LMFDB - Embedded newform 144.12.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,12,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("144.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	144.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:	$110.641418001$
Analytic rank:	$0$
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} - 829x - 6375 $ x^3 - 829*x - 6375
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			1.2
Root			$-23.6534$ 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)

$f(q)$	$=$	$q-1973.64 q^{5} -55801.0 q^{7} +O(q^{10})$	$q-1973.64 q^{5} -55801.0 q^{7} +838258. q^{11} -610726. q^{13} +2.80987e6 q^{17} -1.12790e7 q^{19} -1.99673e7 q^{23} -4.49329e7 q^{25} -1.15370e8 q^{29} -1.51288e7 q^{31} +1.10131e8 q^{35} +2.30525e8 q^{37} +3.56388e8 q^{41} +7.19149e8 q^{43} +7.29690e8 q^{47} +1.13642e9 q^{49} -5.76098e9 q^{53} -1.65442e9 q^{55} +6.44443e9 q^{59} -8.76581e9 q^{61} +1.20535e9 q^{65} -1.88138e9 q^{67} +2.25098e10 q^{71} +2.47385e10 q^{73} -4.67756e10 q^{77} +2.12043e10 q^{79} +3.57526e10 q^{83} -5.54568e9 q^{85} -9.42944e10 q^{89} +3.40791e10 q^{91} +2.22608e10 q^{95} +8.97811e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 1584 q^{5} + 17796 q^{7}+O(q^{10}) $ 3 * q + 1584 * q^5 + 17796 * q^7	$ 3 q + 1584 q^{5} + 17796 q^{7} - 238272 q^{11} - 49458 q^{13} - 766368 q^{17} + 4590936 q^{19} + 13188480 q^{23} + 18034281 q^{25} + 8005200 q^{29} - 10185564 q^{31} + 235065408 q^{35} - 73215510 q^{37} - 1032516960 q^{41} - 28458216 q^{43} + 2840606592 q^{47} - 109741845 q^{49} - 6755828784 q^{53} + 846590976 q^{55} + 15928599936 q^{59} + 5136563970 q^{61} - 37415401248 q^{65} + 6119899728 q^{67} + 58699811328 q^{71} + 2561705778 q^{73} - 86561454336 q^{77} + 17842143972 q^{79} + 134316444096 q^{83} + 43096069632 q^{85} - 152402442048 q^{89} + 56268770664 q^{91} + 207495767424 q^{95} + 106677733482 q^{97}+O(q^{100}) $ 3 * q + 1584 * q^5 + 17796 * q^7 - 238272 * q^11 - 49458 * q^13 - 766368 * q^17 + 4590936 * q^19 + 13188480 * q^23 + 18034281 * q^25 + 8005200 * q^29 - 10185564 * q^31 + 235065408 * q^35 - 73215510 * q^37 - 1032516960 * q^41 - 28458216 * q^43 + 2840606592 * q^47 - 109741845 * q^49 - 6755828784 * q^53 + 846590976 * q^55 + 15928599936 * q^59 + 5136563970 * q^61 - 37415401248 * q^65 + 6119899728 * q^67 + 58699811328 * q^71 + 2561705778 * q^73 - 86561454336 * q^77 + 17842143972 * q^79 + 134316444096 * q^83 + 43096069632 * q^85 - 152402442048 * q^89 + 56268770664 * q^91 + 207495767424 * q^95 + 106677733482 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1973.64	−0.282444	−0.141222	−	0.989978i	$-0.545103\pi$
$5$	−1973.64	−0.282444	−0.141222	+	0.989978i	$0.545103\pi$
$6$	0	0
$7$	−55801.0	−1.25488	−0.627441	−	0.778664i	$-0.715897\pi$
$7$	−55801.0	−1.25488	−0.627441	+	0.778664i	$0.715897\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	838258.	1.56934	0.784671	−	0.619912i	$-0.212832\pi$
$11$	838258.	1.56934	0.784671	+	0.619912i	$0.212832\pi$
$12$	0	0
$13$	−610726.	−0.456203	−0.228101	−	0.973637i	$-0.573252\pi$
$13$	−610726.	−0.456203	−0.228101	+	0.973637i	$0.573252\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.80987e6	0.479974	0.239987	−	0.970776i	$-0.422857\pi$
$17$	2.80987e6	0.479974	0.239987	+	0.970776i	$0.422857\pi$
$18$	0	0
$19$	−1.12790e7	−1.04503	−0.522513	−	0.852631i	$-0.675006\pi$
$19$	−1.12790e7	−1.04503	−0.522513	+	0.852631i	$0.675006\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.99673e7	−0.646867	−0.323434	−	0.946251i	$-0.604837\pi$
$23$	−1.99673e7	−0.646867	−0.323434	+	0.946251i	$0.604837\pi$
$24$	0	0
$25$	−4.49329e7	−0.920225
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.15370e8	−1.04448	−0.522242	−	0.852797i	$-0.674904\pi$
$29$	−1.15370e8	−1.04448	−0.522242	+	0.852797i	$0.674904\pi$
$30$	0	0
$31$	−1.51288e7	−0.0949105	−0.0474553	−	0.998873i	$-0.515111\pi$
$31$	−1.51288e7	−0.0949105	−0.0474553	+	0.998873i	$0.515111\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.10131e8	0.354434
$36$	0	0
$37$	2.30525e8	0.546523	0.273262	−	0.961940i	$-0.411898\pi$
$37$	2.30525e8	0.546523	0.273262	+	0.961940i	$0.411898\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.56388e8	0.480409	0.240205	−	0.970722i	$-0.422785\pi$
$41$	3.56388e8	0.480409	0.240205	+	0.970722i	$0.422785\pi$
$42$	0	0
$43$	7.19149e8	0.746006	0.373003	−	0.927830i	$-0.378328\pi$
$43$	7.19149e8	0.746006	0.373003	+	0.927830i	$0.378328\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.29690e8	0.464088	0.232044	−	0.972705i	$-0.425459\pi$
$47$	7.29690e8	0.464088	0.232044	+	0.972705i	$0.425459\pi$
$48$	0	0
$49$	1.13642e9	0.574727
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.76098e9	−1.89225	−0.946126	−	0.323800i	$-0.895040\pi$
$53$	−5.76098e9	−1.89225	−0.946126	+	0.323800i	$0.895040\pi$
$54$	0	0
$55$	−1.65442e9	−0.443252
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.44443e9	1.17354	0.586771	−	0.809753i	$-0.300399\pi$
