LMFDB - Embedded newform 192.12.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,12,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, 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("192.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	192.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:	$147.521890667$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1945}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 486 $ x^2 - x - 486
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-21.5511$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+243.000 q^{3} +3700.71 q^{5} +72358.6 q^{7} +59049.0 q^{9} +O(q^{10})$	$q+243.000 q^{3} +3700.71 q^{5} +72358.6 q^{7} +59049.0 q^{9} -1.03599e6 q^{11} +447532. q^{13} +899273. q^{15} +966903. q^{17} +180409. q^{19} +1.75831e7 q^{21} -1.67982e7 q^{23} -3.51329e7 q^{25} +1.43489e7 q^{27} -1.48887e8 q^{29} -2.16841e8 q^{31} -2.51746e8 q^{33} +2.67778e8 q^{35} -6.70953e8 q^{37} +1.08750e8 q^{39} -4.72819e8 q^{41} -1.14447e9 q^{43} +2.18523e8 q^{45} +1.31669e7 q^{47} +3.25844e9 q^{49} +2.34957e8 q^{51} +4.04271e9 q^{53} -3.83390e9 q^{55} +4.38395e7 q^{57} +6.23772e9 q^{59} -6.61870e9 q^{61} +4.27270e9 q^{63} +1.65619e9 q^{65} -1.97892e10 q^{67} -4.08196e9 q^{69} +2.10433e10 q^{71} -1.27113e10 q^{73} -8.53729e9 q^{75} -7.49628e10 q^{77} -1.16162e10 q^{79} +3.48678e9 q^{81} +5.54557e10 q^{83} +3.57823e9 q^{85} -3.61796e10 q^{87} +5.43356e10 q^{89} +3.23828e10 q^{91} -5.26924e10 q^{93} +6.67643e8 q^{95} +8.98530e10 q^{97} -6.11742e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 486 q^{3} - 5300 q^{5} + 38872 q^{7} + 118098 q^{9}+O(q^{10}) $ 2 * q + 486 * q^3 - 5300 * q^5 + 38872 * q^7 + 118098 * q^9	$ 2 q + 486 q^{3} - 5300 q^{5} + 38872 q^{7} + 118098 q^{9} - 1047400 q^{11} + 22900 q^{13} - 1287900 q^{15} + 8733300 q^{17} + 7346600 q^{19} + 9445896 q^{21} - 6711744 q^{23} - 2948210 q^{25} + 28697814 q^{27} - 180180692 q^{29} - 211581400 q^{31} - 254518200 q^{33} + 569181200 q^{35} - 112222700 q^{37} + 5564700 q^{39} - 726961180 q^{41} + 216408856 q^{43} - 312959700 q^{45} + 2174779088 q^{47} + 2402462706 q^{49} + 2122191900 q^{51} + 112024700 q^{53} - 3731208560 q^{55} + 1785223800 q^{57} - 3243949400 q^{59} - 16526230620 q^{61} + 2295352728 q^{63} + 5478177080 q^{65} - 20772619112 q^{67} - 1630953792 q^{69} + 20637101600 q^{71} - 18548203500 q^{73} - 716415030 q^{75} - 74580754400 q^{77} - 28230083800 q^{79} + 6973568802 q^{81} + 7189282056 q^{83} - 66324859080 q^{85} - 43783908156 q^{87} + 103679180788 q^{89} + 46602268400 q^{91} - 51414280200 q^{93} - 63833162000 q^{95} + 199614486500 q^{97} - 61847922600 q^{99}+O(q^{100}) $ 2 * q + 486 * q^3 - 5300 * q^5 + 38872 * q^7 + 118098 * q^9 - 1047400 * q^11 + 22900 * q^13 - 1287900 * q^15 + 8733300 * q^17 + 7346600 * q^19 + 9445896 * q^21 - 6711744 * q^23 - 2948210 * q^25 + 28697814 * q^27 - 180180692 * q^29 - 211581400 * q^31 - 254518200 * q^33 + 569181200 * q^35 - 112222700 * q^37 + 5564700 * q^39 - 726961180 * q^41 + 216408856 * q^43 - 312959700 * q^45 + 2174779088 * q^47 + 2402462706 * q^49 + 2122191900 * q^51 + 112024700 * q^53 - 3731208560 * q^55 + 1785223800 * q^57 - 3243949400 * q^59 - 16526230620 * q^61 + 2295352728 * q^63 + 5478177080 * q^65 - 20772619112 * q^67 - 1630953792 * q^69 + 20637101600 * q^71 - 18548203500 * q^73 - 716415030 * q^75 - 74580754400 * q^77 - 28230083800 * q^79 + 6973568802 * q^81 + 7189282056 * q^83 - 66324859080 * q^85 - 43783908156 * q^87 + 103679180788 * q^89 + 46602268400 * q^91 - 51414280200 * q^93 - 63833162000 * q^95 + 199614486500 * q^97 - 61847922600 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	0	0
$5$	3700.71	0.529603	0.264801	−	0.964303i	$-0.414694\pi$
$5$	3700.71	0.529603	0.264801	+	0.964303i	$0.414694\pi$
$6$	0	0
$7$	72358.6	1.62724	0.813619	−	0.581399i	$-0.197494\pi$
$7$	72358.6	1.62724	0.813619	+	0.581399i	$0.197494\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	−1.03599e6	−1.93953	−0.969764	−	0.244045i	$-0.921526\pi$
$11$	−1.03599e6	−1.93953	−0.969764	+	0.244045i	$0.921526\pi$
$12$	0	0
$13$	447532.	0.334300	0.167150	−	0.985932i	$-0.446544\pi$
$13$	447532.	0.334300	0.167150	+	0.985932i	$0.446544\pi$
$14$	0	0
$15$	899273.	0.305766
$16$	0	0
$17$	966903.	0.165163	0.0825817	−	0.996584i	$-0.473683\pi$
$17$	966903.	0.165163	0.0825817	+	0.996584i	$0.473683\pi$
$18$	0	0
$19$	180409.	0.0167153	0.00835765	−	0.999965i	$-0.497340\pi$
$19$	180409.	0.0167153	0.00835765	+	0.999965i	$0.497340\pi$
$20$	0	0
$21$	1.75831e7	0.939486
$22$	0	0
$23$	−1.67982e7	−0.544202	−0.272101	−	0.962269i	$-0.587718\pi$
$23$	−1.67982e7	−0.544202	−0.272101	+	0.962269i	$0.587718\pi$
$24$	0	0
$25$	−3.51329e7	−0.719521
$26$	0	0
$27$	1.43489e7	0.192450
$28$	0	0
$29$	−1.48887e8	−1.34793	−0.673967	−	0.738761i	$-0.735411\pi$
$29$	−1.48887e8	−1.34793	−0.673967	+	0.738761i	$0.735411\pi$
$30$	0	0
$31$	−2.16841e8	−1.36036	−0.680178	−	0.733047i	$-0.738098\pi$
$31$	−2.16841e8	−1.36036	−0.680178	+	0.733047i	$0.738098\pi$
$32$	0	0
$33$	−2.51746e8	−1.11979
$34$	0	0
$35$	2.67778e8	0.861789
$36$	0	0
$37$	−6.70953e8	−1.59068	−0.795340	−	0.606164i	$-0.792708\pi$
$37$	−6.70953e8	−1.59068	−0.795340	+	0.606164i	$0.792708\pi$
$38$	0	0
$39$	1.08750e8	0.193008
$40$	0	0
$41$	−4.72819e8	−0.637358	−0.318679	−	0.947863i	$-0.603239\pi$
