978$	0	0
$979$	−1.42010e19	−0.515502
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.50402e19	0.885195	0.442598	−	0.896720i	$-0.354057\pi$
$983$	2.50402e19	0.885195	0.442598	+	0.896720i	$0.354057\pi$
$984$	0	0
$985$	2.23055e19	0.778173
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.79399e18	0.0949401
$990$	0	0
$991$	2.30912e19	0.774404	0.387202	−	0.921995i	$-0.373442\pi$
$991$	2.30912e19	0.774404	0.387202	+	0.921995i	$0.373442\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.16263e19	−1.35993
$996$	0	0
$997$	1.25704e19	0.405349	0.202675	−	0.979246i	$-0.435037\pi$
$997$	1.25704e19	0.405349	0.202675	+	0.979246i	$0.435037\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.14.a.t.1.3		3
3.2	odd	2		144.14.a.u.1.1		3
4.3	odd	2		72.14.a.g.1.3	✓	3
12.11	even	2		72.14.a.h.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.14.a.g.1.3	✓	3	4.3	odd	2
72.14.a.h.1.1	yes	3	12.11	even	2
144.14.a.t.1.3		3	1.1	even	1	trivial
144.14.a.u.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

\(f(q)\)	\(=\)	\(q+28326.9 q^{5} -120030. q^{7} +2.03411e6 q^{11} -6.77159e6 q^{13} -4.68096e7 q^{17} +2.37852e8 q^{19} -3.19781e8 q^{23} -4.18292e8 q^{25} -6.56256e8 q^{29} +2.79615e7 q^{31} -3.40007e9 q^{35} +1.45690e10 q^{37} -2.93648e10 q^{41} -8.73722e9 q^{43} -5.43201e10 q^{47} -8.24818e10 q^{49} -6.14513e10 q^{53} +5.76198e10 q^{55} +2.17020e11 q^{59} +3.22228e10 q^{61} -1.91818e11 q^{65} -3.18737e11 q^{67} +1.22188e12 q^{71} +7.02020e11 q^{73} -2.44153e11 q^{77} +5.01365e11 q^{79} +3.38646e12 q^{83} -1.32597e12 q^{85} -6.98143e12 q^{89} +8.12793e11 q^{91} +6.73761e12 q^{95} -5.81870e12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 50256 q^{5} - 14532 q^{7} + 32448 q^{11} + 12923574 q^{13} + 9909216 q^{17} - 356030136 q^{19} + 253392000 q^{23} + 1405874241 q^{25} + 2200998480 q^{29} + 2967736956 q^{31} + 13208644032 q^{35} - 20166914670 q^{37}+ \cdots - 1700880850134 q^{97}+O(q^{100}) \) 3 * q - 50256 * q^5 - 14532 * q^7 + 32448 * q^11 + 12923574 * q^13 + 9909216 * q^17 - 356030136 * q^19 + 253392000 * q^23 + 1405874241 * q^25 + 2200998480 * q^29 + 2967736956 * q^31 + 13208644032 * q^35 - 20166914670 * q^37 + 31161545760 * q^41 + 9538966728 * q^43 + 111693895296 * q^47 + 94963151355 * q^49 - 33679447728 * q^53 - 191082451968 * q^55 + 299191569024 * q^59 - 72974149830 * q^61 - 1350236874144 * q^65 + 651715795824 * q^67 + 1331692113408 * q^71 - 1221222435774 * q^73 - 6117252995328 * q^77 - 661902595332 * q^79 + 6970892668992 * q^83 + 2740394122752 * q^85 - 15287064432192 * q^89 - 4923933166536 * q^91 + 32496957393024 * q^95 - 1700880850134 * q^97

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