$59$	6.44443e9	1.17354	0.586771	+	0.809753i	$0.300399\pi$
$60$	0	0
$61$	−8.76581e9	−1.32886	−0.664428	−	0.747352i	$-0.731325\pi$
$61$	−8.76581e9	−1.32886	−0.664428	+	0.747352i	$0.731325\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.20535e9	0.128852
$66$	0	0
$67$	−1.88138e9	−0.170241	−0.0851207	−	0.996371i	$-0.527128\pi$
$67$	−1.88138e9	−0.170241	−0.0851207	+	0.996371i	$0.527128\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.25098e10	1.48065	0.740323	−	0.672251i	$-0.234673\pi$
$71$	2.25098e10	1.48065	0.740323	+	0.672251i	$0.234673\pi$
$72$	0	0
$73$	2.47385e10	1.39668	0.698342	−	0.715764i	$-0.253921\pi$
$73$	2.47385e10	1.39668	0.698342	+	0.715764i	$0.253921\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.67756e10	−1.96934
$78$	0	0
$79$	2.12043e10	0.775310	0.387655	−	0.921805i	$-0.373285\pi$
$79$	2.12043e10	0.775310	0.387655	+	0.921805i	$0.373285\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.57526e10	0.996271	0.498136	−	0.867099i	$-0.334018\pi$
$83$	3.57526e10	0.996271	0.498136	+	0.867099i	$0.334018\pi$
$84$	0	0
$85$	−5.54568e9	−0.135566
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.42944e10	−1.78995	−0.894975	−	0.446116i	$-0.852807\pi$
$89$	−9.42944e10	−1.78995	−0.894975	+	0.446116i	$0.852807\pi$
$90$	0	0
$91$	3.40791e10	0.572480
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.22608e10	0.295162
$96$	0	0
$97$	8.97811e10	1.06155	0.530775	−	0.847513i	$-0.321901\pi$
$97$	8.97811e10	1.06155	0.530775	+	0.847513i	$0.321901\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.39692e10	0.889647	0.444824	−	0.895618i	$-0.353266\pi$
$101$	9.39692e10	0.889647	0.444824	+	0.895618i	$0.353266\pi$
$102$	0	0
$103$	2.20454e10	0.187376	0.0936880	−	0.995602i	$-0.470134\pi$
$103$	2.20454e10	0.187376	0.0936880	+	0.995602i	$0.470134\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.64712e11	1.13531	0.567655	−	0.823267i	$-0.307851\pi$
$107$	1.64712e11	1.13531	0.567655	+	0.823267i	$0.307851\pi$
$108$	0	0
$109$	−1.04195e11	−0.648634	−0.324317	−	0.945949i	$-0.605134\pi$
$109$	−1.04195e11	−0.648634	−0.324317	+	0.945949i	$0.605134\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.46771e11	−0.749391	−0.374696	−	0.927148i	$-0.622253\pi$
$113$	−1.46771e11	−0.749391	−0.374696	+	0.927148i	$0.622253\pi$
$114$	0	0
$115$	3.94082e10	0.182704
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.56794e11	−0.602310
$120$	0	0
$121$	4.17364e11	1.46284
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.85050e11	0.542357
$126$	0	0
$127$	4.29925e11	1.15471	0.577355	−	0.816493i	$-0.304085\pi$
$127$	4.29925e11	1.15471	0.577355	+	0.816493i	$0.304085\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.10895e11	−1.38349	−0.691743	−	0.722144i	$-0.743157\pi$
$131$	−6.10895e11	−1.38349	−0.691743	+	0.722144i	$0.743157\pi$
$132$	0	0
$133$	6.29381e11	1.31138
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.00365e12	1.77673	0.888363	−	0.459142i	$-0.151843\pi$
$137$	1.00365e12	1.77673	0.888363	+	0.459142i	$0.151843\pi$
$138$	0	0
$139$	1.13039e12	1.84777	0.923887	−	0.382666i	$-0.124994\pi$
$139$	1.13039e12	1.84777	0.923887	+	0.382666i	$0.124994\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.11946e11	−0.715938
$144$	0	0
$145$	2.27698e11	0.295009
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.66255e10	0.0854769	0.0427384	−	0.999086i	$-0.486392\pi$
$149$	7.66255e10	0.0854769	0.0427384	+	0.999086i	$0.486392\pi$
$150$	0	0
$151$	3.08902e11	0.320219	0.160110	−	0.987099i	$-0.448815\pi$
$151$	3.08902e11	0.320219	0.160110	+	0.987099i	$0.448815\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.98587e10	0.0268069
$156$	0	0
$157$	2.17480e12	1.81958	0.909791	−	0.415068i	$-0.136242\pi$
$157$	2.17480e12	1.81958	0.909791	+	0.415068i	$0.136242\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.11419e12	0.811742
$162$	0	0
$163$	1.65478e11	0.112644	0.0563220	−	0.998413i	$-0.482063\pi$
$163$	1.65478e11	0.112644	0.0563220	+	0.998413i	$0.482063\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.29165e12	−1.36523	−0.682617	−	0.730776i	$-0.739158\pi$
$167$	−2.29165e12	−1.36523	−0.682617	+	0.730776i	$0.739158\pi$
$168$	0	0
$169$	−1.41917e12	−0.791879
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.43042e11	−0.168304	−0.0841520	−	0.996453i	$-0.526818\pi$
$173$	−3.43042e11	−0.168304	−0.0841520	+	0.996453i	$0.526818\pi$
$174$	0	0
$175$	2.50730e12	1.15477
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.40809e11	−0.341984	−0.170992	−	0.985272i	$-0.554697\pi$
$179$	−8.40809e11	−0.341984	−0.170992	+	0.985272i	$0.554697\pi$
$180$	0	0
$181$	3.46806e12	1.32695	0.663475	−	0.748198i	$-0.269081\pi$
$181$	3.46806e12	1.32695	0.663475	+	0.748198i	$0.269081\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.54973e11	−0.154362
$186$	0	0
$187$	2.35540e12	0.753244
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.44861e12	1.83562	0.917810	−	0.397021i	$-0.129956\pi$