$41$	−4.72819e8	−0.637358	−0.318679	+	0.947863i	$0.603239\pi$
$42$	0	0
$43$	−1.14447e9	−1.18721	−0.593607	−	0.804755i	$-0.702297\pi$
$43$	−1.14447e9	−1.18721	−0.593607	+	0.804755i	$0.702297\pi$
$44$	0	0
$45$	2.18523e8	0.176534
$46$	0	0
$47$	1.31669e7	0.00837425	0.00418712	−	0.999991i	$-0.498667\pi$
$47$	1.31669e7	0.00837425	0.00418712	+	0.999991i	$0.498667\pi$
$48$	0	0
$49$	3.25844e9	1.64790
$50$	0	0
$51$	2.34957e8	0.0953572
$52$	0	0
$53$	4.04271e9	1.32787	0.663935	−	0.747791i	$-0.268885\pi$
$53$	4.04271e9	1.32787	0.663935	+	0.747791i	$0.268885\pi$
$54$	0	0
$55$	−3.83390e9	−1.02718
$56$	0	0
$57$	4.38395e7	0.00965059
$58$	0	0
$59$	6.23772e9	1.13590	0.567949	−	0.823064i	$-0.307737\pi$
$59$	6.23772e9	1.13590	0.567949	+	0.823064i	$0.307737\pi$
$60$	0	0
$61$	−6.61870e9	−1.00336	−0.501682	−	0.865052i	$-0.667285\pi$
$61$	−6.61870e9	−1.00336	−0.501682	+	0.865052i	$0.667285\pi$
$62$	0	0
$63$	4.27270e9	0.542412
$64$	0	0
$65$	1.65619e9	0.177046
$66$	0	0
$67$	−1.97892e10	−1.79067	−0.895336	−	0.445391i	$-0.853065\pi$
$67$	−1.97892e10	−1.79067	−0.895336	+	0.445391i	$0.853065\pi$
$68$	0	0
$69$	−4.08196e9	−0.314195
$70$	0	0
$71$	2.10433e10	1.38418	0.692090	−	0.721811i	$-0.256690\pi$
$71$	2.10433e10	1.38418	0.692090	+	0.721811i	$0.256690\pi$
$72$	0	0
$73$	−1.27113e10	−0.717652	−0.358826	−	0.933404i	$-0.616823\pi$
$73$	−1.27113e10	−0.717652	−0.358826	+	0.933404i	$0.616823\pi$
$74$	0	0
$75$	−8.53729e9	−0.415416
$76$	0	0
$77$	−7.49628e10	−3.15607
$78$	0	0
$79$	−1.16162e10	−0.424731	−0.212365	−	0.977190i	$-0.568117\pi$
$79$	−1.16162e10	−0.424731	−0.212365	+	0.977190i	$0.568117\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	5.54557e10	1.54531	0.772657	−	0.634824i	$-0.218927\pi$
$83$	5.54557e10	1.54531	0.772657	+	0.634824i	$0.218927\pi$
$84$	0	0
$85$	3.57823e9	0.0874710
$86$	0	0
$87$	−3.61796e10	−0.778230
$88$	0	0
$89$	5.43356e10	1.03143	0.515715	−	0.856760i	$-0.327526\pi$
$89$	5.43356e10	1.03143	0.515715	+	0.856760i	$0.327526\pi$
$90$	0	0
$91$	3.23828e10	0.543985
$92$	0	0
$93$	−5.26924e10	−0.785402
$94$	0	0
$95$	6.67643e8	0.00885247
$96$	0	0
$97$	8.98530e10	1.06240	0.531200	−	0.847247i	$-0.321741\pi$
$97$	8.98530e10	1.06240	0.531200	+	0.847247i	$0.321741\pi$
$98$	0	0
$99$	−6.11742e10	−0.646509
$100$	0	0
$101$	2.25134e10	0.213144	0.106572	−	0.994305i	$-0.466013\pi$
$101$	2.25134e10	0.213144	0.106572	+	0.994305i	$0.466013\pi$
$102$	0	0
$103$	9.54872e9	0.0811597	0.0405798	−	0.999176i	$-0.487079\pi$
$103$	9.54872e9	0.0811597	0.0405798	+	0.999176i	$0.487079\pi$
$104$	0	0
$105$	6.50701e10	0.497554
$106$	0	0
$107$	−9.38434e10	−0.646835	−0.323417	−	0.946256i	$-0.604832\pi$
$107$	−9.38434e10	−0.646835	−0.323417	+	0.946256i	$0.604832\pi$
$108$	0	0
$109$	−1.91213e11	−1.19034	−0.595171	−	0.803599i	$-0.702916\pi$
$109$	−1.91213e11	−1.19034	−0.595171	+	0.803599i	$0.702916\pi$
$110$	0	0
$111$	−1.63042e11	−0.918379
$112$	0	0
$113$	4.95153e10	0.252818	0.126409	−	0.991978i	$-0.459655\pi$
$113$	4.95153e10	0.252818	0.126409	+	0.991978i	$0.459655\pi$
$114$	0	0
$115$	−6.21653e10	−0.288211
$116$	0	0
$117$	2.64263e10	0.111433
$118$	0	0
$119$	6.99637e10	0.268760
$120$	0	0
$121$	7.87965e11	2.76177
$122$	0	0
$123$	−1.14895e11	−0.367979
$124$	0	0
$125$	−3.10715e11	−0.910663
$126$	0	0
$127$	−1.97557e11	−0.530607	−0.265303	−	0.964165i	$-0.585472\pi$
$127$	−1.97557e11	−0.530607	−0.265303	+	0.964165i	$0.585472\pi$
$128$	0	0
$129$	−2.78107e11	−0.685438
$130$	0	0
$131$	−1.83115e11	−0.414697	−0.207348	−	0.978267i	$-0.566483\pi$
$131$	−1.83115e11	−0.414697	−0.207348	+	0.978267i	$0.566483\pi$
$132$	0	0
$133$	1.30542e10	0.0271998
$134$	0	0
$135$	5.31011e10	0.101922
$136$	0	0
$137$	6.28201e11	1.11208	0.556040	−	0.831156i	$-0.312320\pi$
$137$	6.28201e11	1.11208	0.556040	+	0.831156i	$0.312320\pi$
$138$	0	0
$139$	−4.54003e11	−0.742125	−0.371063	−	0.928608i	$-0.621006\pi$
$139$	−4.54003e11	−0.742125	−0.371063	+	0.928608i	$0.621006\pi$
$140$	0	0
$141$	3.19956e9	0.00483487
$142$	0	0
$143$	−4.63639e11	−0.648383
$144$	0	0
$145$	−5.50989e11	−0.713870
$146$	0	0
$147$	7.91800e11	0.951416
$148$	0	0
$149$	−6.73907e10	−0.0751753	−0.0375877	−	0.999293i	$-0.511967\pi$
$149$	−6.73907e10	−0.0751753	−0.0375877	+	0.999293i	$0.511967\pi$
$150$	0	0
$151$	1.18702e11	0.123051	0.0615255	−	0.998106i	$-0.480403\pi$
$151$	1.18702e11	0.123051	0.0615255	+	0.998106i	$0.480403\pi$
$152$	0	0
$153$	5.70947e10	0.0550545
$154$	0	0
$155$	−8.02467e11	−0.720448
$156$	0	0
$157$	4.26981e11	0.357240	0.178620	−	0.983918i	$-0.442837\pi$
$157$	4.26981e11	0.357240	0.178620	+	0.983918i	$0.442837\pi$
$158$	0	0
$159$	9.82379e11	0.766646
$160$	0	0
$161$	−1.21549e12	−0.885545
$162$	0	0
$163$	−1.59306e12	−1.08443	−0.542213	−	0.840241i	$-0.682413\pi$
$163$	−1.59306e12	−1.08443	−0.542213	+	0.840241i	$0.682413\pi$
$164$	0	0
$165$	−9.31638e11	−0.593042
$166$	0	0
$167$	−2.86964e12	−1.70957	−0.854784	−	0.518984i	$-0.826310\pi$
$167$	−2.86964e12	−1.70957	−0.854784	+	0.518984i	$0.826310\pi$
$168$	0	0
$169$	−1.59188e12	−0.888244
$170$	0	0
$171$	1.06530e10	0.00557177
$172$	0	0
$173$	9.44648e11	0.463464	0.231732	−	0.972780i	$-0.425561\pi$
$173$	9.44648e11	0.463464	0.231732	+	0.972780i	$0.425561\pi$
$174$	0	0
$175$	−2.54216e12	−1.17083
$176$	0	0
$177$	1.51576e12	0.655811
$178$	0	0