$191$	6.44861e12	1.83562	0.917810	+	0.397021i	$0.129956\pi$
$192$	0	0
$193$	5.43570e12	1.46113	0.730567	−	0.682841i	$-0.239256\pi$
$193$	5.43570e12	1.46113	0.730567	+	0.682841i	$0.239256\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.12393e11	−0.0510006	−0.0255003	−	0.999675i	$-0.508118\pi$
$197$	−2.12393e11	−0.0510006	−0.0255003	+	0.999675i	$0.508118\pi$
$198$	0	0
$199$	4.44182e12	1.00895	0.504475	−	0.863427i	$-0.331686\pi$
$199$	4.44182e12	1.00895	0.504475	+	0.863427i	$0.331686\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.43773e12	1.31070
$204$	0	0
$205$	−7.03381e11	−0.135689
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.45474e12	−1.64000
$210$	0	0
$211$	2.97268e12	0.489323	0.244661	−	0.969609i	$-0.421323\pi$
$211$	2.97268e12	0.489323	0.244661	+	0.969609i	$0.421323\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.41934e12	−0.210705
$216$	0	0
$217$	8.44200e11	0.119101
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.71606e12	−0.218965
$222$	0	0
$223$	−1.23437e13	−1.49889	−0.749444	−	0.662067i	$-0.769679\pi$
$223$	−1.23437e13	−1.49889	−0.749444	+	0.662067i	$0.769679\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.17307e12	−0.569647	−0.284824	−	0.958580i	$-0.591935\pi$
$227$	−5.17307e12	−0.569647	−0.284824	+	0.958580i	$0.591935\pi$
$228$	0	0
$229$	−5.74042e12	−0.602349	−0.301175	−	0.953569i	$-0.597379\pi$
$229$	−5.74042e12	−0.602349	−0.301175	+	0.953569i	$0.597379\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.16855e13	−1.11478	−0.557391	−	0.830250i	$-0.688197\pi$
$233$	−1.16855e13	−1.11478	−0.557391	+	0.830250i	$0.688197\pi$
$234$	0	0
$235$	−1.44014e12	−0.131079
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.80369e13	1.49614	0.748071	−	0.663619i	$-0.230980\pi$
$239$	1.80369e13	1.49614	0.748071	+	0.663619i	$0.230980\pi$
$240$	0	0
$241$	−3.09827e12	−0.245485	−0.122742	−	0.992439i	$-0.539169\pi$
$241$	−3.09827e12	−0.245485	−0.122742	+	0.992439i	$0.539169\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.24289e12	−0.162328
$246$	0	0
$247$	6.88840e12	0.476744
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.03760e13	0.657390	0.328695	−	0.944436i	$-0.393391\pi$
$251$	1.03760e13	0.657390	0.328695	+	0.944436i	$0.393391\pi$
$252$	0	0
$253$	−1.67377e13	−1.01516
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.51225e13	1.39776	0.698878	−	0.715241i	$-0.253683\pi$
$257$	2.51225e13	1.39776	0.698878	+	0.715241i	$0.253683\pi$
$258$	0	0
$259$	−1.28635e13	−0.685822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.18918e13	−1.07282	−0.536408	−	0.843959i	$-0.680219\pi$
$263$	−2.18918e13	−1.07282	−0.536408	+	0.843959i	$0.680219\pi$
$264$	0	0
$265$	1.13701e13	0.534456
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.09258e12	−0.133870	−0.0669350	−	0.997757i	$-0.521322\pi$
$269$	−3.09258e12	−0.133870	−0.0669350	+	0.997757i	$0.521322\pi$
$270$	0	0
$271$	−2.73592e13	−1.13703	−0.568515	−	0.822673i	$-0.692482\pi$
$271$	−2.73592e13	−1.13703	−0.568515	+	0.822673i	$0.692482\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.76653e13	−1.44415
$276$	0	0
$277$	−1.72091e13	−0.634046	−0.317023	−	0.948418i	$-0.602683\pi$
$277$	−1.72091e13	−0.634046	−0.317023	+	0.948418i	$0.602683\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.79474e13	1.29210	0.646051	−	0.763294i	$-0.276419\pi$
$281$	3.79474e13	1.29210	0.646051	+	0.763294i	$0.276419\pi$
$282$	0	0
$283$	1.11476e12	0.0365054	0.0182527	−	0.999833i	$-0.494190\pi$
$283$	1.11476e12	0.0365054	0.0182527	+	0.999833i	$0.494190\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.98868e13	−0.602856
$288$	0	0
$289$	−2.63765e13	−0.769625
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.47446e12	−0.0398896	−0.0199448	−	0.999801i	$-0.506349\pi$
$293$	−1.47446e12	−0.0398896	−0.0199448	+	0.999801i	$0.506349\pi$
$294$	0	0
$295$	−1.27190e13	−0.331460
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.21945e13	0.295103
$300$	0	0
$301$	−4.01292e13	−0.936149
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.73005e13	0.375328
$306$	0	0
$307$	−7.12340e13	−1.49082	−0.745412	−	0.666605i	$-0.767747\pi$
$307$	−7.12340e13	−1.49082	−0.745412	+	0.666605i	$0.767747\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.61508e13	−0.509687	−0.254844	−	0.966982i	$-0.582024\pi$
$311$	−2.61508e13	−0.509687	−0.254844	+	0.966982i	$0.582024\pi$
$312$	0	0
$313$	−7.70099e13	−1.44895	−0.724473	−	0.689303i	$-0.757917\pi$
$313$	−7.70099e13	−1.44895	−0.724473	+	0.689303i	$0.757917\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.95199e13	1.39524	0.697622	−	0.716466i	$-0.254242\pi$
$317$	7.95199e13	1.39524	0.697622	+	0.716466i	$0.254242\pi$
$318$	0	0
$319$	−9.67094e13	−1.63915
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.16927e13	−0.501586
$324$	0	0
$325$	2.74417e13	0.419809
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.07174e13	−0.582375
$330$	0	0
$331$	8.88258e13	1.22881	0.614406	−	0.788990i	$-0.289396\pi$
$331$	8.88258e13	1.22881	0.614406	+	0.788990i	$0.289396\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.71317e12	0.0480837
$336$	0	0
$337$	−7.05469e13	−0.884125	−0.442062	−	0.896984i	$-0.645753\pi$
$337$	−7.05469e13	−0.884125	−0.442062	+	0.896984i	$0.645753\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.26818e13	−0.148947
$342$	0	0
$343$	4.69233e13	0.533668
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.45983e13	−0.155772	−0.0778859	−	0.996962i	$-0.524817\pi$
$347$	−1.45983e13	−0.155772	−0.0778859	+	0.996962i	$0.524817\pi$
$348$	0	0
$349$	1.66156e14	1.71781	0.858905	−	0.512134i	$-0.171145\pi$
$349$	1.66156e14	1.71781	0.858905	+	0.512134i	$0.171145\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.88853e14	−1.83385	−0.916923	−	0.399063i	$-0.869335\pi$
$353$	−1.88853e14	−1.83385	−0.916923	+	0.399063i	$0.869335\pi$
$354$	0	0
$355$	−4.44263e13	−0.418200
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.04450e14	0.924465	0.462233	−	0.886759i	$-0.347048\pi$
$359$	1.04450e14	0.924465	0.462233	+	0.886759i	$0.347048\pi$
$360$	0	0
$361$	1.07265e13	0.0920807
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.88250e13	−0.394486
$366$	0	0
$367$	−7.25880e13	−0.569117	−0.284558	−	0.958659i	$-0.591847\pi$
$367$	−7.25880e13	−0.569117	−0.284558	+	0.958659i	$0.591847\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.21468e14	2.37455
$372$	0	0
$373$	−1.01857e14	−0.730456	−0.365228	−	0.930918i	$-0.619009\pi$
$373$	−1.01857e14	−0.730456	−0.365228	+	0.930918i	$0.619009\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.04592e13	0.476497
$378$	0	0
$379$	2.69386e14	1.76954	0.884768	−	0.466031i	$-0.154317\pi$
$379$	2.69386e14	1.76954	0.884768	+	0.466031i	$0.154317\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.81124e13	−0.360309	−0.180155	−	0.983638i	$-0.557660\pi$
$383$	−5.81124e13	−0.360309	−0.180155	+	0.983638i	$0.557660\pi$
$384$	0	0
$385$	9.23182e13	0.556228
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.77446e13	−0.556378	−0.278189	−	0.960526i	$-0.589734\pi$
$389$	−9.77446e13	−0.556378	−0.278189	+	0.960526i	$0.589734\pi$
$390$	0	0
$391$	−5.61055e13	−0.310480
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.18497e13	−0.218982
$396$	0	0
$397$	−7.53335e13	−0.383389	−0.191695	−	0.981455i	$-0.561398\pi$
$397$	−7.53335e13	−0.383389	−0.191695	+	0.981455i	$0.561398\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.23796e13	0.348596	0.174298	−	0.984693i	$-0.444234\pi$
$401$	7.23796e13	0.348596	0.174298	+	0.984693i	$0.444234\pi$
$402$	0	0
$403$	9.23953e12	0.0432984
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.93239e14	0.857682
$408$	0	0
$409$	2.96840e14	1.28246	0.641230	−	0.767349i	$-0.278424\pi$
$409$	2.96840e14	1.28246	0.641230	+	0.767349i	$0.278424\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.59606e14	−1.47266
$414$	0	0
$415$	−7.05626e13	−0.281391
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.61517e13	−0.250244	−0.125122	−	0.992141i	$-0.539932\pi$
$419$	−6.61517e13	−0.250244	−0.125122	+	0.992141i	$0.539932\pi$
$420$	0	0
$421$	1.18955e14	0.438360	0.219180	−	0.975684i	$-0.429662\pi$
$421$	1.18955e14	0.438360	0.219180	+	0.975684i	$0.429662\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.26256e14	−0.441684
$426$	0	0
$427$	4.89141e14	1.66756
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.03192e14	−0.334212	−0.167106	−	0.985939i	$-0.553442\pi$
$431$	−1.03192e14	−0.334212	−0.167106	+	0.985939i	$0.553442\pi$
$432$	0	0
$433$	3.89828e14	1.23081	0.615404	−	0.788212i	$-0.288993\pi$
$433$	3.89828e14	1.23081	0.615404	+	0.788212i	$0.288993\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.25211e14	0.675994
$438$	0	0
$439$	5.04732e14	1.47743	0.738714	−	0.674020i	$-0.235434\pi$
$439$	5.04732e14	1.47743	0.738714	+	0.674020i	$0.235434\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.69734e14	−1.30807	−0.654036	−	0.756463i	$-0.726926\pi$
$443$	−4.69734e14	−1.30807	−0.654036	+	0.756463i	$0.726926\pi$
$444$	0	0
$445$	1.86103e14	0.505561
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.54770e14	1.17608	0.588041	−	0.808831i	$-0.299900\pi$
$449$	4.54770e14	1.17608	0.588041	+	0.808831i	$0.299900\pi$
$450$	0	0
$451$	2.98745e14	0.753927
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.72599e13	−0.161694
$456$	0	0
$457$	1.81666e14	0.426319	0.213160	−	0.977017i	$-0.431625\pi$
$457$	1.81666e14	0.426319	0.213160	+	0.977017i	$0.431625\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.98314e14	1.33836	0.669182	−	0.743099i	$-0.266645\pi$
$461$	5.98314e14	1.33836	0.669182	+	0.743099i	$0.266645\pi$
$462$	0	0
$463$	−5.05645e14	−1.10446	−0.552230	−	0.833692i	$-0.686223\pi$
$463$	−5.05645e14	−1.10446	−0.552230	+	0.833692i	$0.686223\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.64772e13	0.180160	0.0900801	−	0.995935i	$-0.471288\pi$
$467$	8.64772e13	0.180160	0.0900801	+	0.995935i	$0.471288\pi$
$468$	0	0
$469$	1.04983e14	0.213633