$179$	1.03716e11	0.0421846	0.0210923	−	0.999778i	$-0.493286\pi$
$179$	1.03716e11	0.0421846	0.0210923	+	0.999778i	$0.493286\pi$
$180$	0	0
$181$	3.88807e12	1.48765	0.743826	−	0.668373i	$-0.233009\pi$
$181$	3.88807e12	1.48765	0.743826	+	0.668373i	$0.233009\pi$
$182$	0	0
$183$	−1.60835e12	−0.579293
$184$	0	0
$185$	−2.48300e12	−0.842428
$186$	0	0
$187$	−1.00170e12	−0.320339
$188$	0	0
$189$	1.03827e12	0.313162
$190$	0	0
$191$	−5.02318e12	−1.42986	−0.714932	−	0.699194i	$-0.753543\pi$
$191$	−5.02318e12	−1.42986	−0.714932	+	0.699194i	$0.753543\pi$
$192$	0	0
$193$	9.75193e11	0.262135	0.131068	−	0.991373i	$-0.458159\pi$
$193$	9.75193e11	0.262135	0.131068	+	0.991373i	$0.458159\pi$
$194$	0	0
$195$	4.02453e11	0.102217
$196$	0	0
$197$	2.81611e12	0.676216	0.338108	−	0.941107i	$-0.390213\pi$
$197$	2.81611e12	0.676216	0.338108	+	0.941107i	$0.390213\pi$
$198$	0	0
$199$	5.97846e12	1.35799	0.678996	−	0.734142i	$-0.262415\pi$
$199$	5.97846e12	1.35799	0.678996	+	0.734142i	$0.262415\pi$
$200$	0	0
$201$	−4.80877e12	−1.03385
$202$	0	0
$203$	−1.07733e13	−2.19341
$204$	0	0
$205$	−1.74976e12	−0.337546
$206$	0	0
$207$	−9.91917e11	−0.181401
$208$	0	0
$209$	−1.86902e11	−0.0324198
$210$	0	0
$211$	−1.16625e13	−1.91972	−0.959862	−	0.280473i	$-0.909509\pi$
$211$	−1.16625e13	−1.91972	−0.959862	+	0.280473i	$0.909509\pi$
$212$	0	0
$213$	5.11352e12	0.799157
$214$	0	0
$215$	−4.23536e12	−0.628752
$216$	0	0
$217$	−1.56903e13	−2.21362
$218$	0	0
$219$	−3.08884e12	−0.414337
$220$	0	0
$221$	4.32720e11	0.0552141
$222$	0	0
$223$	−5.47931e12	−0.665348	−0.332674	−	0.943042i	$-0.607951\pi$
$223$	−5.47931e12	−0.665348	−0.332674	+	0.943042i	$0.607951\pi$
$224$	0	0
$225$	−2.07456e12	−0.239840
$226$	0	0
$227$	7.92971e12	0.873203	0.436601	−	0.899655i	$-0.356182\pi$
$227$	7.92971e12	0.873203	0.436601	+	0.899655i	$0.356182\pi$
$228$	0	0
$229$	−1.43526e13	−1.50604	−0.753019	−	0.657998i	$-0.771403\pi$
$229$	−1.43526e13	−1.50604	−0.753019	+	0.657998i	$0.771403\pi$
$230$	0	0
$231$	−1.82160e13	−1.82216
$232$	0	0
$233$	−1.04880e13	−1.00054	−0.500270	−	0.865870i	$-0.666766\pi$
$233$	−1.04880e13	−1.00054	−0.500270	+	0.865870i	$0.666766\pi$
$234$	0	0
$235$	4.87269e10	0.00443502
$236$	0	0
$237$	−2.82273e12	−0.245218
$238$	0	0
$239$	−2.19376e12	−0.181971	−0.0909853	−	0.995852i	$-0.529002\pi$
$239$	−2.19376e12	−0.181971	−0.0909853	+	0.995852i	$0.529002\pi$
$240$	0	0
$241$	1.67370e13	1.32612	0.663061	−	0.748566i	$-0.269257\pi$
$241$	1.67370e13	1.32612	0.663061	+	0.748566i	$0.269257\pi$
$242$	0	0
$243$	8.47289e11	0.0641500
$244$	0	0
$245$	1.20585e13	0.872732
$246$	0	0
$247$	8.07390e10	0.00558792
$248$	0	0
$249$	1.34757e13	0.892187
$250$	0	0
$251$	8.55537e11	0.0542043	0.0271021	−	0.999633i	$-0.491372\pi$
$251$	8.55537e11	0.0542043	0.0271021	+	0.999633i	$0.491372\pi$
$252$	0	0
$253$	1.74028e13	1.05549
$254$	0	0
$255$	8.69509e11	0.0505014
$256$	0	0
$257$	3.35801e13	1.86831	0.934157	−	0.356863i	$-0.116154\pi$
$257$	3.35801e13	1.86831	0.934157	+	0.356863i	$0.116154\pi$
$258$	0	0
$259$	−4.85492e13	−2.58841
$260$	0	0
$261$	−8.79165e12	−0.449311
$262$	0	0
$263$	4.27665e12	0.209579	0.104789	−	0.994494i	$-0.466583\pi$
$263$	4.27665e12	0.209579	0.104789	+	0.994494i	$0.466583\pi$
$264$	0	0
$265$	1.49609e13	0.703243
$266$	0	0
$267$	1.32036e13	0.595496
$268$	0	0
$269$	−5.98975e12	−0.259281	−0.129641	−	0.991561i	$-0.541382\pi$
$269$	−5.98975e12	−0.259281	−0.129641	+	0.991561i	$0.541382\pi$
$270$	0	0
$271$	−6.73797e12	−0.280026	−0.140013	−	0.990150i	$-0.544714\pi$
$271$	−6.73797e12	−0.280026	−0.140013	+	0.990150i	$0.544714\pi$
$272$	0	0
$273$	7.86902e12	0.314070
$274$	0	0
$275$	3.63973e13	1.39553
$276$	0	0
$277$	1.07764e13	0.397041	0.198520	−	0.980097i	$-0.436386\pi$
$277$	1.07764e13	0.397041	0.198520	+	0.980097i	$0.436386\pi$
$278$	0	0
$279$	−1.28043e13	−0.453452
$280$	0	0
$281$	3.69195e13	1.25710	0.628552	−	0.777768i	$-0.283648\pi$
$281$	3.69195e13	1.25710	0.628552	+	0.777768i	$0.283648\pi$
$282$	0	0
$283$	−3.10491e13	−1.01677	−0.508386	−	0.861129i	$-0.669758\pi$
$283$	−3.10491e13	−1.01677	−0.508386	+	0.861129i	$0.669758\pi$
$284$	0	0
$285$	1.62237e11	0.00511098
$286$	0	0
$287$	−3.42125e13	−1.03713
$288$	0	0
$289$	−3.33370e13	−0.972721
$290$	0	0
$291$	2.18343e13	0.613377
$292$	0	0
$293$	−5.52302e13	−1.49419	−0.747093	−	0.664720i	$-0.768551\pi$
$293$	−5.52302e13	−1.49419	−0.747093	+	0.664720i	$0.768551\pi$
$294$	0	0
$295$	2.30840e13	0.601575
$296$	0	0
$297$	−1.48653e13	−0.373262
$298$	0	0
$299$	−7.51774e12	−0.181926
$300$	0	0
$301$	−8.28125e13	−1.93188
$302$	0	0
$303$	5.47076e12	0.123059
$304$	0	0
$305$	−2.44939e13	−0.531385
$306$	0	0
$307$	−7.43204e13	−1.55542	−0.777709	−	0.628624i	$-0.783618\pi$
$307$	−7.43204e13	−1.55542	−0.777709	+	0.628624i	$0.783618\pi$
$308$	0	0
$309$	2.32034e12	0.0468576
$310$	0	0
$311$	−3.89430e13	−0.759010	−0.379505	−	0.925190i	$-0.623906\pi$
$311$	−3.89430e13	−0.759010	−0.379505	+	0.925190i	$0.623906\pi$
$312$	0	0
$313$	3.90171e13	0.734110	0.367055	−	0.930199i	$-0.380366\pi$
$313$	3.90171e13	0.734110	0.367055	+	0.930199i	$0.380366\pi$
$314$	0	0
$315$	1.58120e13	0.287263
$316$	0	0
$317$	−4.65971e13	−0.817585	−0.408792	−	0.912627i	$-0.634050\pi$
$317$	−4.65971e13	−0.817585	−0.408792	+	0.912627i	$0.634050\pi$
$318$	0	0
$319$	1.54246e14	2.61436
$320$	0	0
$321$	−2.28040e13	−0.373450
$322$	0	0
$323$	1.74438e11	0.00276076
$324$	0	0
$325$	−1.57231e13	−0.240536
$326$	0	0
$327$	−4.64648e13	−0.687245
$328$	0	0
$329$	9.52739e11	0.0136269
$330$	0	0
$331$	6.99455e13	0.967623	0.483811	−	0.875172i	$-0.339252\pi$
$331$	6.99455e13	0.967623	0.483811	+	0.875172i	$0.339252\pi$
$332$	0	0
$333$	−3.96191e13	−0.530227
$334$	0	0
$335$	−7.32340e13	−0.948345
$336$	0	0
$337$	1.11801e14	1.40114	0.700569	−	0.713585i	$-0.252930\pi$
$337$	1.11801e14	1.40114	0.700569	+	0.713585i	$0.252930\pi$
$338$	0	0
$339$	1.20322e13	0.145965
$340$	0	0
$341$	2.24646e14	2.63845
$342$	0	0
$343$	9.26994e13	1.05429
$344$	0	0
$345$	−1.51062e13	−0.166398
$346$	0	0
$347$	−3.63901e13	−0.388303	−0.194152	−	0.980972i	$-0.562195\pi$
$347$	−3.63901e13	−0.388303	−0.194152	+	0.980972i	$0.562195\pi$
$348$	0	0
$349$	−8.52083e12	−0.0880932	−0.0440466	−	0.999029i	$-0.514025\pi$
$349$	−8.52083e12	−0.0880932	−0.0440466	+	0.999029i	$0.514025\pi$
$350$	0	0
$351$	6.42160e12	0.0643360
$352$	0	0
$353$	−4.50603e13	−0.437555	−0.218778	−	0.975775i	$-0.570207\pi$
$353$	−4.50603e13	−0.437555	−0.218778	+	0.975775i	$0.570207\pi$
$354$	0	0
$355$	7.78751e13	0.733065
$356$	0	0
$357$	1.70012e13	0.155169
$358$	0	0
$359$	1.06639e12	0.00943840	0.00471920	−	0.999989i	$-0.498498\pi$
$359$	1.06639e12	0.00943840	0.00471920	+	0.999989i	$0.498498\pi$
$360$	0	0
$361$	−1.16458e14	−0.999721
$362$	0	0
$363$	1.91475e14	1.59451
$364$	0	0
$365$	−4.70408e13	−0.380070
$366$	0	0
$367$	2.34488e14	1.83848	0.919238	−	0.393702i	$-0.128806\pi$
$367$	2.34488e14	1.83848	0.919238	+	0.393702i	$0.128806\pi$
$368$	0	0
$369$	−2.79195e13	−0.212453
$370$	0	0
$371$	2.92525e14	2.16076
$372$	0	0
$373$	2.10203e14	1.50744	0.753721	−	0.657195i	$-0.228257\pi$
$373$	2.10203e14	1.50744	0.753721	+	0.657195i	$0.228257\pi$
$374$	0	0
$375$	−7.55038e13	−0.525771
$376$	0	0
$377$	−6.66319e13	−0.450614
$378$	0	0
$379$	−5.01106e13	−0.329165	−0.164583	−	0.986363i	$-0.552628\pi$
$379$	−5.01106e13	−0.329165	−0.164583	+	0.986363i	$0.552628\pi$
$380$	0	0
$381$	−4.80064e13	−0.306346
$382$	0	0
$383$	−1.72523e13	−0.106968	−0.0534839	−	0.998569i	$-0.517033\pi$
$383$	−1.72523e13	−0.106968	−0.0534839	+	0.998569i	$0.517033\pi$
$384$	0	0
$385$	−2.77416e14	−1.67146
$386$	0	0
$387$	−6.75800e13	−0.395738
$388$	0	0
$389$	−6.19041e13	−0.352368	−0.176184	−	0.984357i	$-0.556375\pi$
$389$	−6.19041e13	−0.352368	−0.176184	+	0.984357i	$0.556375\pi$
$390$	0	0
$391$	−1.62422e13	−0.0898822
$392$	0	0
$393$	−4.44968e13	−0.239425
$394$	0	0
$395$	−4.29881e13	−0.224938
$396$	0	0
$397$	1.69202e14	0.861110	0.430555	−	0.902564i	$-0.358318\pi$
$397$	1.69202e14	0.861110	0.430555	+	0.902564i	$0.358318\pi$
$398$	0	0
$399$	3.17216e12	0.0157038
$400$	0	0
$401$	3.29537e14	1.58712	0.793560	−	0.608492i	$-0.208225\pi$
$401$	3.29537e14	1.58712	0.793560	+	0.608492i	$0.208225\pi$
$402$	0	0
$403$	−9.70435e13	−0.454767
$404$	0	0
$405$	1.29036e13	0.0588447
$406$	0	0
$407$	6.95101e14	3.08517
$408$	0	0
$409$	−3.14267e14	−1.35775	−0.678876	−	0.734253i	$-0.737533\pi$
$409$	−3.14267e14	−1.35775	−0.678876	+	0.734253i	$0.737533\pi$
$410$	0	0
$411$	1.52653e14	0.642059
$412$	0	0
$413$	4.51352e14	1.84838
$414$	0	0
$415$	2.05225e14	0.818402
$416$	0	0
$417$	−1.10323e14	−0.428466
$418$	0	0
$419$	3.34666e14	1.26600	0.633002	−	0.774150i	$-0.281823\pi$
$419$	3.34666e14	1.26600	0.633002	+	0.774150i	$0.281823\pi$
$420$	0	0
$421$	−2.04256e14	−0.752704	−0.376352	−	0.926477i	$-0.622822\pi$
$421$	−2.04256e14	−0.752704	−0.376352	+	0.926477i	$0.622822\pi$
$422$	0	0
$423$	7.77493e11	0.00279142
$424$	0	0
$425$	−3.39701e13	−0.118839
$426$	0	0
$427$	−4.78920e14	−1.63271
$428$	0	0
$429$	−1.12664e14	−0.374344
$430$	0	0
$431$	−3.34783e14	−1.08427	−0.542136	−	0.840291i	$-0.682384\pi$
$431$	−3.34783e14	−1.08427	−0.542136	+	0.840291i	$0.682384\pi$
$432$	0	0
$433$	4.97010e14	1.56921	0.784606	−	0.619995i	$-0.212865\pi$
$433$	4.97010e14	1.56921	0.784606	+	0.619995i	$0.212865\pi$
$434$	0	0
$435$	−1.33890e14	−0.412153
$436$	0	0
$437$	−3.03055e12	−0.00909650
$438$	0	0
$439$	−5.20500e14	−1.52358	−0.761791	−	0.647823i	$-0.775680\pi$
$439$	−5.20500e14	−1.52358	−0.761791	+	0.647823i	$0.775680\pi$
$440$	0	0
$441$	1.92408e14	0.549300
$442$	0	0
$443$	4.95757e14	1.38054	0.690269	−	0.723553i	$-0.257492\pi$
$443$	4.95757e14	1.38054	0.690269	+	0.723553i	$0.257492\pi$
$444$	0	0
$445$	2.01080e14	0.546248
$446$	0	0
$447$	−1.63759e13	−0.0434025
$448$	0	0
$449$	−5.58259e14	−1.44371	−0.721857	−	0.692042i	$-0.756711\pi$
$449$	−5.58259e14	−1.44371	−0.721857	+	0.692042i	$0.756711\pi$
$450$	0	0
$451$	4.89836e14	1.23617
$452$	0	0
$453$	2.88446e13	0.0710436
$454$	0	0
$455$	1.19839e14	0.288096
$456$	0	0
$457$	6.01951e13	0.141261	0.0706305	−	0.997503i	$-0.477499\pi$
$457$	6.01951e13	0.141261	0.0706305	+	0.997503i	$0.477499\pi$
$458$	0	0
$459$	1.38740e13	0.0317857
$460$	0	0
$461$	5.66904e14	1.26810	0.634051	−	0.773291i	$-0.281391\pi$
$461$	5.66904e14	1.26810	0.634051	+	0.773291i	$0.281391\pi$
$462$	0	0
$463$	5.23760e14	1.14403	0.572015	−	0.820243i	$-0.306162\pi$
$463$	5.23760e14	1.14403	0.572015	+	0.820243i	$0.306162\pi$
$464$	0	0
$465$	−1.94999e14	−0.415951
$466$	0	0
$467$	−1.49157e14	−0.310743	−0.155371	−	0.987856i	$-0.549657\pi$
$467$	−1.49157e14	−0.310743	−0.155371	+	0.987856i	$0.549657\pi$
$468$	0	0
$469$	−1.43192e15	−2.91385
$470$	0	0
$471$	1.03756e14	0.206253