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.02832e14	1.17074
$474$	0	0
$475$	5.06800e14	0.961660
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.97061e14	−0.900668	−0.450334	−	0.892860i	$-0.648695\pi$
$479$	−4.97061e14	−0.900668	−0.450334	+	0.892860i	$0.648695\pi$
$480$	0	0
$481$	−1.40788e14	−0.249325
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.77195e14	−0.299829
$486$	0	0
$487$	−6.55930e14	−1.08505	−0.542523	−	0.840041i	$-0.682531\pi$
$487$	−6.55930e14	−1.08505	−0.542523	+	0.840041i	$0.682531\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.91963e14	−0.778009	−0.389004	−	0.921236i	$-0.627181\pi$
$491$	−4.91963e14	−0.778009	−0.389004	+	0.921236i	$0.627181\pi$
$492$	0	0
$493$	−3.24174e14	−0.501326
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.25607e15	−1.85803
$498$	0	0
$499$	1.00037e15	1.44746	0.723730	−	0.690083i	$-0.242426\pi$
$499$	1.00037e15	1.44746	0.723730	+	0.690083i	$0.242426\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.02633e13	0.0557551	0.0278776	−	0.999611i	$-0.491125\pi$
$503$	4.02633e13	0.0557551	0.0278776	+	0.999611i	$0.491125\pi$
$504$	0	0
$505$	−1.85461e14	−0.251276
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.29605e15	1.68142	0.840709	−	0.541488i	$-0.182139\pi$
$509$	1.29605e15	1.68142	0.840709	+	0.541488i	$0.182139\pi$
$510$	0	0
$511$	−1.38043e15	−1.75267
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.35097e13	−0.0529233
$516$	0	0
$517$	6.11668e14	0.728313
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.62447e14	0.299526	0.149763	−	0.988722i	$-0.452149\pi$
$521$	2.62447e14	0.299526	0.149763	+	0.988722i	$0.452149\pi$
$522$	0	0
$523$	−2.61154e13	−0.0291835	−0.0145917	−	0.999894i	$-0.504645\pi$
$523$	−2.61154e13	−0.0291835	−0.0145917	+	0.999894i	$0.504645\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.25099e13	−0.0455546
$528$	0	0
$529$	−5.54119e14	−0.581563
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.17655e14	−0.219164
$534$	0	0
$535$	−3.25082e14	−0.320662
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.52615e14	0.901943
$540$	0	0
$541$	−3.00607e14	−0.278878	−0.139439	−	0.990231i	$-0.544530\pi$
$541$	−3.00607e14	−0.278878	−0.139439	+	0.990231i	$0.544530\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.05643e14	0.183203
$546$	0	0
$547$	8.07253e14	0.704822	0.352411	−	0.935845i	$-0.385362\pi$
$547$	8.07253e14	0.704822	0.352411	+	0.935845i	$0.385362\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.30126e15	1.09151
$552$	0	0
$553$	−1.18322e15	−0.972922
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.96905e15	−1.55616	−0.778078	−	0.628168i	$-0.783805\pi$
$557$	−1.96905e15	−1.55616	−0.778078	+	0.628168i	$0.783805\pi$
$558$	0	0
$559$	−4.39203e14	−0.340330
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.79968e14	−0.357615	−0.178808	−	0.983884i	$-0.557224\pi$
$563$	−4.79968e14	−0.357615	−0.178808	+	0.983884i	$0.557224\pi$
$564$	0	0
$565$	2.89673e14	0.211661
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.04880e15	−1.44007	−0.720033	−	0.693940i	$-0.755873\pi$
$569$	−2.04880e15	−1.44007	−0.720033	+	0.693940i	$0.755873\pi$
$570$	0	0
$571$	1.76429e14	0.121638	0.0608192	−	0.998149i	$-0.480629\pi$
$571$	1.76429e14	0.121638	0.0608192	+	0.998149i	$0.480629\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.97186e14	0.595264
$576$	0	0
$577$	−1.00079e15	−0.651439	−0.325720	−	0.945466i	$-0.605607\pi$
$577$	−1.00079e15	−0.651439	−0.325720	+	0.945466i	$0.605607\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.99503e15	−1.25020
$582$	0	0
$583$	−4.82918e15	−2.96959
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.89286e15	1.71324	0.856620	−	0.515948i	$-0.172560\pi$
$587$	2.89286e15	1.71324	0.856620	+	0.515948i	$0.172560\pi$
$588$	0	0
$589$	1.70638e14	0.0991840
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.57904e15	0.884283	0.442142	−	0.896945i	$-0.354219\pi$
$593$	1.57904e15	0.884283	0.442142	+	0.896945i	$0.354219\pi$
$594$	0	0
$595$	3.09454e14	0.170119
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.16850e15	−1.67883	−0.839414	−	0.543493i	$-0.817102\pi$
$599$	−3.16850e15	−1.67883	−0.839414	+	0.543493i	$0.817102\pi$
$600$	0	0
$601$	−2.82722e15	−1.47079	−0.735394	−	0.677640i	$-0.763003\pi$
$601$	−2.82722e15	−1.47079	−0.735394	+	0.677640i	$0.763003\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.23727e14	−0.413170
$606$	0	0
$607$	−1.68353e15	−0.829245	−0.414622	−	0.909994i	$-0.636086\pi$
$607$	−1.68353e15	−0.829245	−0.414622	+	0.909994i	$0.636086\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.45641e14	−0.211718
$612$	0	0
$613$	−4.01898e15	−1.87535	−0.937677	−	0.347509i	$-0.887028\pi$
$613$	−4.01898e15	−1.87535	−0.937677	+	0.347509i	$0.887028\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.78146e15	1.25229	0.626144	−	0.779707i	$-0.284632\pi$
$617$	2.78146e15	1.25229	0.626144	+	0.779707i	$0.284632\pi$
$618$	0	0
$619$	1.34314e15	0.594048	0.297024	−	0.954870i	$-0.404006\pi$