$472$	0	0
$473$	1.18566e15	2.30264
$474$	0	0
$475$	−6.33830e12	−0.0120270
$476$	0	0
$477$	2.38718e14	0.442623
$478$	0	0
$479$	4.11276e14	0.745225	0.372613	−	0.927987i	$-0.378462\pi$
$479$	4.11276e14	0.745225	0.372613	+	0.927987i	$0.378462\pi$
$480$	0	0
$481$	−3.00273e14	−0.531764
$482$	0	0
$483$	−2.95365e14	−0.511270
$484$	0	0
$485$	3.32520e14	0.562650
$486$	0	0
$487$	1.09682e14	0.181437	0.0907186	−	0.995877i	$-0.471084\pi$
$487$	1.09682e14	0.181437	0.0907186	+	0.995877i	$0.471084\pi$
$488$	0	0
$489$	−3.87113e14	−0.626093
$490$	0	0
$491$	−4.72537e14	−0.747288	−0.373644	−	0.927572i	$-0.621892\pi$
$491$	−4.72537e14	−0.747288	−0.373644	+	0.927572i	$0.621892\pi$
$492$	0	0
$493$	−1.43960e14	−0.222629
$494$	0	0
$495$	−2.26388e14	−0.342393
$496$	0	0
$497$	1.52266e15	2.25239
$498$	0	0
$499$	5.74148e14	0.830750	0.415375	−	0.909650i	$-0.363650\pi$
$499$	5.74148e14	0.830750	0.415375	+	0.909650i	$0.363650\pi$
$500$	0	0
$501$	−6.97322e14	−0.987020
$502$	0	0
$503$	−1.08687e15	−1.50505	−0.752527	−	0.658562i	$-0.771165\pi$
$503$	−1.08687e15	−1.50505	−0.752527	+	0.658562i	$0.771165\pi$
$504$	0	0
$505$	8.33156e13	0.112882
$506$	0	0
$507$	−3.86826e14	−0.512828
$508$	0	0
$509$	−1.21857e15	−1.58089	−0.790444	−	0.612534i	$-0.790150\pi$
$509$	−1.21857e15	−1.58089	−0.790444	+	0.612534i	$0.790150\pi$
$510$	0	0
$511$	−9.19771e14	−1.16779
$512$	0	0
$513$	2.58868e12	0.00321686
$514$	0	0
$515$	3.53370e13	0.0429824
$516$	0	0
$517$	−1.36408e13	−0.0162421
$518$	0	0
$519$	2.29549e14	0.267581
$520$	0	0
$521$	2.69864e14	0.307990	0.153995	−	0.988072i	$-0.450786\pi$
$521$	2.69864e14	0.307990	0.153995	+	0.988072i	$0.450786\pi$
$522$	0	0
$523$	1.16264e15	1.29922	0.649612	−	0.760266i	$-0.274931\pi$
$523$	1.16264e15	1.29922	0.649612	+	0.760266i	$0.274931\pi$
$524$	0	0
$525$	−6.17746e14	−0.675980
$526$	0	0
$527$	−2.09665e14	−0.224681
$528$	0	0
$529$	−6.70630e14	−0.703845
$530$	0	0
$531$	3.68331e14	0.378633
$532$	0	0
$533$	−2.11602e14	−0.213068
$534$	0	0
$535$	−3.47287e14	−0.342565
$536$	0	0
$537$	2.52030e13	0.0243553
$538$	0	0
$539$	−3.37571e15	−3.19615
$540$	0	0
$541$	−8.25701e14	−0.766016	−0.383008	−	0.923745i	$-0.625112\pi$
$541$	−8.25701e14	−0.766016	−0.383008	+	0.923745i	$0.625112\pi$
$542$	0	0
$543$	9.44800e14	0.858896
$544$	0	0
$545$	−7.07624e14	−0.630408
$546$	0	0
$547$	−5.74544e13	−0.0501641	−0.0250820	−	0.999685i	$-0.507985\pi$
$547$	−5.74544e13	−0.0501641	−0.0250820	+	0.999685i	$0.507985\pi$
$548$	0	0
$549$	−3.90828e14	−0.334455
$550$	0	0
$551$	−2.68607e13	−0.0225311
$552$	0	0
$553$	−8.40529e14	−0.691138
$554$	0	0
$555$	−6.03370e14	−0.486376
$556$	0	0
$557$	7.26265e14	0.573973	0.286987	−	0.957935i	$-0.407346\pi$
$557$	7.26265e14	0.573973	0.286987	+	0.957935i	$0.407346\pi$
$558$	0	0
$559$	−5.12188e14	−0.396885
$560$	0	0
$561$	−2.43414e14	−0.184948
$562$	0	0
$563$	−1.71049e15	−1.27446	−0.637228	−	0.770676i	$-0.719919\pi$
$563$	−1.71049e15	−1.27446	−0.637228	+	0.770676i	$0.719919\pi$
$564$	0	0
$565$	1.83242e14	0.133893
$566$	0	0
$567$	2.52299e14	0.180804
$568$	0	0
$569$	1.09455e15	0.769343	0.384671	−	0.923054i	$-0.374315\pi$
$569$	1.09455e15	0.769343	0.384671	+	0.923054i	$0.374315\pi$
$570$	0	0
$571$	−1.11533e15	−0.768965	−0.384482	−	0.923132i	$-0.625620\pi$
$571$	−1.11533e15	−0.768965	−0.384482	+	0.923132i	$0.625620\pi$
$572$	0	0
$573$	−1.22063e15	−0.825533
$574$	0	0
$575$	5.90169e14	0.391565
$576$	0	0
$577$	−2.38973e14	−0.155554	−0.0777771	−	0.996971i	$-0.524782\pi$
$577$	−2.38973e14	−0.155554	−0.0777771	+	0.996971i	$0.524782\pi$
$578$	0	0
$579$	2.36972e14	0.151344
$580$	0	0
$581$	4.01270e15	2.51459
$582$	0	0
$583$	−4.18821e15	−2.57544
$584$	0	0
$585$	9.77962e13	0.0590153
$586$	0	0
$587$	1.38562e15	0.820606	0.410303	−	0.911949i	$-0.365423\pi$
$587$	1.38562e15	0.820606	0.410303	+	0.911949i	$0.365423\pi$
$588$	0	0
$589$	−3.91202e13	−0.0227388
$590$	0	0
$591$	6.84315e14	0.390414
$592$	0	0
$593$	1.16303e15	0.651313	0.325656	−	0.945488i	$-0.394415\pi$
$593$	1.16303e15	0.651313	0.325656	+	0.945488i	$0.394415\pi$
$594$	0	0
$595$	2.58916e14	0.142336
$596$	0	0
$597$	1.45277e15	0.784037
$598$	0	0
$599$	−1.93685e15	−1.02624	−0.513119	−	0.858317i	$-0.671510\pi$
$599$	−1.93685e15	−1.02624	−0.513119	+	0.858317i	$0.671510\pi$
$600$	0	0
$601$	1.80344e15	0.938194	0.469097	−	0.883147i	$-0.344580\pi$
$601$	1.80344e15	0.938194	0.469097	+	0.883147i	$0.344580\pi$
$602$	0	0
$603$	−1.16853e15	−0.596891
$604$	0	0
$605$	2.91603e15	1.46264
$606$	0	0
$607$	−3.61637e15	−1.78129	−0.890646	−	0.454696i	$-0.849748\pi$
$607$	−3.61637e15	−1.78129	−0.890646	+	0.454696i	$0.849748\pi$
$608$	0	0
$609$	−2.61791e15	−1.26637
$610$	0	0
$611$	5.89262e12	0.00279951
$612$	0	0
$613$	1.52906e15	0.713496	0.356748	−	0.934201i	$-0.383886\pi$
$613$	1.52906e15	0.713496	0.356748	+	0.934201i	$0.383886\pi$
$614$	0	0
$615$	−4.25193e14	−0.194882
$616$	0	0
$617$	2.55216e15	1.14905	0.574527	−	0.818486i	$-0.305186\pi$
$617$	2.55216e15	1.14905	0.574527	+	0.818486i	$0.305186\pi$
$618$	0	0
$619$	−3.80917e14	−0.168474	−0.0842368	−	0.996446i	$-0.526845\pi$
$619$	−3.80917e14	−0.168474	−0.0842368	+	0.996446i	$0.526845\pi$
$620$	0	0
$621$	−2.41036e14	−0.104732
$622$	0	0
$623$	3.93165e15	1.67838
$624$	0	0
$625$	5.65605e14	0.237232
$626$	0	0
$627$	−4.54173e13	−0.0187176
$628$	0	0
$629$	−6.48747e14	−0.262722
$630$	0	0
$631$	−3.56890e14	−0.142027	−0.0710137	−	0.997475i	$-0.522623\pi$
$631$	−3.56890e14	−0.142027	−0.0710137	+	0.997475i	$0.522623\pi$
$632$	0	0
$633$	−2.83399e15	−1.10835
$634$	0	0
$635$	−7.31102e14	−0.281011
$636$	0	0
$637$	1.45826e15	0.550892
$638$	0	0
$639$	1.24258e15	0.461393
$640$	0	0
$641$	−2.54768e15	−0.929877	−0.464938	−	0.885343i	$-0.653923\pi$
$641$	−2.54768e15	−0.929877	−0.464938	+	0.885343i	$0.653923\pi$
$642$	0	0
$643$	1.63356e15	0.586103	0.293052	−	0.956097i	$-0.405329\pi$
$643$	1.63356e15	0.586103	0.293052	+	0.956097i	$0.405329\pi$
$644$	0	0
$645$	−1.02919e15	−0.363010
$646$	0	0
$647$	−4.25249e15	−1.47458	−0.737292	−	0.675574i	$-0.763896\pi$
$647$	−4.25249e15	−1.47458	−0.737292	+	0.675574i	$0.763896\pi$
$648$	0	0
$649$	−6.46221e15	−2.20311
$650$	0	0
$651$	−3.81275e15	−1.27804
$652$	0	0
$653$	2.44142e15	0.804675	0.402337	−	0.915491i	$-0.368198\pi$
$653$	2.44142e15	0.804675	0.402337	+	0.915491i	$0.368198\pi$
$654$	0	0
$655$	−6.77654e14	−0.219625
$656$	0	0
$657$	−7.50589e14	−0.239217
$658$	0	0
$659$	2.38613e14	0.0747867	0.0373934	−	0.999301i	$-0.488095\pi$
$659$	2.38613e14	0.0747867	0.0373934	+	0.999301i	$0.488095\pi$
$660$	0	0
$661$	−4.87186e15	−1.50171	−0.750857	−	0.660465i	$-0.770359\pi$
$661$	−4.87186e15	−1.50171	−0.750857	+	0.660465i	$0.770359\pi$
$662$	0	0
$663$	1.05151e14	0.0318779
$664$	0	0
$665$	4.83097e13	0.0144051
$666$	0	0
$667$	2.50104e15	0.733548
$668$	0	0
$669$	−1.33147e15	−0.384139
$670$	0	0
$671$	6.85692e15	1.94605
$672$	0	0
$673$	−3.52255e15	−0.983499	−0.491749	−	0.870737i	$-0.663642\pi$
$673$	−3.52255e15	−0.983499	−0.491749	+	0.870737i	$0.663642\pi$
$674$	0	0
$675$	−5.04118e14	−0.138472
$676$	0	0
$677$	2.58634e15	0.698952	0.349476	−	0.936945i	$-0.386360\pi$
$677$	2.58634e15	0.698952	0.349476	+	0.936945i	$0.386360\pi$
$678$	0	0
$679$	6.50163e15	1.72878
$680$	0	0
$681$	1.92692e15	0.504144
$682$	0	0
$683$	1.30583e15	0.336180	0.168090	−	0.985772i	$-0.446240\pi$
$683$	1.30583e15	0.336180	0.168090	+	0.985772i	$0.446240\pi$
$684$	0	0
$685$	2.32479e15	0.588960
$686$	0	0
$687$	−3.48769e15	−0.869512
$688$	0	0
$689$	1.80924e15	0.443906
$690$	0	0
$691$	1.18257e15	0.285559	0.142780	−	0.989754i	$-0.454396\pi$
$691$	1.18257e15	0.285559	0.142780	+	0.989754i	$0.454396\pi$
$692$	0	0
$693$	−4.42648e15	−1.05202
$694$	0	0
$695$	−1.68013e15	−0.393031
$696$	0	0
$697$	−4.57170e14	−0.105268
$698$	0	0
$699$	−2.54858e15	−0.577662
$700$	0	0
$701$	3.03869e15	0.678012	0.339006	−	0.940784i	$-0.389909\pi$
$701$	3.03869e15	0.678012	0.339006	+	0.940784i	$0.389909\pi$
$702$	0	0
$703$	−1.21046e14	−0.0265887
$704$	0	0
$705$	1.18406e13	0.00256056
$706$	0	0
$707$	1.62904e15	0.346836
$708$	0	0
$709$	−2.59417e15	−0.543807	−0.271903	−	0.962325i	$-0.587653\pi$
$709$	−2.59417e15	−0.543807	−0.271903	+	0.962325i	$0.587653\pi$
$710$	0	0
$711$	−6.85923e14	−0.141577
$712$	0	0
$713$	3.64255e15	0.740308
$714$	0	0
$715$	−1.71579e15	−0.343385
$716$	0	0
$717$	−5.33084e14	−0.105061
$718$	0	0
$719$	4.81042e15	0.933629	0.466815	−	0.884355i	$-0.345402\pi$
$719$	4.81042e15	0.933629	0.466815	+	0.884355i	$0.345402\pi$
$720$	0	0
$721$	6.90932e14	0.132066
$722$	0	0
$723$	4.06708e15	0.765636
$724$	0	0
$725$	5.23084e15	0.969867
$726$	0	0
$727$	−8.01188e15	−1.46317	−0.731586	−	0.681749i	$-0.761219\pi$
$727$	−8.01188e15	−1.46317	−0.731586	+	0.681749i	$0.761219\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−1.10659e15	−0.196084
$732$	0	0
$733$	−3.63527e15	−0.634549	−0.317275	−	0.948334i	$-0.602768\pi$
$733$	−3.63527e15	−0.634549	−0.317275	+	0.948334i	$0.602768\pi$
$734$	0	0
$735$	2.93022e15	0.503872
$736$	0	0
$737$	2.05014e16	3.47306
$738$	0	0
$739$	8.07406e15	1.34756	0.673779	−	0.738933i	$-0.264670\pi$
$739$	8.07406e15	1.34756	0.673779	+	0.738933i	$0.264670\pi$
$740$	0	0
$741$	1.96196e13	0.00322619
$742$	0	0
$743$	−6.46070e15	−1.04675	−0.523373	−	0.852104i	$-0.675326\pi$
$743$	−6.46070e15	−1.04675	−0.523373	+	0.852104i	$0.675326\pi$
$744$	0	0
$745$	−2.49393e14	−0.0398130
$746$	0	0
$747$	3.27460e15	0.515105
$748$	0	0
$749$	−6.79038e15	−1.05255
$750$	0	0
$751$	1.19982e16	1.83272	0.916360	−	0.400356i	$-0.131114\pi$
$751$	1.19982e16	1.83272	0.916360	+	0.400356i	$0.131114\pi$
$752$	0	0
$753$	2.07896e14	0.0312948
$754$	0	0
$755$	4.39282e14	0.0651682
$756$	0	0
$757$	8.11408e14	0.118635	0.0593174	−	0.998239i	$-0.481108\pi$
$757$	8.11408e14	0.118635	0.0593174	+	0.998239i	$0.481108\pi$
$758$	0	0
$759$	4.22888e15	0.609390
$760$	0	0
$761$	−8.21569e15	−1.16689	−0.583443	−	0.812154i	$-0.698295\pi$
$761$	−8.21569e15	−1.16689	−0.583443	+	0.812154i	$0.698295\pi$
$762$	0	0
$763$	−1.38359e16	−1.93697
$764$	0	0
$765$	2.11291e14	0.0291570
$766$	0	0
$767$	2.79158e15	0.379730
$768$	0	0
$769$	4.70956e15	0.631518	0.315759	−	0.948839i	$-0.397741\pi$
$769$	4.70956e15	0.631518	0.315759	+	0.948839i	$0.397741\pi$
$770$	0	0
$771$	8.15996e15	1.07867
$772$	0	0
$773$	−1.17494e16	−1.53118	−0.765592	−	0.643327i	$-0.777554\pi$
$773$	−1.17494e16	−1.53118	−0.765592	+	0.643327i	$0.777554\pi$
$774$	0	0
$775$	7.61826e15	0.978805
$776$	0	0
$777$	−1.17975e16	−1.49442
$778$	0	0
$779$	−8.53009e13	−0.0106536
$780$	0	0
$781$	−2.18006e16	−2.68466
$782$	0	0
$783$	−2.13637e15	−0.259410
$784$	0	0
$785$	1.58013e15	0.189195
$786$	0	0
$787$	1.33025e16	1.57063	0.785314	−	0.619097i	$-0.212501\pi$