$619$	1.34314e15	0.594048	0.297024	+	0.954870i	$0.404006\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.26172e15	2.24617
$624$	0	0
$625$	1.82877e15	0.767040
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.47747e14	0.262317
$630$	0	0
$631$	−4.37222e15	−1.73997	−0.869983	−	0.493082i	$-0.835870\pi$
$631$	−4.37222e15	−1.73997	−0.869983	+	0.493082i	$0.835870\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.48518e14	−0.326141
$636$	0	0
$637$	−6.94042e14	−0.262192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.41904e15	−1.24791	−0.623957	−	0.781459i	$-0.714476\pi$
$641$	−3.41904e15	−1.24791	−0.623957	+	0.781459i	$0.714476\pi$
$642$	0	0
$643$	4.34807e15	1.56004	0.780022	−	0.625753i	$-0.215208\pi$
$643$	4.34807e15	1.56004	0.780022	+	0.625753i	$0.215208\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.18491e15	−0.757636	−0.378818	−	0.925471i	$-0.623669\pi$
$647$	−2.18491e15	−0.757636	−0.378818	+	0.925471i	$0.623669\pi$
$648$	0	0
$649$	5.40209e15	1.84169
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.16104e15	1.37145	0.685725	−	0.727861i	$-0.259485\pi$
$653$	4.16104e15	1.37145	0.685725	+	0.727861i	$0.259485\pi$
$654$	0	0
$655$	1.20569e15	0.390758
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.37735e14	−0.168538	−0.0842692	−	0.996443i	$-0.526856\pi$
$659$	−5.37735e14	−0.168538	−0.0842692	+	0.996443i	$0.526856\pi$
$660$	0	0
$661$	−1.32255e15	−0.407666	−0.203833	−	0.979006i	$-0.565340\pi$
$661$	−1.32255e15	−0.407666	−0.203833	+	0.979006i	$0.565340\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.24217e15	−0.370393
$666$	0	0
$667$	2.30361e15	0.675643
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.34800e15	−2.08543
$672$	0	0
$673$	−7.25732e13	−0.0202625	−0.0101313	−	0.999949i	$-0.503225\pi$
$673$	−7.25732e13	−0.0202625	−0.0101313	+	0.999949i	$0.503225\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.00943e15	0.813292	0.406646	−	0.913586i	$-0.366698\pi$
$677$	3.00943e15	0.813292	0.406646	+	0.913586i	$0.366698\pi$
$678$	0	0
$679$	−5.00987e15	−1.33212
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.63684e15	−0.421399	−0.210699	−	0.977551i	$-0.567574\pi$
$683$	−1.63684e15	−0.421399	−0.210699	+	0.977551i	$0.567574\pi$
$684$	0	0
$685$	−1.98085e15	−0.501826
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.51838e15	0.863250
$690$	0	0
$691$	5.77526e15	1.39458	0.697289	−	0.716790i	$-0.254390\pi$
$691$	5.77526e15	1.39458	0.697289	+	0.716790i	$0.254390\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.23099e15	−0.521893
$696$	0	0
$697$	1.00140e15	0.230584
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.63213e15	0.364171	0.182085	−	0.983283i	$-0.441715\pi$
$701$	1.63213e15	0.364171	0.182085	+	0.983283i	$0.441715\pi$
$702$	0	0
$703$	−2.60010e15	−0.571131
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.24357e15	−1.11640
$708$	0	0
$709$	7.13860e15	1.49644	0.748219	−	0.663452i	$-0.230909\pi$
$709$	7.13860e15	1.49644	0.748219	+	0.663452i	$0.230909\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.02080e14	0.0613945
$714$	0	0
$715$	1.01040e15	0.202213
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.80685e15	1.32110	0.660552	−	0.750780i	$-0.270322\pi$
$719$	6.80685e15	1.32110	0.660552	+	0.750780i	$0.270322\pi$
$720$	0	0
$721$	−1.23016e15	−0.235134
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.18388e15	0.961161
$726$	0	0
$727$	−2.01758e15	−0.368462	−0.184231	−	0.982883i	$-0.558979\pi$
$727$	−2.01758e15	−0.368462	−0.184231	+	0.982883i	$0.558979\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.02072e15	0.358064
$732$	0	0
$733$	7.79870e14	0.136129	0.0680644	−	0.997681i	$-0.478318\pi$
$733$	7.79870e14	0.136129	0.0680644	+	0.997681i	$0.478318\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.57708e15	−0.267167
$738$	0	0
$739$	1.19425e16	1.99321	0.996603	−	0.0823516i	$-0.0262431\pi$
$739$	1.19425e16	1.99321	0.996603	+	0.0823516i	$0.0262431\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.22771e15	0.522945	0.261473	−	0.965211i	$-0.415792\pi$
$743$	3.22771e15	0.522945	0.261473	+	0.965211i	$0.415792\pi$
$744$	0	0
$745$	−1.51231e14	−0.0241425
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.19109e15	−1.42468
$750$	0	0
$751$	1.06165e15	0.162167	0.0810836	−	0.996707i	$-0.474162\pi$
$751$	1.06165e15	0.162167	0.0810836	+	0.996707i	$0.474162\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.09661e14	−0.0904441
$756$	0	0
$757$	3.65045e15	0.533726	0.266863	−	0.963734i	$-0.414013\pi$
$757$	3.65045e15	0.533726	0.266863	+	0.963734i	$0.414013\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.81802e15	1.11040	0.555202	−	0.831715i	$-0.312641\pi$
$761$	7.81802e15	1.11040	0.555202	+	0.831715i	$0.312641\pi$
$762$	0	0
$763$	5.81416e15	0.813958
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.93578e15	−0.535373
$768$	0	0
$769$	2.82275e15	0.378510	0.189255	−	0.981928i	$-0.439393\pi$