$787$	1.33025e16	1.57063	0.785314	+	0.619097i	$0.212501\pi$
$788$	0	0
$789$	1.03923e15	0.121000
$790$	0	0
$791$	3.58286e15	0.411395
$792$	0	0
$793$	−2.96208e15	−0.335424
$794$	0	0
$795$	3.63550e15	0.406018
$796$	0	0
$797$	1.20782e16	1.33040	0.665199	−	0.746666i	$-0.268347\pi$
$797$	1.20782e16	1.33040	0.665199	+	0.746666i	$0.268347\pi$
$798$	0	0
$799$	1.27311e13	0.00138312
$800$	0	0
$801$	3.20846e15	0.343810
$802$	0	0
$803$	1.31688e16	1.39191
$804$	0	0
$805$	−4.49819e15	−0.468987
$806$	0	0
$807$	−1.45551e15	−0.149696
$808$	0	0
$809$	5.02426e15	0.509748	0.254874	−	0.966974i	$-0.417966\pi$
$809$	5.02426e15	0.509748	0.254874	+	0.966974i	$0.417966\pi$
$810$	0	0
$811$	−3.39996e13	−0.00340298	−0.00170149	−	0.999999i	$-0.500542\pi$
$811$	−3.39996e13	−0.00340298	−0.00170149	+	0.999999i	$0.500542\pi$
$812$	0	0
$813$	−1.63733e15	−0.161673
$814$	0	0
$815$	−5.89544e15	−0.574314
$816$	0	0
$817$	−2.06474e14	−0.0198446
$818$	0	0
$819$	1.91217e15	0.181328
$820$	0	0
$821$	1.80167e16	1.68573	0.842863	−	0.538129i	$-0.180869\pi$
$821$	1.80167e16	1.68573	0.842863	+	0.538129i	$0.180869\pi$
$822$	0	0
$823$	−5.17666e15	−0.477915	−0.238957	−	0.971030i	$-0.576806\pi$
$823$	−5.17666e15	−0.477915	−0.238957	+	0.971030i	$0.576806\pi$
$824$	0	0
$825$	8.84455e15	0.805710
$826$	0	0
$827$	4.37881e15	0.393619	0.196810	−	0.980442i	$-0.436942\pi$
$827$	4.37881e15	0.393619	0.196810	+	0.980442i	$0.436942\pi$
$828$	0	0
$829$	−1.83589e15	−0.162853	−0.0814265	−	0.996679i	$-0.525948\pi$
$829$	−1.83589e15	−0.162853	−0.0814265	+	0.996679i	$0.525948\pi$
$830$	0	0
$831$	2.61867e15	0.229232
$832$	0	0
$833$	3.15059e15	0.272173
$834$	0	0
$835$	−1.06197e16	−0.905392
$836$	0	0
$837$	−3.11144e15	−0.261801
$838$	0	0
$839$	−4.24026e15	−0.352129	−0.176064	−	0.984379i	$-0.556337\pi$
$839$	−4.24026e15	−0.352129	−0.176064	+	0.984379i	$0.556337\pi$
$840$	0	0
$841$	9.96693e15	0.816927
$842$	0	0
$843$	8.97144e15	0.725789
$844$	0	0
$845$	−5.89107e15	−0.470416
$846$	0	0
$847$	5.70160e16	4.49405
$848$	0	0
$849$	−7.54493e15	−0.587033
$850$	0	0
$851$	1.12708e16	0.865651
$852$	0	0
$853$	2.06183e16	1.56327	0.781634	−	0.623737i	$-0.214386\pi$
$853$	2.06183e16	1.56327	0.781634	+	0.623737i	$0.214386\pi$
$854$	0	0
$855$	3.94236e13	0.00295082
$856$	0	0
$857$	1.18051e16	0.872316	0.436158	−	0.899870i	$-0.356339\pi$
$857$	1.18051e16	0.872316	0.436158	+	0.899870i	$0.356339\pi$
$858$	0	0
$859$	−2.58163e16	−1.88335	−0.941676	−	0.336522i	$-0.890749\pi$
$859$	−2.58163e16	−1.88335	−0.941676	+	0.336522i	$0.890749\pi$
$860$	0	0
$861$	−8.31363e15	−0.598789
$862$	0	0
$863$	8.55796e15	0.608570	0.304285	−	0.952581i	$-0.401582\pi$
$863$	8.55796e15	0.608570	0.304285	+	0.952581i	$0.401582\pi$
$864$	0	0
$865$	3.49587e15	0.245452
$866$	0	0
$867$	−8.10089e15	−0.561601
$868$	0	0
$869$	1.20342e16	0.823777
$870$	0	0
$871$	−8.85629e15	−0.598621
$872$	0	0
$873$	5.30573e15	0.354133
$874$	0	0
$875$	−2.24829e16	−1.48186
$876$	0	0
$877$	1.91250e16	1.24481	0.622405	−	0.782695i	$-0.286156\pi$
$877$	1.91250e16	1.24481	0.622405	+	0.782695i	$0.286156\pi$
$878$	0	0
$879$	−1.34209e16	−0.862669
$880$	0	0
$881$	−2.10140e16	−1.33395	−0.666977	−	0.745079i	$-0.732412\pi$
$881$	−2.10140e16	−1.33395	−0.666977	+	0.745079i	$0.732412\pi$
$882$	0	0
$883$	2.22222e16	1.39317	0.696585	−	0.717474i	$-0.254702\pi$
$883$	2.22222e16	1.39317	0.696585	+	0.717474i	$0.254702\pi$
$884$	0	0
$885$	5.60941e15	0.347319
$886$	0	0
$887$	−2.18293e15	−0.133494	−0.0667469	−	0.997770i	$-0.521262\pi$
$887$	−2.18293e15	−0.133494	−0.0667469	+	0.997770i	$0.521262\pi$
$888$	0	0
$889$	−1.42950e16	−0.863423
$890$	0	0
$891$	−3.61228e15	−0.215503
$892$	0	0
$893$	2.37543e12	0.000139978 0
$894$	0	0
$895$	3.83823e14	0.0223411
$896$	0	0
$897$	−1.82681e15	−0.105035
$898$	0	0
$899$	3.22849e16	1.83367
$900$	0	0
$901$	3.90891e15	0.219315
$902$	0	0
$903$	−2.01234e16	−1.11537
$904$	0	0
$905$	1.43886e16	0.787864
$906$	0	0
$907$	−2.06657e16	−1.11792	−0.558959	−	0.829195i	$-0.688799\pi$
$907$	−2.06657e16	−1.11792	−0.558959	+	0.829195i	$0.688799\pi$
$908$	0	0
$909$	1.32939e15	0.0710481
$910$	0	0
$911$	−3.53802e16	−1.86814	−0.934069	−	0.357093i	$-0.883768\pi$
$911$	−3.53802e16	−1.86814	−0.934069	+	0.357093i	$0.883768\pi$
$912$	0	0
$913$	−5.74516e16	−2.99718
$914$	0	0
$915$	−5.95202e15	−0.306795
$916$	0	0
$917$	−1.32499e16	−0.674810
$918$	0	0
$919$	−1.42662e16	−0.717916	−0.358958	−	0.933354i	$-0.616868\pi$
$919$	−1.42662e16	−0.717916	−0.358958	+	0.933354i	$0.616868\pi$
$920$	0	0
$921$	−1.80599e16	−0.898021
$922$	0	0
$923$	9.41754e15	0.462731
$924$	0	0
$925$	2.35725e16	1.14453
$926$	0	0
$927$	5.63842e14	0.0270532
$928$	0	0
$929$	−1.33625e16	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$929$	−1.33625e16	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$930$	0	0
$931$	5.87853e14	0.0275452
$932$	0	0
$933$	−9.46315e15	−0.438215
$934$	0	0
$935$	−3.70701e15	−0.169652
$936$	0	0
$937$	2.55253e16	1.15452	0.577262	−	0.816559i	$-0.304121\pi$
$937$	2.55253e16	1.15452	0.577262	+	0.816559i	$0.304121\pi$
$938$	0	0
$939$	9.48115e15	0.423838
$940$	0	0
$941$	−8.09251e15	−0.357553	−0.178776	−	0.983890i	$-0.557214\pi$
$941$	−8.09251e15	−0.357553	−0.178776	+	0.983890i	$0.557214\pi$
$942$	0	0
$943$	7.94251e15	0.346851
$944$	0	0
$945$	3.84232e15	0.165851
$946$	0	0
$947$	−1.57497e16	−0.671965	−0.335983	−	0.941868i	$-0.609068\pi$
$947$	−1.57497e16	−0.671965	−0.335983	+	0.941868i	$0.609068\pi$
$948$	0	0
$949$	−5.68871e15	−0.239911
$950$	0	0
$951$	−1.13231e16	−0.472033
$952$	0	0
$953$	−4.17771e16	−1.72158	−0.860791	−	0.508958i	$-0.830031\pi$
$953$	−4.17771e16	−1.72158	−0.860791	+	0.508958i	$0.830031\pi$
$954$	0	0
$955$	−1.85893e16	−0.757260
$956$	0	0
$957$	3.74818e16	1.50940
$958$	0	0
$959$	4.54558e16	1.80962
$960$	0	0
$961$	2.16117e16	0.850570
$962$	0	0
$963$	−5.54136e15	−0.215612
$964$	0	0
$965$	3.60891e15	0.138827
$966$	0	0
$967$	−1.23895e16	−0.471205	−0.235602	−	0.971850i	$-0.575706\pi$
$967$	−1.23895e16	−0.471205	−0.235602	+	0.971850i	$0.575706\pi$
$968$	0	0
$969$	4.23885e13	0.00159392
$970$	0	0
$971$	−1.25057e16	−0.464945	−0.232473	−	0.972603i	$-0.574682\pi$
$971$	−1.25057e16	−0.464945	−0.232473	+	0.972603i	$0.574682\pi$
$972$	0	0
$973$	−3.28510e16	−1.20761
$974$	0	0
$975$	−3.82071e15	−0.138873
$976$	0	0
$977$	−2.22032e16	−0.797988	−0.398994	−	0.916954i	$-0.630641\pi$
$977$	−2.22032e16	−0.797988	−0.398994	+	0.916954i	$0.630641\pi$
$978$	0	0
$979$	−5.62912e16	−2.00049
$980$	0	0
$981$	−1.12909e16	−0.396781
$982$	0	0
$983$	−1.91680e16	−0.666090	−0.333045	−	0.942911i	$-0.608076\pi$
$983$	−1.91680e16	−0.666090	−0.333045	+	0.942911i	$0.608076\pi$
$984$	0	0
$985$	1.04216e16	0.358126
$986$	0	0
$987$	2.31516e14	0.00786749
$988$	0	0
$989$	1.92251e16	0.646084
$990$	0	0
$991$	−2.50183e16	−0.831482	−0.415741	−	0.909483i	$-0.636478\pi$
$991$	−2.50183e16	−0.831482	−0.415741	+	0.909483i	$0.636478\pi$
$992$	0	0
$993$	1.69968e16	0.558657
$994$	0	0
$995$	2.21245e16	0.719196
$996$	0	0
$997$	−1.15404e16	−0.371019	−0.185510	−	0.982642i	$-0.559394\pi$
$997$	−1.15404e16	−0.371019	−0.185510	+	0.982642i	$0.559394\pi$
$998$	0	0
$999$	−9.62745e15	−0.306126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.12.a.x.1.2		2
4.3	odd	2		192.12.a.u.1.2		2
8.3	odd	2		96.12.a.f.1.1	yes	2
8.5	even	2		96.12.a.d.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.12.a.d.1.1	✓	2	8.5	even	2
96.12.a.f.1.1	yes	2	8.3	odd	2
192.12.a.u.1.2		2	4.3	odd	2
192.12.a.x.1.2		2	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q+243.000 q^{3} +3700.71 q^{5} +72358.6 q^{7} +59049.0 q^{9} +O(q^{10})\)	\(q+243.000 q^{3} +3700.71 q^{5} +72358.6 q^{7} +59049.0 q^{9} -1.03599e6 q^{11} +447532. q^{13} +899273. q^{15} +966903. q^{17} +180409. q^{19} +1.75831e7 q^{21} -1.67982e7 q^{23} -3.51329e7 q^{25} +1.43489e7 q^{27} -1.48887e8 q^{29} -2.16841e8 q^{31} -2.51746e8 q^{33} +2.67778e8 q^{35} -6.70953e8 q^{37} +1.08750e8 q^{39} -4.72819e8 q^{41} -1.14447e9 q^{43} +2.18523e8 q^{45} +1.31669e7 q^{47} +3.25844e9 q^{49} +2.34957e8 q^{51} +4.04271e9 q^{53} -3.83390e9 q^{55} +4.38395e7 q^{57} +6.23772e9 q^{59} -6.61870e9 q^{61} +4.27270e9 q^{63} +1.65619e9 q^{65} -1.97892e10 q^{67} -4.08196e9 q^{69} +2.10433e10 q^{71} -1.27113e10 q^{73} -8.53729e9 q^{75} -7.49628e10 q^{77} -1.16162e10 q^{79} +3.48678e9 q^{81} +5.54557e10 q^{83} +3.57823e9 q^{85} -3.61796e10 q^{87} +5.43356e10 q^{89} +3.23828e10 q^{91} -5.26924e10 q^{93} +6.67643e8 q^{95} +8.98530e10 q^{97} -6.11742e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 486 q^{3} - 5300 q^{5} + 38872 q^{7} + 118098 q^{9}+O(q^{10}) \) 2 * q + 486 * q^3 - 5300 * q^5 + 38872 * q^7 + 118098 * q^9	\( 2 q + 486 q^{3} - 5300 q^{5} + 38872 q^{7} + 118098 q^{9} - 1047400 q^{11} + 22900 q^{13} - 1287900 q^{15} + 8733300 q^{17} + 7346600 q^{19} + 9445896 q^{21} - 6711744 q^{23} - 2948210 q^{25} + 28697814 q^{27} - 180180692 q^{29} - 211581400 q^{31} - 254518200 q^{33} + 569181200 q^{35} - 112222700 q^{37} + 5564700 q^{39} - 726961180 q^{41} + 216408856 q^{43} - 312959700 q^{45} + 2174779088 q^{47} + 2402462706 q^{49} + 2122191900 q^{51} + 112024700 q^{53} - 3731208560 q^{55} + 1785223800 q^{57} - 3243949400 q^{59} - 16526230620 q^{61} + 2295352728 q^{63} + 5478177080 q^{65} - 20772619112 q^{67} - 1630953792 q^{69} + 20637101600 q^{71} - 18548203500 q^{73} - 716415030 q^{75} - 74580754400 q^{77} - 28230083800 q^{79} + 6973568802 q^{81} + 7189282056 q^{83} - 66324859080 q^{85} - 43783908156 q^{87} + 103679180788 q^{89} + 46602268400 q^{91} - 51414280200 q^{93} - 63833162000 q^{95} + 199614486500 q^{97} - 61847922600 q^{99}+O(q^{100}) \) 2 * q + 486 * q^3 - 5300 * q^5 + 38872 * q^7 + 118098 * q^9 - 1047400 * q^11 + 22900 * q^13 - 1287900 * q^15 + 8733300 * q^17 + 7346600 * q^19 + 9445896 * q^21 - 6711744 * q^23 - 2948210 * q^25 + 28697814 * q^27 - 180180692 * q^29 - 211581400 * q^31 - 254518200 * q^33 + 569181200 * q^35 - 112222700 * q^37 + 5564700 * q^39 - 726961180 * q^41 + 216408856 * q^43 - 312959700 * q^45 + 2174779088 * q^47 + 2402462706 * q^49 + 2122191900 * q^51 + 112024700 * q^53 - 3731208560 * q^55 + 1785223800 * q^57 - 3243949400 * q^59 - 16526230620 * q^61 + 2295352728 * q^63 + 5478177080 * q^65 - 20772619112 * q^67 - 1630953792 * q^69 + 20637101600 * q^71 - 18548203500 * q^73 - 716415030 * q^75 - 74580754400 * q^77 - 28230083800 * q^79 + 6973568802 * q^81 + 7189282056 * q^83 - 66324859080 * q^85 - 43783908156 * q^87 + 103679180788 * q^89 + 46602268400 * q^91 - 51414280200 * q^93 - 63833162000 * q^95 + 199614486500 * q^97 - 61847922600 * q^99

Introduction

L-functions

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