$769$	2.82275e15	0.378510	0.189255	+	0.981928i	$0.439393\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.59128e15	−1.24994	−0.624970	−	0.780649i	$-0.714889\pi$
$773$	−9.59128e15	−1.24994	−0.624970	+	0.780649i	$0.714889\pi$
$774$	0	0
$775$	6.79779e14	0.0873390
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.01971e15	−0.502040
$780$	0	0
$781$	1.88690e16	2.32364
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.29227e15	−0.513930
$786$	0	0
$787$	6.46710e15	0.763570	0.381785	−	0.924251i	$-0.375310\pi$
$787$	6.46710e15	0.763570	0.381785	+	0.924251i	$0.375310\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.18996e15	0.940397
$792$	0	0
$793$	5.35350e15	0.606228
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.95629e15	0.215483	0.107742	−	0.994179i	$-0.465638\pi$
$797$	1.95629e15	0.215483	0.107742	+	0.994179i	$0.465638\pi$
$798$	0	0
$799$	2.05034e15	0.222750
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.07373e16	2.19188
$804$	0	0
$805$	−2.19901e15	−0.229272
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.57213e15	−0.869705	−0.434852	−	0.900502i	$-0.643199\pi$
$809$	−8.57213e15	−0.869705	−0.434852	+	0.900502i	$0.643199\pi$
$810$	0	0
$811$	1.61738e15	0.161881	0.0809406	−	0.996719i	$-0.474208\pi$
$811$	1.61738e15	0.161881	0.0809406	+	0.996719i	$0.474208\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.26593e14	−0.0318156
$816$	0	0
$817$	−8.11131e15	−0.779597
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.42885e16	−1.33690	−0.668450	−	0.743757i	$-0.733042\pi$
$821$	−1.42885e16	−1.33690	−0.668450	+	0.743757i	$0.733042\pi$
$822$	0	0
$823$	−1.57793e15	−0.145677	−0.0728383	−	0.997344i	$-0.523206\pi$
$823$	−1.57793e15	−0.145677	−0.0728383	+	0.997344i	$0.523206\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.61323e15	0.234907	0.117454	−	0.993078i	$-0.462527\pi$
$827$	2.61323e15	0.234907	0.117454	+	0.993078i	$0.462527\pi$
$828$	0	0
$829$	−1.13560e16	−1.00734	−0.503668	−	0.863897i	$-0.668016\pi$
$829$	−1.13560e16	−1.00734	−0.503668	+	0.863897i	$0.668016\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.19320e15	0.275854
$834$	0	0
$835$	4.52288e15	0.385602
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.78938e16	1.48598	0.742990	−	0.669303i	$-0.233407\pi$
$839$	1.78938e16	1.48598	0.742990	+	0.669303i	$0.233407\pi$
$840$	0	0
$841$	1.10962e15	0.0909484
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.80094e15	0.223662
$846$	0	0
$847$	−2.32893e16	−1.83569
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.60295e15	−0.353528
$852$	0	0
$853$	5.11907e15	0.388125	0.194063	−	0.980989i	$-0.437834\pi$
$853$	5.11907e15	0.388125	0.194063	+	0.980989i	$0.437834\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.04520e16	−0.772332	−0.386166	−	0.922429i	$-0.626201\pi$
$857$	−1.04520e16	−0.772332	−0.386166	+	0.922429i	$0.626201\pi$
$858$	0	0
$859$	6.75264e15	0.492619	0.246309	−	0.969191i	$-0.420782\pi$
$859$	6.75264e15	0.492619	0.246309	+	0.969191i	$0.420782\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.20983e16	−1.57144	−0.785721	−	0.618580i	$-0.787708\pi$
$863$	−2.20983e16	−1.57144	−0.785721	+	0.618580i	$0.787708\pi$
$864$	0	0
$865$	6.77042e14	0.0475365
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.77747e16	1.21673
$870$	0	0
$871$	1.14901e15	0.0776646
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.03260e16	−0.680593
$876$	0	0
$877$	−4.13928e15	−0.269419	−0.134709	−	0.990885i	$-0.543010\pi$
$877$	−4.13928e15	−0.269419	−0.134709	+	0.990885i	$0.543010\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.38433e15	−0.0878767	−0.0439383	−	0.999034i	$-0.513991\pi$
$881$	−1.38433e15	−0.0878767	−0.0439383	+	0.999034i	$0.513991\pi$
$882$	0	0
$883$	−1.49361e16	−0.936384	−0.468192	−	0.883627i	$-0.655094\pi$
$883$	−1.49361e16	−0.936384	−0.468192	+	0.883627i	$0.655094\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.19717e16	−0.732107	−0.366054	−	0.930594i	$-0.619291\pi$
$887$	−1.19717e16	−0.732107	−0.366054	+	0.930594i	$0.619291\pi$
$888$	0	0
$889$	−2.39903e16	−1.44902
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.23020e15	−0.484984
$894$	0	0
$895$	1.65945e15	0.0965914
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.74540e15	0.0991326
$900$	0	0
$901$	−1.61876e16	−0.908232
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.84470e15	−0.374789
$906$	0	0
$907$	−7.00230e14	−0.0378792	−0.0189396	−	0.999821i	$-0.506029\pi$
$907$	−7.00230e14	−0.0378792	−0.0189396	+	0.999821i	$0.506029\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.22392e16	1.17427	0.587135	−	0.809489i	$-0.300256\pi$
$911$	2.22392e16	1.17427	0.587135	+	0.809489i	$0.300256\pi$
$912$	0	0
$913$	2.99699e16	1.56349
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.40885e16	1.73611
$918$	0	0
$919$	−7.74531e15	−0.389765	−0.194883	−	0.980827i	$-0.562433\pi$
$919$	−7.74531e15	−0.389765	−0.194883	+	0.980827i	$0.562433\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.37473e16	−0.675475
$924$	0	0
$925$	−1.03582e16	−0.502925
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.24994e16	1.54095	0.770476	−	0.637468i	$-0.220018\pi$
$929$	3.24994e16	1.54095	0.770476	+	0.637468i	$0.220018\pi$
$930$	0	0
$931$	−1.28178e16	−0.600605
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.64871e15	−0.212749
$936$	0	0
$937$	8.46971e15	0.383090	0.191545	−	0.981484i	$-0.438650\pi$
$937$	8.46971e15	0.383090	0.191545	+	0.981484i	$0.438650\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.93549e16	0.855162	0.427581	−	0.903977i	$-0.359366\pi$
$941$	1.93549e16	0.855162	0.427581	+	0.903977i	$0.359366\pi$
$942$	0	0
$943$	−7.11608e15	−0.310761
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.27974e15	0.310593	0.155296	−	0.987868i	$-0.450367\pi$
$947$	7.27974e15	0.310593	0.155296	+	0.987868i	$0.450367\pi$
$948$	0	0
$949$	−1.51085e16	−0.637171
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.22292e15	0.132812	0.0664062	−	0.997793i	$-0.478847\pi$
$953$	3.22292e15	0.132812	0.0664062	+	0.997793i	$0.478847\pi$
$954$	0	0
$955$	−1.27272e16	−0.518460
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.60048e16	−2.22958
$960$	0	0
$961$	−2.51796e16	−0.990992
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.07281e16	−0.412689
$966$	0	0
$967$	3.17472e16	1.20743	0.603713	−	0.797202i	$-0.293687\pi$
$967$	3.17472e16	1.20743	0.603713	+	0.797202i	$0.293687\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.71269e15	−0.175212	−0.0876058	−	0.996155i	$-0.527922\pi$
$971$	−4.71269e15	−0.175212	−0.0876058	+	0.996155i	$0.527922\pi$
$972$	0	0
$973$	−6.30771e16	−2.31874
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.15716e14	−0.0329110	−0.0164555	−	0.999865i	$-0.505238\pi$
$977$	−9.15716e14	−0.0329110	−0.0164555	+	0.999865i	$0.505238\pi$
$978$	0	0
$979$	−7.90430e16	−2.80905
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.33048e16	−1.15734	−0.578672	−	0.815560i	$-0.696429\pi$
$983$	−3.33048e16	−1.15734	−0.578672	+	0.815560i	$0.696429\pi$
$984$	0	0
$985$	4.19187e14	0.0144048
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.43594e16	−0.482567
$990$	0	0
$991$	−1.64763e16	−0.547589	−0.273794	−	0.961788i	$-0.588279\pi$
$991$	−1.64763e16	−0.547589	−0.273794	+	0.961788i	$0.588279\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.76656e15	−0.284972
$996$	0	0
$997$	1.11920e16	0.359818	0.179909	−	0.983683i	$-0.442420\pi$
$997$	1.11920e16	0.359818	0.179909	+	0.983683i	$0.442420\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.12.a.t.1.2		3
3.2	odd	2		144.12.a.s.1.2		3
4.3	odd	2		72.12.a.h.1.2	yes	3
12.11	even	2		72.12.a.g.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.12.a.g.1.2	✓	3	12.11	even	2
72.12.a.h.1.2	yes	3	4.3	odd	2
144.12.a.s.1.2		3	3.2	odd	2
144.12.a.t.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

\(f(q)\)	\(=\)	\(q-1973.64 q^{5} -55801.0 q^{7} +O(q^{10})\)	\(q-1973.64 q^{5} -55801.0 q^{7} +838258. q^{11} -610726. q^{13} +2.80987e6 q^{17} -1.12790e7 q^{19} -1.99673e7 q^{23} -4.49329e7 q^{25} -1.15370e8 q^{29} -1.51288e7 q^{31} +1.10131e8 q^{35} +2.30525e8 q^{37} +3.56388e8 q^{41} +7.19149e8 q^{43} +7.29690e8 q^{47} +1.13642e9 q^{49} -5.76098e9 q^{53} -1.65442e9 q^{55} +6.44443e9 q^{59} -8.76581e9 q^{61} +1.20535e9 q^{65} -1.88138e9 q^{67} +2.25098e10 q^{71} +2.47385e10 q^{73} -4.67756e10 q^{77} +2.12043e10 q^{79} +3.57526e10 q^{83} -5.54568e9 q^{85} -9.42944e10 q^{89} +3.40791e10 q^{91} +2.22608e10 q^{95} +8.97811e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 1584 q^{5} + 17796 q^{7}+O(q^{10}) \) 3 * q + 1584 * q^5 + 17796 * q^7	\( 3 q + 1584 q^{5} + 17796 q^{7} - 238272 q^{11} - 49458 q^{13} - 766368 q^{17} + 4590936 q^{19} + 13188480 q^{23} + 18034281 q^{25} + 8005200 q^{29} - 10185564 q^{31} + 235065408 q^{35} - 73215510 q^{37} - 1032516960 q^{41} - 28458216 q^{43} + 2840606592 q^{47} - 109741845 q^{49} - 6755828784 q^{53} + 846590976 q^{55} + 15928599936 q^{59} + 5136563970 q^{61} - 37415401248 q^{65} + 6119899728 q^{67} + 58699811328 q^{71} + 2561705778 q^{73} - 86561454336 q^{77} + 17842143972 q^{79} + 134316444096 q^{83} + 43096069632 q^{85} - 152402442048 q^{89} + 56268770664 q^{91} + 207495767424 q^{95} + 106677733482 q^{97}+O(q^{100}) \) 3 * q + 1584 * q^5 + 17796 * q^7 - 238272 * q^11 - 49458 * q^13 - 766368 * q^17 + 4590936 * q^19 + 13188480 * q^23 + 18034281 * q^25 + 8005200 * q^29 - 10185564 * q^31 + 235065408 * q^35 - 73215510 * q^37 - 1032516960 * q^41 - 28458216 * q^43 + 2840606592 * q^47 - 109741845 * q^49 - 6755828784 * q^53 + 846590976 * q^55 + 15928599936 * q^59 + 5136563970 * q^61 - 37415401248 * q^65 + 6119899728 * q^67 + 58699811328 * q^71 + 2561705778 * q^73 - 86561454336 * q^77 + 17842143972 * q^79 + 134316444096 * q^83 + 43096069632 * q^85 - 152402442048 * q^89 + 56268770664 * q^91 + 207495767424 * q^95 + 106677733482 * q^97

Introduction

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